[
    {
        "formal": "MeasureTheory.integral_simpleFunc_larger_space ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192\u209b F hf_int : Integrable \u2191f \u22a2 \u222b (x : \u03b2), \u2191f x \u2202\u03bc = \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x})) \u2022 x ** simp_rw [\u2190 f.coe_toLargerSpace_eq hm] ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192\u209b F hf_int : Integrable \u2191f \u22a2 \u222b (x : \u03b2), \u2191(SimpleFunc.toLargerSpace hm f) x \u2202\u03bc =     \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x ** have hf_int : Integrable (f.toLargerSpace hm) \u03bc := by rwa [SimpleFunc.coe_toLargerSpace_eq] ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192\u209b F hf_int\u271d : Integrable \u2191f hf_int : Integrable \u2191(SimpleFunc.toLargerSpace hm f) \u22a2 \u222b (x : \u03b2), \u2191(SimpleFunc.toLargerSpace hm f) x \u2202\u03bc =     \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x ** rw [SimpleFunc.integral_eq_sum _ hf_int] ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192\u209b F hf_int\u271d : Integrable \u2191f hf_int : Integrable \u2191(SimpleFunc.toLargerSpace hm f) \u22a2 \u2211 x in SimpleFunc.range (SimpleFunc.toLargerSpace hm f),       ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x =     \u2211 x in SimpleFunc.range f, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(SimpleFunc.toLargerSpace hm f) \u207b\u00b9' {x})) \u2022 x ** congr 1 ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192\u209b F hf_int : Integrable \u2191f \u22a2 Integrable \u2191(SimpleFunc.toLargerSpace hm f) ** rwa [SimpleFunc.coe_toLargerSpace_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Martingale.stoppedValue_min_ae_eq_condexp ** \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 (stoppedValue f fun x => min (\u03c3 x) (\u03c4 x)) =\u1d50[\u03bc] \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3] ** refine'\n  (h.stoppedValue_ae_eq_condexp_of_le h\u03c4 (h\u03c3.min h\u03c4) (fun x => min_le_right _ _) h\u03c4_le).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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[\u03bc]     \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3] ** refine' ae_of_ae_restrict_of_ae_restrict_compl {x | \u03c3 x \u2264 \u03c4 x} _ _ ** 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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c3 x \u2264 \u03c4 x},     (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x ** exact condexp_min_stopping_time_ae_eq_restrict_le h\u03c3 h\u03c4 ** 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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 \u03bc[stoppedValue f         \u03c4|IsStoppingTime.measurableSpace         (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]     \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3] ** apply Filter.EventuallyEq.trans _ ((condexp_min_stopping_time_ae_eq_restrict_le h\u03c4 h\u03c3).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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this :   \u03bc[stoppedValue f         \u03c4|IsStoppingTime.measurableSpace         (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]     \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3] \u22a2 \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c3 x \u2264 \u03c4 x}\u1d9c,     (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x ** rw [ae_restrict_iff' (h\u03c3.measurableSpace_le _ (h\u03c3.measurableSet_le_stopping_time h\u03c4).compl)] ** \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this :   \u03bc[stoppedValue f         \u03c4|IsStoppingTime.measurableSpace         (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]     \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3] \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc,     x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u2192       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =         (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x ** rw [Filter.EventuallyEq, ae_restrict_iff'] at this ** \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this :   \u2200\u1d50 (x : \u03a9) \u2202\u03bc,     x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =         (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc,     x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u2192       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =         (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x  \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this :   \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x},     (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x \u22a2 MeasurableSet {x | \u03c4 x \u2264 \u03c3 x} ** swap ** \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this :   \u2200\u1d50 (x : \u03a9) \u2202\u03bc,     x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =         (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc,     x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u2192       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =         (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x ** filter_upwards [this] with x hx hx_mem ** case h \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this :   \u2200\u1d50 (x : \u03a9) \u2202\u03bc,     x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =         (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x x : \u03a9 hx :   \u03c4 x \u2264 \u03c3 x \u2192     (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x hx_mem : x \u2208 {\u03c9 | \u03c3 \u03c9 \u2264 \u03c4 \u03c9}\u1d9c \u22a2 (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =     (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x ** simp only [Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hx_mem ** case h \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this :   \u2200\u1d50 (x : \u03a9) \u2202\u03bc,     x \u2208 {x | \u03c4 x \u2264 \u03c3 x} \u2192       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =         (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x x : \u03a9 hx :   \u03c4 x \u2264 \u03c3 x \u2192     (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x hx_mem : \u03c4 x < \u03c3 x \u22a2 (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =     (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x ** exact hx hx_mem.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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this :   \u2200\u1d50 (x : \u03a9) \u2202Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x},     (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))]) x =       (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) x \u22a2 MeasurableSet {x | \u03c4 x \u2264 \u03c3 x} ** 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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 \u03a9 \u2192 E ** exact stoppedValue f \u03c4 ** \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 \u03bc[stoppedValue f         \u03c4|IsStoppingTime.measurableSpace         (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]     \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9))] ** rw [IsStoppingTime.measurableSpace_min h\u03c3, IsStoppingTime.measurableSpace_min h\u03c4, inf_comm] ** \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) h1 : \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c4] = stoppedValue f \u03c4 \u22a2 \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c4] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]     \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3] ** rw [h1] ** \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) h1 : \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c4] = stoppedValue f \u03c4 \u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}] \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3] ** exact (condexp_stoppedValue_stopping_time_ae_eq_restrict_le h h\u03c4 h\u03c3 h\u03c4_le).symm ** \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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c4] = stoppedValue f \u03c4 ** refine' condexp_of_stronglyMeasurable h\u03c4.measurableSpace_le _ _ ** 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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 StronglyMeasurable (stoppedValue f \u03c4) ** refine' Measurable.stronglyMeasurable _ ** 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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 Measurable (stoppedValue f \u03c4) ** exact measurable_stoppedValue h.adapted.progMeasurable_of_discrete h\u03c4 ** 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\u271d : \u03b9 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 n : \u03b9 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_sf_min :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 Integrable (stoppedValue f \u03c4) ** exact integrable_stoppedValue \u03b9 h\u03c4 h.integrable h\u03c4_le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.lintegral_mono ** \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\u209b \u211d\u22650\u221e hfg : f \u2264 g h\u03bc\u03bd : \u03bc \u2264 \u03bd \u22a2 lintegral (f \u2294 g) \u03bc = lintegral g \u03bc ** rw [sup_of_le_right hfg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.restrict_singleton ** \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 \u22a2 restrict \u03bc {a} = \u2191\u2191\u03bc {a} \u2022 dirac a ** ext1 s hs ** case h \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** by_cases ha : a \u2208 s ** case pos \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : a \u2208 s \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** have : s \u2229 {a} = {a} := by simpa ** case pos \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : a \u2208 s this : s \u2229 {a} = {a} \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** simp [*] ** \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : a \u2208 s \u22a2 s \u2229 {a} = {a} ** simpa ** case neg \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : \u00aca \u2208 s \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** have : s \u2229 {a} = \u2205 := inter_singleton_eq_empty.2 ha ** case neg \u03b1 : Type u_1 \u03b2 : Type ?u.7488 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 \u03bc : Measure \u03b1 a : \u03b1 s : Set \u03b1 hs : MeasurableSet s ha : \u00aca \u2208 s this : s \u2229 {a} = \u2205 \u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s ** simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.val_one ** n : \u2115 inst\u271d : Fact (1 < n) \u22a2 val 1 = 1 ** rw [val_one_eq_one_mod] ** n : \u2115 inst\u271d : Fact (1 < n) \u22a2 1 % n = 1 ** exact Nat.mod_eq_of_lt Fact.out ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.snorm'_lim_eq_lintegral_liminf ** \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 \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G p : \u211d hp_nonneg : 0 \u2264 p f_lim : \u03b1 \u2192 G 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 = (\u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc) ^ (1 / p) ** suffices h_no_pow :\n    (\u222b\u207b a, (\u2016f_lim a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) = \u222b\u207b a, atTop.liminf fun m => (\u2016f m a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc by\n  rw [snorm', h_no_pow] ** \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 \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G p : \u211d hp_nonneg : 0 \u2264 p f_lim : \u03b1 \u2192 G h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f_lim a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc ** refine' lintegral_congr_ae (h_lim.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\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G p : \u211d hp_nonneg : 0 \u2264 p f_lim : \u03b1 \u2192 G h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) a : \u03b1 ha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a)) \u22a2 (fun a => \u2191\u2016f_lim a\u2016\u208a ^ p) a = (fun a => liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop) a ** dsimp only ** \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 \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G p : \u211d hp_nonneg : 0 \u2264 p f_lim : \u03b1 \u2192 G h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) a : \u03b1 ha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a)) \u22a2 \u2191\u2016f_lim a\u2016\u208a ^ p = liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop ** rw [Tendsto.liminf_eq] ** \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 \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G p : \u211d hp_nonneg : 0 \u2264 p f_lim : \u03b1 \u2192 G h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) a : \u03b1 ha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a)) \u22a2 Tendsto (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop (\ud835\udcdd (\u2191\u2016f_lim a\u2016\u208a ^ p)) ** simp_rw [ENNReal.coe_rpow_of_nonneg _ hp_nonneg, ENNReal.tendsto_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\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G p : \u211d hp_nonneg : 0 \u2264 p f_lim : \u03b1 \u2192 G h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) a : \u03b1 ha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a)) \u22a2 Tendsto (fun a_1 => \u2016f a_1 a\u2016\u208a ^ p) atTop (\ud835\udcdd (\u2016f_lim a\u2016\u208a ^ p)) ** refine' ((NNReal.continuous_rpow_const hp_nonneg).tendsto \u2016f_lim a\u2016\u208a).comp _ ** \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 \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G p : \u211d hp_nonneg : 0 \u2264 p f_lim : \u03b1 \u2192 G h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) a : \u03b1 ha : Tendsto (fun n => f n a) atTop (\ud835\udcdd (f_lim a)) \u22a2 Tendsto (fun a_1 => \u2016f a_1 a\u2016\u208a) atTop (\ud835\udcdd \u2016f_lim a\u2016\u208a) ** exact (continuous_nnnorm.tendsto (f_lim a)).comp 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\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G p : \u211d hp_nonneg : 0 \u2264 p f_lim : \u03b1 \u2192 G h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) h_no_pow : \u222b\u207b (a : \u03b1), \u2191\u2016f_lim a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc \u22a2 snorm' f_lim p \u03bc = (\u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc) ^ (1 / p) ** rw [snorm', h_no_pow] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.mem_insert_of_mem ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** match e : zoom (cmp v) t with\n| (nil, p) =>\n  let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_nil ht e\n  simp [\u2190 mem_toList, h\u2081] at h\n  simp [\u2190 mem_toList, h\u2082]; cases h <;> simp [*]\n| (node .., p) =>\n  let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_node ht e\n  simp [\u2190 mem_toList, h\u2081] at h\n  simp [\u2190 mem_toList, h\u2082]; rcases h with h|h|h <;> simp [*]\n  exact .inr (Path.zoom_zoomed\u2081 e) ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t p : Path \u03b1 e : zoom (cmp v) t Path.root = (nil, p) \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_nil ht e ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t p : Path \u03b1 e : zoom (cmp v) t Path.root = (nil, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** simp [\u2190 mem_toList, h\u2081] at h ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n p : Path \u03b1 e : zoom (cmp v) t Path.root = (nil, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** simp [\u2190 mem_toList, h\u2082] ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n p : Path \u03b1 e : zoom (cmp v) t Path.root = (nil, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d \u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq ** cases h <;> simp [*] ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_node ht e ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n h : v' \u2208 t c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** simp [\u2190 mem_toList, h\u2081] at h ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d \u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq ** simp [\u2190 mem_toList, h\u2082] ** \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d \u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq ** rcases h with h|h|h <;> simp [*] ** case inr.inl \u03b1 : Type u_1 c : RBColor n : Nat v' : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 p : Path \u03b1 e : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p) w\u271d\u00b9 w\u271d : List \u03b1 h\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d h\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d h : v' = v\u271d \u22a2 (v\u271d \u2208 w\u271d\u00b9 \u2228 v\u271d = v \u2228 v\u271d \u2208 w\u271d) \u2228 cmp v v\u271d = Ordering.eq ** exact .inr (Path.zoom_zoomed\u2081 e) ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Pos.zero_addChar_eq ** c : Char \u22a2 0 + c = { byteIdx := csize c } ** rw [\u2190 zero_addChar_byteIdx] ** Qed",
        "informal": ""
    },
    {
        "formal": "Partrec.fix ** \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 \u2295 \u03b1 hf : Partrec f \u22a2 Partrec (PFun.fix f) ** let F : \u03b1 \u2192 \u2115 \u2192. Sum \u03c3 \u03b1 := fun a n =>\n  n.rec (some (Sum.inr a)) fun _ IH => IH.bind fun s => Sum.casesOn s (fun _ => Part.some s) f ** \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 \u2295 \u03b1 hf : Partrec f F : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=   fun a n =>     Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n \u22a2 Partrec (PFun.fix f) ** have hF : Partrec\u2082 F :=\n  Partrec.nat_rec snd (sum_inr.comp fst).partrec\n    (sum_casesOn_right (snd.comp snd) (snd.comp <| snd.comp fst).to\u2082 (hf.comp snd).to\u2082).to\u2082 ** \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 \u2295 \u03b1 hf : Partrec f F : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=   fun a n =>     Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n hF : Partrec\u2082 F \u22a2 Partrec (PFun.fix f) ** let p a n := @Part.map _ Bool (fun s => Sum.casesOn s (fun _ => true) fun _ => false) (F a n) ** \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 \u2295 \u03b1 hf : Partrec f F : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=   fun a n =>     Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n hF : Partrec\u2082 F p : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n) \u22a2 Partrec (PFun.fix f) ** have hp : Partrec\u2082 p :=\n  hF.map ((sum_casesOn Computable.id (const true).to\u2082 (const false).to\u2082).comp snd).to\u2082 ** \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 \u2295 \u03b1 hf : Partrec f F : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=   fun a n =>     Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n hF : Partrec\u2082 F p : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n) hp : Partrec\u2082 p \u22a2 Partrec (PFun.fix f) ** exact (hp.rfind.bind (hF.bind (sum_casesOn_right snd snd.to\u2082 none.to\u2082).to\u2082).to\u2082).of_eq fun a =>\n  ext fun b => by simp; apply fix_aux f ** \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 \u2295 \u03b1 hf : Partrec f F : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=   fun a n =>     Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n hF : Partrec\u2082 F p : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n) hp : Partrec\u2082 p a : \u03b1 b : \u03c3 \u22a2 (b \u2208       Part.bind (Nat.rfind (p a)) fun b =>         Part.bind (F (a, b).1 (a, b).2) fun b_1 =>           Sum.casesOn ((a, b), b_1).2 (fun b_2 => Part.some (((a, b), b_1), b_2).2) fun b => Part.none) \u2194     b \u2208 PFun.fix f a ** simp ** \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 \u2295 \u03b1 hf : Partrec f F : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=   fun a n =>     Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n hF : Partrec\u2082 F p : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n) hp : Partrec\u2082 p a : \u03b1 b : \u03c3 \u22a2 (\u2203 a_1,       ((\u2203 a_2,             Sum.inl a_2 \u2208               Nat.rec (Part.some (Sum.inr a))                 (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) a_1) \u2227           \u2200 {m : \u2115},             m < a_1 \u2192               \u2203 b,                 Sum.inr b \u2208                   Nat.rec (Part.some (Sum.inr a))                     (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m) \u2227         Sum.inl b \u2208           Nat.rec (Part.some (Sum.inr a))             (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) a_1) \u2194     b \u2208 PFun.fix f a ** apply fix_aux f ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.pdf.IsUniform.integral_eq ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 \u22a2 \u222b (x : \u03a9), X x \u2202\u2119 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x ** haveI := hasPDF hns hnt huX ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 this : HasPDF X \u2119 \u22a2 \u222b (x : \u03a9), X x \u2202\u2119 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x ** haveI := huX.isProbabilityMeasure hns hnt hms ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 this\u271d : HasPDF X \u2119 this : IsProbabilityMeasure \u2119 \u22a2 \u222b (x : \u03a9), X x \u2202\u2119 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x ** rw [\u2190 integral_mul_eq_integral] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 this\u271d : HasPDF X \u2119 this : IsProbabilityMeasure \u2119 \u22a2 \u222b (x : \u211d), x * ENNReal.toReal (pdf (fun x => X x) \u2119 x) = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x ** rw [integral_congr_ae (Filter.EventuallyEq.mul (ae_eq_refl _) (pdf_toReal_ae_eq huX))] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 this\u271d\u00b9 : HasPDF X \u2119 this\u271d : IsProbabilityMeasure \u2119 this :   \u2200 (x : \u211d),     x * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) =       x * Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x \u22a2 \u222b (a : \u211d), a * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) a) =     ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x ** simp_rw [this, \u2190 s.indicator_mul_right fun x => x, integral_indicator hms] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 this\u271d\u00b9 : HasPDF X \u2119 this\u271d : IsProbabilityMeasure \u2119 this :   \u2200 (x : \u211d),     x * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) =       x * Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x \u22a2 \u222b (a : \u211d) in s, a * (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) a = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x ** change \u222b x in s, x * (volume s)\u207b\u00b9.toReal \u2022 (1 : \u211d) = _ ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 this\u271d\u00b9 : HasPDF X \u2119 this\u271d : IsProbabilityMeasure \u2119 this :   \u2200 (x : \u211d),     x * ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) =       x * Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x \u22a2 \u222b (x : \u211d) in s, x * ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1 = ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 * \u222b (x : \u211d) in s, x ** rw [integral_mul_right, mul_comm, smul_eq_mul, mul_one] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 this\u271d : HasPDF X \u2119 this : IsProbabilityMeasure \u2119 x : \u211d \u22a2 ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) = Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x ** by_cases hx : x \u2208 s ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 this\u271d : HasPDF X \u2119 this : IsProbabilityMeasure \u2119 x : \u211d hx : x \u2208 s \u22a2 ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) = Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x ** simp [Set.indicator_of_mem hx] ** case neg \u03a9 : Type u_1 E : Type u_2 inst\u271d : 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 hnt : \u2191\u2191volume s \u2260 \u22a4 huX : IsUniform X s \u2119 this\u271d : HasPDF X \u2119 this : IsProbabilityMeasure \u2119 x : \u211d hx : \u00acx \u2208 s \u22a2 ENNReal.toReal (Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x) = Set.indicator s (ENNReal.toReal (\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x ** simp [Set.indicator_of_not_mem hx] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.bodd_neg ** n : \u2124 \u22a2 bodd (-n) = bodd n ** cases n with\n| ofNat =>\n  rw [\u2190negOfNat_eq, bodd_negOfNat]\n  simp\n| negSucc n =>\n  rw [neg_negSucc, bodd_coe, Nat.bodd_succ]\n  change (!Nat.bodd n) = !(bodd n)\n  rw [bodd_coe] ** case ofNat a\u271d : \u2115 \u22a2 bodd (-ofNat a\u271d) = bodd (ofNat a\u271d) ** rw [\u2190negOfNat_eq, bodd_negOfNat] ** case ofNat a\u271d : \u2115 \u22a2 Nat.bodd a\u271d = bodd (ofNat a\u271d) ** simp ** case negSucc n : \u2115 \u22a2 bodd (- -[n+1]) = bodd -[n+1] ** rw [neg_negSucc, bodd_coe, Nat.bodd_succ] ** case negSucc n : \u2115 \u22a2 (!Nat.bodd n) = bodd -[n+1] ** change (!Nat.bodd n) = !(bodd n) ** case negSucc n : \u2115 \u22a2 (!Nat.bodd n) = !bodd \u2191n ** rw [bodd_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.measurable_measure_condCdf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) \u22a2 Measurable fun a => StieltjesFunction.measure (condCdf \u03c1 a) ** rw [Measure.measurable_measure] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) \u22a2 \u2200 (s : Set \u211d), MeasurableSet s \u2192 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s ** refine' fun s hs => ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s \u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s ** refine' MeasurableSpace.induction_on_inter\n  (C := fun s => Measurable fun b \u21a6 StieltjesFunction.measure (condCdf \u03c1 b) s)\n  (borel_eq_generateFrom_Iic \u211d) isPiSystem_Iic _ _ _ _ hs ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s \u22a2 (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) \u2205 ** simp only [measure_empty, measurable_const] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s \u22a2 \u2200 (t : Set \u211d), t \u2208 range Iic \u2192 (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t ** rintro S \u27e8u, rfl\u27e9 ** case refine'_2.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s u : \u211d \u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (Iic u) ** simp_rw [measure_condCdf_Iic \u03c1 _ u] ** case refine'_2.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s u : \u211d \u22a2 Measurable fun b => ENNReal.ofReal (\u2191(condCdf \u03c1 b) u) ** exact (measurable_condCdf \u03c1 u).ennreal_ofReal ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s \u22a2 \u2200 (t : Set \u211d),     MeasurableSet t \u2192       (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t \u2192         (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t\u1d9c ** intro t ht ht_cd_meas ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t ht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t \u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\u1d9c ** have :\n  (fun a => (condCdf \u03c1 a).measure t\u1d9c) =\n    (fun a => (condCdf \u03c1 a).measure univ) - fun a => (condCdf \u03c1 a).measure t := by\n  ext1 a\n  rw [measure_compl ht (measure_ne_top (condCdf \u03c1 a).measure _), Pi.sub_apply] ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t ht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t this :   (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =     (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t \u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\u1d9c ** simp_rw [this, measure_condCdf_univ \u03c1] ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t ht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t this :   (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =     (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t \u22a2 Measurable ((fun a => 1) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t) ** exact Measurable.sub measurable_const ht_cd_meas ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t ht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t \u22a2 (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =     (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t ** ext1 a ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t ht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t a : \u03b1 \u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c =     ((fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a =>         \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t)       a ** rw [measure_compl ht (measure_ne_top (condCdf \u03c1 a).measure _), Pi.sub_apply] ** case refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s \u22a2 \u2200 (f : \u2115 \u2192 Set \u211d),     Pairwise (Disjoint on f) \u2192       (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192         (\u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)) \u2192           (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (\u22c3 i, f i) ** intro f hf_disj hf_meas hf_cd_meas ** case refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s f : \u2115 \u2192 Set \u211d hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_cd_meas : \u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i) \u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (\u22c3 i, f i) ** simp_rw [measure_iUnion hf_disj hf_meas] ** case refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u211d hs : MeasurableSet s f : \u2115 \u2192 Set \u211d hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_cd_meas : \u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i) \u22a2 Measurable fun b => \u2211' (i : \u2115), \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (f i) ** exact Measurable.ennreal_tsum hf_cd_meas ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.liftOn\u2082_toFinmap ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v \u03b3 : Type u_1 s\u2081 s\u2082 : AList \u03b2 f : AList \u03b2 \u2192 AList \u03b2 \u2192 \u03b3 H : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : AList \u03b2), a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082 \u22a2 liftOn\u2082 \u27e6s\u2081\u27e7 \u27e6s\u2082\u27e7 f H = f s\u2081 s\u2082 ** cases s\u2081 ** case mk \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v \u03b3 : Type u_1 s\u2082 : AList \u03b2 f : AList \u03b2 \u2192 AList \u03b2 \u2192 \u03b3 H : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : AList \u03b2), a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082 entries\u271d : List (Sigma \u03b2) nodupKeys\u271d : NodupKeys entries\u271d \u22a2 liftOn\u2082 \u27e6{ entries := entries\u271d, nodupKeys := nodupKeys\u271d }\u27e7 \u27e6s\u2082\u27e7 f H =     f { entries := entries\u271d, nodupKeys := nodupKeys\u271d } s\u2082 ** cases s\u2082 ** case mk.mk \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v \u03b3 : Type u_1 f : AList \u03b2 \u2192 AList \u03b2 \u2192 \u03b3 H : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : AList \u03b2), a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082 entries\u271d\u00b9 : List (Sigma \u03b2) nodupKeys\u271d\u00b9 : NodupKeys entries\u271d\u00b9 entries\u271d : List (Sigma \u03b2) nodupKeys\u271d : NodupKeys entries\u271d \u22a2 liftOn\u2082 \u27e6{ entries := entries\u271d\u00b9, nodupKeys := nodupKeys\u271d\u00b9 }\u27e7 \u27e6{ entries := entries\u271d, nodupKeys := nodupKeys\u271d }\u27e7 f H =     f { entries := entries\u271d\u00b9, nodupKeys := nodupKeys\u271d\u00b9 } { entries := entries\u271d, nodupKeys := nodupKeys\u271d } ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.setToSimpleFunc_const ** \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 x : F m : MeasurableSpace \u03b1 \u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) x ** cases h\u03b1 : isEmpty_or_nonempty \u03b1 ** 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 x : F m : MeasurableSpace \u03b1 h\u271d : IsEmpty \u03b1 h\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) \u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) x ** have h_univ_empty : (univ : Set \u03b1) = \u2205 := Subsingleton.elim _ _ ** 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 x : F m : MeasurableSpace \u03b1 h\u271d : IsEmpty \u03b1 h\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) h_univ_empty : Set.univ = \u2205 \u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) x ** rw [h_univ_empty, hT_empty] ** 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 x : F m : MeasurableSpace \u03b1 h\u271d : IsEmpty \u03b1 h\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) h_univ_empty : Set.univ = \u2205 \u22a2 setToSimpleFunc T (const \u03b1 x) = \u21910 x ** simp only [setToSimpleFunc, ContinuousLinearMap.zero_apply, sum_empty,\n  range_eq_empty_of_isEmpty] ** 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 x : F m : MeasurableSpace \u03b1 h\u271d : Nonempty \u03b1 h\u03b1 : (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) = (_ : IsEmpty \u03b1 \u2228 Nonempty \u03b1) \u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) x ** exact setToSimpleFunc_const' T x ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** induction' f using MeasureTheory.SimpleFunc.induction with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing L ** case h_ind \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** simp only [hs, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero, univ_inter,\n  piecewise_eq_indicator, lintegral_indicator, lintegral_const, Measure.restrict_apply',\n  ENNReal.coe_indicator, Function.const_apply] at hL ** case h_ind \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** have c_ne_zero : c \u2260 0 := by\n  intro hc\n  simp only [hc, ENNReal.coe_zero, zero_mul, not_lt_zero] at hL ** case h_ind \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 this : L / \u2191c < \u2191\u2191\u03bc s \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** obtain \u27e8t, ht, ts, mlt, t_top\u27e9 :\n  \u2203 t : Set \u03b1, MeasurableSet t \u2227 t \u2286 s \u2227 L / \u2191c < \u03bc t \u2227 \u03bc t < \u221e :=\n  Measure.exists_subset_measure_lt_top hs this ** case h_ind.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 this : L / \u2191c < \u2191\u2191\u03bc s t : Set \u03b1 ht : MeasurableSet t ts : t \u2286 s mlt : L / \u2191c < \u2191\u2191\u03bc t t_top : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** refine' \u27e8piecewise t ht (const \u03b1 c) (const \u03b1 0), fun x => _, _, _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s \u22a2 c \u2260 0 ** intro hc ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s hc : c = 0 \u22a2 False ** simp only [hc, ENNReal.coe_zero, zero_mul, not_lt_zero] at hL ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 \u22a2 L / \u2191c < \u2191\u2191\u03bc s ** rwa [ENNReal.div_lt_iff, mul_comm] ** case h0 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 \u22a2 \u2191c \u2260 0 \u2228 L \u2260 0 ** simp only [c_ne_zero, Ne.def, coe_eq_zero, not_false_iff, true_or_iff] ** case ht \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 \u22a2 \u2191c \u2260 \u22a4 \u2228 L \u2260 \u22a4 ** simp only [Ne.def, coe_ne_top, not_false_iff, true_or_iff] ** case h_ind.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 this : L / \u2191c < \u2191\u2191\u03bc s t : Set \u03b1 ht : MeasurableSet t ts : t \u2286 s mlt : L / \u2191c < \u2191\u2191\u03bc t t_top : \u2191\u2191\u03bc t < \u22a4 x : \u03b1 \u22a2 \u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x ** refine indicator_le_indicator_of_subset ts (fun x => ?_) x ** case h_ind.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 this : L / \u2191c < \u2191\u2191\u03bc s t : Set \u03b1 ht : MeasurableSet t ts : t \u2286 s mlt : L / \u2191c < \u2191\u2191\u03bc t t_top : \u2191\u2191\u03bc t < \u22a4 x\u271d x : \u03b1 \u22a2 0 \u2264 \u2191(const \u03b1 c) x ** exact zero_le _ ** case h_ind.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 this : L / \u2191c < \u2191\u2191\u03bc s t : Set \u03b1 ht : MeasurableSet t ts : t \u2286 s mlt : L / \u2191c < \u2191\u2191\u03bc t t_top : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc < \u22a4 ** simp only [ht, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero, univ_inter,\n  piecewise_eq_indicator, ENNReal.coe_indicator, Function.const_apply, lintegral_indicator,\n  lintegral_const, Measure.restrict_apply', ENNReal.mul_lt_top ENNReal.coe_ne_top t_top.ne] ** case h_ind.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 this : L / \u2191c < \u2191\u2191\u03bc s t : Set \u03b1 ht : MeasurableSet t ts : t \u2286 s mlt : L / \u2191c < \u2191\u2191\u03bc t t_top : \u2191\u2191\u03bc t < \u22a4 \u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc ** simp only [ht, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero,\n  piecewise_eq_indicator, ENNReal.coe_indicator, Function.const_apply, lintegral_indicator,\n  lintegral_const, Measure.restrict_apply', univ_inter] ** case h_ind.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 this : L / \u2191c < \u2191\u2191\u03bc s t : Set \u03b1 ht : MeasurableSet t ts : t \u2286 s mlt : L / \u2191c < \u2191\u2191\u03bc t t_top : \u2191\u2191\u03bc t < \u22a4 \u22a2 L < \u2191c * \u2191\u2191\u03bc t ** rwa [mul_comm, \u2190 ENNReal.div_lt_iff] ** case h_ind.intro.intro.intro.intro.refine'_3.h0 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 this : L / \u2191c < \u2191\u2191\u03bc s t : Set \u03b1 ht : MeasurableSet t ts : t \u2286 s mlt : L / \u2191c < \u2191\u2191\u03bc t t_top : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u2191c \u2260 0 \u2228 L \u2260 0 ** simp only [c_ne_zero, Ne.def, coe_eq_zero, not_false_iff, true_or_iff] ** case h_ind.intro.intro.intro.intro.refine'_3.ht \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s L : \u211d\u22650\u221e hL : L < \u2191c * \u2191\u2191\u03bc s c_ne_zero : c \u2260 0 this : L / \u2191c < \u2191\u2191\u03bc s t : Set \u03b1 ht : MeasurableSet t ts : t \u2286 s mlt : L / \u2191c < \u2191\u2191\u03bc t t_top : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u2191c \u2260 \u22a4 \u2228 L \u2260 \u22a4 ** simp only [Ne.def, coe_ne_top, not_false_iff, true_or_iff] ** case h_add \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** replace hL : L < \u222b\u207b x, f\u2081 x \u2202\u03bc + \u222b\u207b x, f\u2082 x \u2202\u03bc ** case h_add \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** by_cases hf\u2081 : \u222b\u207b x, f\u2081 x \u2202\u03bc = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** by_cases hf\u2082 : \u222b\u207b x, f\u2082 x \u2202\u03bc = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** obtain \u27e8L\u2081, L\u2082, hL\u2081, hL\u2082, hL\u27e9 :\n  \u2203 L\u2081 L\u2082 : \u211d\u22650\u221e, (L\u2081 < \u222b\u207b x, f\u2081 x \u2202\u03bc) \u2227 (L\u2082 < \u222b\u207b x, f\u2082 x \u2202\u03bc) \u2227 L < L\u2081 + L\u2082 :=\n  ENNReal.exists_lt_add_of_lt_add hL hf\u2081 hf\u2082 ** case neg.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 L\u2081 L\u2082 : \u211d\u22650\u221e hL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc hL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hL : L < L\u2081 + L\u2082 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** rcases h\u2081 hL\u2081 with \u27e8g\u2081, g\u2081_le, g\u2081_top, hg\u2081\u27e9 ** case neg.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 L\u2081 L\u2082 : \u211d\u22650\u221e hL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc hL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hL : L < L\u2081 + L\u2082 g\u2081 : \u03b1 \u2192\u209b \u211d\u22650 g\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x g\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4 hg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** rcases h\u2082 hL\u2082 with \u27e8g\u2082, g\u2082_le, g\u2082_top, hg\u2082\u27e9 ** case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 L\u2081 L\u2082 : \u211d\u22650\u221e hL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc hL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hL : L < L\u2081 + L\u2082 g\u2081 : \u03b1 \u2192\u209b \u211d\u22650 g\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x g\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4 hg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc g\u2082 : \u03b1 \u2192\u209b \u211d\u22650 g\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x g\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4 hg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** refine' \u27e8g\u2081 + g\u2082, fun x => add_le_add (g\u2081_le x) (g\u2082_le x), _, _\u27e9 ** case hL \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc ** rwa [\u2190 lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** simp only [hf\u2081, zero_add] at hL ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** rcases h\u2082 hL with \u27e8g, g_le, g_top, gL\u27e9 ** case pos.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc g : \u03b1 \u2192\u209b \u211d\u22650 g_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x g_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 gL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** refine' \u27e8g, fun x => (g_le x).trans _, g_top, gL\u27e9 ** case pos.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc g : \u03b1 \u2192\u209b \u211d\u22650 g_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x g_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 gL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc x : \u03b1 \u22a2 \u2191f\u2082 x \u2264 \u2191(f\u2081 + f\u2082) x ** simp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_left, zero_le'] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** simp only [hf\u2082, add_zero] at hL ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** rcases h\u2081 hL with \u27e8g, g_le, g_top, gL\u27e9 ** case pos.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc g : \u03b1 \u2192\u209b \u211d\u22650 g_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x g_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 gL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** refine' \u27e8g, fun x => (g_le x).trans _, g_top, gL\u27e9 ** case pos.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 hL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc g : \u03b1 \u2192\u209b \u211d\u22650 g_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x g_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 gL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc x : \u03b1 \u22a2 \u2191f\u2081 x \u2264 \u2191(f\u2081 + f\u2082) x ** simp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_right, zero_le'] ** case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 L\u2081 L\u2082 : \u211d\u22650\u221e hL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc hL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hL : L < L\u2081 + L\u2082 g\u2081 : \u03b1 \u2192\u209b \u211d\u22650 g\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x g\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4 hg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc g\u2082 : \u03b1 \u2192\u209b \u211d\u22650 g\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x g\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4 hg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc < \u22a4 ** apply lt_of_le_of_lt _ (add_lt_top.2 \u27e8g\u2081_top, g\u2082_top\u27e9) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 L\u2081 L\u2082 : \u211d\u22650\u221e hL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc hL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hL : L < L\u2081 + L\u2082 g\u2081 : \u03b1 \u2192\u209b \u211d\u22650 g\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x g\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4 hg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc g\u2082 : \u03b1 \u2192\u209b \u211d\u22650 g\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x g\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4 hg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc ** rw [\u2190 lintegral_add_left g\u2081.measurable.coe_nnreal_ennreal] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 L\u2081 L\u2082 : \u211d\u22650\u221e hL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc hL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hL : L < L\u2081 + L\u2082 g\u2081 : \u03b1 \u2192\u209b \u211d\u22650 g\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x g\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4 hg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc g\u2082 : \u03b1 \u2192\u209b \u211d\u22650 g\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x g\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4 hg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(\u2191g\u2081 a) + \u2191(\u2191g\u2082 a) \u2202\u03bc ** exact le_rfl ** case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 L\u2081 L\u2082 : \u211d\u22650\u221e hL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc hL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hL : L < L\u2081 + L\u2082 g\u2081 : \u03b1 \u2192\u209b \u211d\u22650 g\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x g\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4 hg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc g\u2082 : \u03b1 \u2192\u209b \u211d\u22650 g\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x g\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4 hg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc ** apply hL.trans ((ENNReal.add_lt_add hg\u2081 hg\u2082).trans_le _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 L\u2081 L\u2082 : \u211d\u22650\u221e hL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc hL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hL : L < L\u2081 + L\u2082 g\u2081 : \u03b1 \u2192\u209b \u211d\u22650 g\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x g\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4 hg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc g\u2082 : \u03b1 \u2192\u209b \u211d\u22650 g\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x g\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4 hg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc ** rw [\u2190 lintegral_add_left g\u2081.measurable.coe_nnreal_ennreal] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 L\u271d : \u211d\u22650\u221e hL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082) h\u2081 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc h\u2082 :   \u2200 {L : \u211d\u22650\u221e},     L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192       \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc L : \u211d\u22650\u221e hL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0 hf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0 L\u2081 L\u2082 : \u211d\u22650\u221e hL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc hL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc hL : L < L\u2081 + L\u2082 g\u2081 : \u03b1 \u2192\u209b \u211d\u22650 g\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x g\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4 hg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc g\u2082 : \u03b1 \u2192\u209b \u211d\u22650 g\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x g\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4 hg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191g\u2081 a) + \u2191(\u2191g\u2082 a) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc ** simp only [coe_add, Pi.add_apply, ENNReal.coe_add, le_rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.rnDeriv_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc r : \u211d \u22a2 rnDeriv (r \u2022 s) \u03bc =\u1da0[ae \u03bc] r \u2022 rnDeriv s \u03bc ** refine'\n  Integrable.ae_eq_of_withDensity\u1d65_eq (integrable_rnDeriv _ _)\n    ((integrable_rnDeriv _ _).smul r) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc r : \u211d \u22a2 withDensity\u1d65 \u03bc (rnDeriv (r \u2022 s) \u03bc) = withDensity\u1d65 \u03bc (r \u2022 rnDeriv s \u03bc) ** rw [withDensity\u1d65_smul (rnDeriv s \u03bc) (r : \u211d), \u2190 add_right_inj ((r \u2022 s).singularPart \u03bc),\n  singularPart_add_withDensity_rnDeriv_eq, singularPart_smul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc r : \u211d \u22a2 r \u2022 s = r \u2022 singularPart s \u03bc + r \u2022 withDensity\u1d65 \u03bc (rnDeriv s \u03bc) ** rw [\u2190 smul_add, singularPart_add_withDensity_rnDeriv_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_pi_Ioo_toReal ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211d h : a \u2264 b \u22a2 ENNReal.toReal (\u2191\u2191volume (Set.pi univ fun i => Ioo (a i) (b i))) = \u220f i : \u03b9, (b i - a i) ** simp only [volume_pi_Ioo, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))] ** Qed",
        "informal": ""
    },
    {
        "formal": "BinaryHeap.size_insert ** \u03b1 : Type u_1 lt : \u03b1 \u2192 \u03b1 \u2192 Bool self : BinaryHeap \u03b1 lt x : \u03b1 \u22a2 size (insert self x) = size self + 1 ** simp [insert, size, size_heapifyUp] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.lcm_mul_right ** m n k : \u2124 \u22a2 lcm (m * n) (k * n) = lcm m k * natAbs n ** simp_rw [Int.lcm, natAbs_mul, Nat.lcm_mul_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Nat.map_swap_antidiagonal ** n : \u2115 \u22a2 (map { toFun := Prod.swap, inj' := (_ : Injective Prod.swap) } (antidiagonal n)).val = (antidiagonal n).val ** simp [antidiagonal, Multiset.Nat.map_swap_antidiagonal] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.TProd.mk_elim ** \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 hnd : Nodup l h : \u2200 (i : \u03b9), i \u2208 l v : TProd \u03b1 l i : \u03b9 hi : i \u2208 l \u22a2 TProd.elim (TProd.mk l (TProd.elim' h v)) hi = TProd.elim v hi ** simp [elim_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.card_fintype_uIcc ** a b : \u2124 \u22a2 Fintype.card \u2191(Set.uIcc a b) = natAbs (b - a) + 1 ** rw [\u2190 card_uIcc, Fintype.card_ofFinset] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_sphere_of_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 s : Set E x : E r : \u211d hr : r \u2260 0 \u22a2 \u2191\u2191\u03bc (sphere x r) = 0 ** rcases hr.lt_or_lt with (h | h) ** 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 : Set E x : E r : \u211d hr : r \u2260 0 h : r < 0 \u22a2 \u2191\u2191\u03bc (sphere x r) = 0 ** simp only [empty_diff, measure_empty, \u2190 closedBall_diff_ball, closedBall_eq_empty.2 h] ** 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 x : E r : \u211d hr : r \u2260 0 h : 0 < r \u22a2 \u2191\u2191\u03bc (sphere x r) = 0 ** rw [\u2190 closedBall_diff_ball,\n  measure_diff ball_subset_closedBall measurableSet_ball measure_ball_lt_top.ne,\n  addHaar_ball_of_pos \u03bc _ h, addHaar_closedBall \u03bc _ h.le, tsub_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.findExistsOneDivLT_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.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i) ** rw [findExistsOneDivLT, 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 MeasureTheory.SignedMeasure.ExistsOneDivLT s i (Nat.find (_ : \u2203 n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n)) ** convert Nat.find_spec (existsNatOneDivLTMeasure_of_not_negative hi) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.measurableSet_limsup ** \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 (limsup s atTop) ** simpa only [\u2190 blimsup_true] using measurableSet_blimsup fun n _ => hs n ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.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 _i : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp E p } \u2192 Prop h_ind :   \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     Disjoint (support f) (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 f} \u22a2 \u2200 (f : { x // x \u2208 Lp E p }), P f ** refine' fun f => (Lp.simpleFunc.denseRange hp_ne_top).induction_on f h_closed _ ** \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 _i : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp E p } \u2192 Prop h_ind :   \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     Disjoint (support f) (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 f} f : { x // x \u2208 Lp E p } \u22a2 \u2200 (a : { x // x \u2208 \u2191(simpleFunc E p \u03bc) }), P \u2191a ** refine' Lp.simpleFunc.induction (\u03b1 := \u03b1) (E := E) (lt_of_lt_of_le zero_lt_one _i.elim).ne'\n  hp_ne_top _ _ ** 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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 _i : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp E p } \u2192 Prop h_ind :   \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     Disjoint (support f) (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 f} f : { x // x \u2208 Lp E p } \u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) ** exact fun c s => h_ind c ** 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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 _i : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp E p } \u2192 Prop h_ind :   \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     Disjoint (support f) (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 f} f : { x // x \u2208 Lp E p } \u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),     Disjoint (support \u2191f) (support \u2191g) \u2192       P \u2191(simpleFunc.toLp f hf) \u2192 P \u2191(simpleFunc.toLp g hg) \u2192 P \u2191(simpleFunc.toLp f hf + simpleFunc.toLp g hg) ** exact fun f g hf hg => h_add hf hg ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.coeFn_comp ** \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 hg : Continuous g f : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 \u2191(comp g hg f) =\u1d50[\u03bc] g \u2218 \u2191f ** rw [comp_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 hg : Continuous g f : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 \u2191(mk (g \u2218 \u2191f) (_ : AEStronglyMeasurable (fun x => g (\u2191f x)) \u03bc)) =\u1d50[\u03bc] g \u2218 \u2191f ** apply coeFn_mk ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.sum_prob_mem_Ioc_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 N : \u2115 hKN : K \u2264 N \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) ** 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 \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) ** haveI : IsProbabilityMeasure \u03c1 := isProbabilityMeasure_map hint.aemeasurable ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 \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) ** have B : \u2200 a b, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b _ in Set.Ioc a b, (1 : \u211d) \u2202\u03c1) := by\n  intro a b\n  rw [ofReal_set_integral_one \u03c1 _,\n    Measure.map_apply_of_aemeasurable hint.aemeasurable measurableSet_Ioc]\n  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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 B : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1) \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) ** calc\n  \u2211 j in range K, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc (j : \u211d) N} =\n      \u2211 j in range K, ENNReal.ofReal (\u222b _ in Set.Ioc (j : \u211d) N, (1 : \u211d) \u2202\u03c1) := by simp_rw [B]\n  _ = ENNReal.ofReal (\u2211 j in range K, \u222b _ in Set.Ioc (j : \u211d) N, (1 : \u211d) \u2202\u03c1) := by\n    rw [ENNReal.ofReal_sum_of_nonneg]\n    simp only [integral_const, Algebra.id.smul_eq_mul, mul_one, ENNReal.toReal_nonneg,\n      imp_true_iff]\n  _ = ENNReal.ofReal (\u2211 j in range K, \u222b _ in (j : \u211d)..N, (1 : \u211d) \u2202\u03c1) := by\n    congr 1\n    refine' sum_congr rfl fun j hj => _\n    rw [intervalIntegral.integral_of_le (Nat.cast_le.2 ((mem_range.1 hj).le.trans hKN))]\n  _ \u2264 ENNReal.ofReal (\ud835\udd3c[X] + 1) := ENNReal.ofReal_le_ofReal 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 = \u2211 j in range K, \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 ** apply sum_congr rfl fun j hj => ?_ ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 j : \u2115 hj : j \u2208 range K \u22a2 \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 = \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 ** rw [intervalIntegral.sum_integral_adjacent_intervals_Ico ((mem_range.1 hj).le.trans hKN)] ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 j : \u2115 hj : j \u2208 range K \u22a2 \u2200 (k : \u2115), k \u2208 Set.Ico j N \u2192 IntervalIntegrable (fun x => 1) \u03c1 \u2191k \u2191(k + 1) ** intro 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 j : \u2115 hj : j \u2208 range K k : \u2115 a\u271d : k \u2208 Set.Ico j N \u22a2 IntervalIntegrable (fun x => 1) \u03c1 \u2191k \u2191(k + 1) ** exact continuous_const.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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2211 j in range K, \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 =     \u2211 i in range N, \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 ** simp_rw [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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2211 x in Finset.sigma (range K) fun a => Ico a N, \u222b (x : \u211d) in \u2191x.snd..\u2191(x.snd + 1), 1 \u2202Measure.map X \u2119 =     \u2211 x in Finset.sigma (range N) fun a => range (min (a + 1) K), \u222b (x : \u211d) in \u2191x.fst..\u2191(x.fst + 1), 1 \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ico a N),     (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range (min (a + 1) 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a N \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208     Finset.sigma (range N) fun a => range (min (a + 1) K) ** simp only [mem_sigma, mem_range, mem_Ico] 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i j : \u2115 hij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a N hij : i < K \u2227 i \u2264 j \u2227 j < N \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208     Finset.sigma (range N) fun a => range (min (a + 1) K) ** simp only [hij, Nat.lt_succ_iff.2 hij.2.1, mem_sigma, mem_range, lt_min_iff, and_self_iff] ** 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)),     (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ico a N ** 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K) \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => Ico a N ** simp only [mem_sigma, mem_range, lt_min_iff] 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i j : \u2115 hij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K) hij : i < N \u2227 j < i + 1 \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 => Ico a N ** simp only [hij, Nat.lt_succ_iff.1 hij.2.1, mem_sigma, mem_range, mem_Ico, 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ico a 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 (min (a + 1) 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a 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 (min (a + 1) 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range (min (a + 1) 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 => Ico a N) =       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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) 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 => Ico a N) =     { 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2211 i in range N, \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264     \u2211 i in range N, (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 ** apply sum_le_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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N \u22a2 \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 ** simp only [Nat.cast_add, Nat.cast_one, sum_const, card_range, nsmul_eq_mul, Nat.cast_min] ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N \u22a2 min (\u2191i + 1) \u2191K * \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119 \u2264 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119 ** refine' mul_le_mul_of_nonneg_right (min_le_left _ _) _ ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N \u22a2 0 \u2264 \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119 ** apply intervalIntegral.integral_nonneg ** case hab \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N \u22a2 \u2191i \u2264 \u2191i + 1 ** simp only [le_add_iff_nonneg_right, zero_le_one] ** case 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N \u22a2 \u2200 (u : \u211d), u \u2208 Set.Icc (\u2191i) (\u2191i + 1) \u2192 0 \u2264 1 ** simp only [zero_le_one, imp_true_iff] ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2211 i in range N, (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u2211 i in range N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1 ** apply sum_le_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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N \u22a2 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1 ** have I : (i : \u211d) \u2264 (i + 1 : \u2115) := by\n  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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N I : \u2191i \u2264 \u2191(i + 1) \u22a2 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1 ** simp_rw [intervalIntegral.integral_of_le I, \u2190 integral_mul_left] ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N I : \u2191i \u2264 \u2191(i + 1) \u22a2 \u222b (a : \u211d) in Set.Ioc \u2191i \u2191(i + 1), (\u2191i + 1) * 1 \u2202Measure.map X \u2119 \u2264     \u222b (x : \u211d) in Set.Ioc \u2191i \u2191(i + 1), x + 1 \u2202Measure.map X \u2119 ** apply set_integral_mono_on ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N \u22a2 \u2191i \u2264 \u2191(i + 1) ** simp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one] ** case 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N I : \u2191i \u2264 \u2191(i + 1) \u22a2 IntegrableOn (fun a => (\u2191i + 1) * 1) (Set.Ioc \u2191i \u2191(i + 1)) ** exact continuous_const.integrableOn_Ioc ** case hg \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N I : \u2191i \u2264 \u2191(i + 1) \u22a2 IntegrableOn (fun a => a + 1) (Set.Ioc \u2191i \u2191(i + 1)) ** exact (continuous_id.add continuous_const).integrableOn_Ioc ** case hs \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N I : \u2191i \u2264 \u2191(i + 1) \u22a2 MeasurableSet (Set.Ioc \u2191i \u2191(i + 1)) ** exact measurableSet_Ioc ** 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N I : \u2191i \u2264 \u2191(i + 1) \u22a2 \u2200 (x : \u211d), x \u2208 Set.Ioc \u2191i \u2191(i + 1) \u2192 (\u2191i + 1) * 1 \u2264 x + 1 ** intro x hx ** 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N I : \u2191i \u2264 \u2191(i + 1) x : \u211d hx : x \u2208 Set.Ioc \u2191i \u2191(i + 1) \u22a2 (\u2191i + 1) * 1 \u2264 x + 1 ** simp only [Nat.cast_add, Nat.cast_one, Set.mem_Ioc] at hx ** 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 i : \u2115 x\u271d : i \u2208 range N I : \u2191i \u2264 \u2191(i + 1) x : \u211d hx : \u2191i < x \u2227 x \u2264 \u2191i + 1 \u22a2 (\u2191i + 1) * 1 \u2264 x + 1 ** simp [hx.1.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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u2211 i in range N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u222b (x : \u211d) in \u21910 ..\u2191N, x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 k : \u2115 x\u271d : k < N \u22a2 IntervalIntegrable (fun x => x + 1) \u03c1 \u2191k \u2191(k + 1) ** exact (continuous_id.add continuous_const).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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x \u2202\u03c1 + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 ** rw [intervalIntegral.integral_add] ** case 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 IntervalIntegrable (fun x => x) \u03c1 0 \u2191N ** exact continuous_id.intervalIntegrable _ _ ** case hg \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 IntervalIntegrable (fun x => 1) \u03c1 0 \u2191N ** exact continuous_const.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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u222b (x : \u211d) in 0 ..\u2191N, x \u2202\u03c1 + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 = (\u222b (a : \u03a9), truncation X (\u2191N) a) + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 ** rw [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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 (\u222b (a : \u03a9), X a) + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 ** refine' add_le_add le_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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 \u2264 1 ** rw [intervalIntegral.integral_of_le (Nat.cast_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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 \u222b (x : \u211d) in Set.Ioc 0 \u2191N, 1 \u2202\u03c1 \u2264 1 ** simp only [integral_const, Measure.restrict_apply', measurableSet_Ioc, Set.univ_inter,\n  Algebra.id.smul_eq_mul, mul_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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 ENNReal.toReal (\u2191\u2191(Measure.map X \u2119) (Set.Ioc 0 \u2191N)) \u2264 1 ** rw [\u2190 ENNReal.one_toReal] ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 \u22a2 ENNReal.toReal (\u2191\u2191(Measure.map X \u2119) (Set.Ioc 0 \u2191N)) \u2264 ENNReal.toReal 1 ** exact ENNReal.toReal_mono ENNReal.one_ne_top prob_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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 \u22a2 \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1) ** intro a b ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 a b : \u211d \u22a2 \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1) ** rw [ofReal_set_integral_one \u03c1 _,\n  Measure.map_apply_of_aemeasurable hint.aemeasurable measurableSet_Ioc] ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 a b : \u211d \u22a2 \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = \u2191\u2191\u2119 (X \u207b\u00b9' Set.Ioc a b) ** 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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 B : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1) \u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} = \u2211 j in range K, ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1) ** simp_rw [B] ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 B : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1) \u22a2 \u2211 j in range K, ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1) =     ENNReal.ofReal (\u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1) ** rw [ENNReal.ofReal_sum_of_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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 B : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1) \u22a2 \u2200 (i : \u2115), i \u2208 range K \u2192 0 \u2264 \u222b (x : \u211d) in Set.Ioc \u2191i \u2191N, 1 \u2202\u03c1 ** simp only [integral_const, Algebra.id.smul_eq_mul, mul_one, ENNReal.toReal_nonneg,\n  imp_true_iff] ** \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 B : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1) \u22a2 ENNReal.ofReal (\u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1) =     ENNReal.ofReal (\u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1) ** congr 1 ** case e_r \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 B : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1) \u22a2 \u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1 = \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 ** refine' sum_congr rfl fun j hj => _ ** case e_r \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 N : \u2115 hKN : K \u2264 N \u03c1 : Measure \u211d := Measure.map X \u2119 this : IsProbabilityMeasure \u03c1 A : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1 B : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1) j : \u2115 hj : j \u2208 range K \u22a2 \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1 = \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 ** rw [intervalIntegral.integral_of_le (Nat.cast_le.2 ((mem_range.1 hj).le.trans hKN))] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.strictMono_iff ** \u03b1 : Type u_1 s t : Finset \u03b1 \u03b2 : Type u_2 inst\u271d : Preorder \u03b2 f : Finset \u03b1 \u2192 \u03b2 \u22a2 StrictMono f \u2194 \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s < f (cons i s hi) ** classical\nsimp only [strictMono_iff_forall_covby, covby_iff, forall_exists_index, and_imp]\naesop ** \u03b1 : Type u_1 s t : Finset \u03b1 \u03b2 : Type u_2 inst\u271d : Preorder \u03b2 f : Finset \u03b1 \u2192 \u03b2 \u22a2 StrictMono f \u2194 \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s < f (cons i s hi) ** simp only [strictMono_iff_forall_covby, covby_iff, forall_exists_index, and_imp] ** \u03b1 : Type u_1 s t : Finset \u03b1 \u03b2 : Type u_2 inst\u271d : Preorder \u03b2 f : Finset \u03b1 \u2192 \u03b2 \u22a2 (\u2200 (a b : Finset \u03b1) (x : \u03b1) (x_1 : \u00acx \u2208 a), b = cons x a x_1 \u2192 f a < f b) \u2194     \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s < f (cons i s hi) ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.coeff_map ** 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 p : MvPolynomial \u03c3 R \u22a2 \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p) ** apply MvPolynomial.induction_on p <;> clear 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 \u22a2 \u2200 (a : R) (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) (\u2191C a)) = \u2191f (coeff m (\u2191C a)) ** intro r m ** 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\u271d : \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 r : R m : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff m (\u2191(map f) (\u2191C r)) = \u2191f (coeff m (\u2191C r)) ** rw [map_C] ** 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\u271d : \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 r : R m : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff m (\u2191C (\u2191f r)) = \u2191f (coeff m (\u2191C r)) ** simp only [coeff_C] ** 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\u271d : \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 r : R m : \u03c3 \u2192\u2080 \u2115 \u22a2 (if 0 = m then \u2191f r else 0) = \u2191f (if 0 = m then r else 0) ** split_ifs ** case neg 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\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 r : R m : \u03c3 \u2192\u2080 \u2115 h\u271d : \u00ac0 = m \u22a2 0 = \u2191f 0 ** rw [f.map_zero] ** case pos 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\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 r : R m : \u03c3 \u2192\u2080 \u2115 h\u271d : 0 = m \u22a2 \u2191f r = \u2191f r ** rfl ** 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 \u22a2 \u2200 (p q : MvPolynomial \u03c3 R),     (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)) \u2192       (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) q) = \u2191f (coeff m q)) \u2192         \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) (p + q)) = \u2191f (coeff m (p + q)) ** intro p q hp hq m ** 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\u271d : \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 p q : MvPolynomial \u03c3 R hp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p) hq : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) q) = \u2191f (coeff m q) m : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff m (\u2191(map f) (p + q)) = \u2191f (coeff m (p + q)) ** simp only [hp, hq, (map f).map_add, coeff_add] ** 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\u271d : \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 p q : MvPolynomial \u03c3 R hp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p) hq : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) q) = \u2191f (coeff m q) m : \u03c3 \u2192\u2080 \u2115 \u22a2 \u2191f (coeff m p) + \u2191f (coeff m q) = \u2191f (coeff m p + coeff m q) ** rw [f.map_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 \u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),     (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p)) \u2192       \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) (p * X n)) = \u2191f (coeff m (p * X n)) ** intro p i hp m ** 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\u271d : \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 p : MvPolynomial \u03c3 R i : \u03c3 hp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p) m : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff m (\u2191(map f) (p * X i)) = \u2191f (coeff m (p * X i)) ** simp only [hp, (map f).map_mul, map_X] ** 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\u271d : \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 p : MvPolynomial \u03c3 R i : \u03c3 hp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p) m : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff m (\u2191(map f) p * X i) = \u2191f (coeff m (p * X i)) ** simp only [hp, mem_support_iff, coeff_mul_X'] ** 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\u271d : \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 p : MvPolynomial \u03c3 R i : \u03c3 hp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p) m : \u03c3 \u2192\u2080 \u2115 \u22a2 (if i \u2208 m.support then \u2191f (coeff (m - fun\u2080 | i => 1) p) else 0) =     \u2191f (if i \u2208 m.support then coeff (m - fun\u2080 | i => 1) p else 0) ** split_ifs ** case neg 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\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 p : MvPolynomial \u03c3 R i : \u03c3 hp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p) m : \u03c3 \u2192\u2080 \u2115 h\u271d : \u00aci \u2208 m.support \u22a2 0 = \u2191f 0 ** rw [f.map_zero] ** case pos 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\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 p : MvPolynomial \u03c3 R i : \u03c3 hp : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(map f) p) = \u2191f (coeff m p) m : \u03c3 \u2192\u2080 \u2115 h\u271d : i \u2208 m.support \u22a2 \u2191f (coeff (m - fun\u2080 | i => 1) p) = \u2191f (coeff (m - fun\u2080 | i => 1) p) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz_estimate ** \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 testAgainstNN \u03bc f \u2264 testAgainstNN \u03bc g + nndist f g * mass \u03bc ** simp only [\u2190 \u03bc.testAgainstNN_const (nndist f g), \u2190 testAgainstNN_add, \u2190 ENNReal.coe_le_coe,\n  BoundedContinuousFunction.coe_add, const_apply, ENNReal.coe_add, Pi.add_apply,\n  coe_nnreal_ennreal_nndist, 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 g : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc \u2264 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191g \u03c9) + edist f g \u2202\u2191\u03bc ** apply lintegral_mono ** case hfg \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 (fun a => \u2191(\u2191f a)) \u2264 fun a => \u2191(\u2191g a) + edist f g ** have le_dist : \u2200 \u03c9, dist (f \u03c9) (g \u03c9) \u2264 nndist f g := BoundedContinuousFunction.dist_coe_le_dist ** case hfg \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 le_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g) \u22a2 (fun a => \u2191(\u2191f a)) \u2264 fun a => \u2191(\u2191g a) + edist f g ** intro \u03c9 ** case hfg \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 le_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g) \u03c9 : \u03a9 \u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9 ** have le' : f \u03c9 \u2264 g \u03c9 + nndist f g := by\n  apply (NNReal.le_add_nndist (f \u03c9) (g \u03c9)).trans\n  rw [add_le_add_iff_left]\n  exact dist_le_coe.mp (le_dist \u03c9) ** case hfg \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 le_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g) \u03c9 : \u03a9 le' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g \u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9 ** have le : (f \u03c9 : \u211d\u22650\u221e) \u2264 (g \u03c9 : \u211d\u22650\u221e) + nndist f g := by\n  rw [\u2190 ENNReal.coe_add];\n  exact ENNReal.coe_mono le' ** case hfg \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 le_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g) \u03c9 : \u03a9 le' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g le : \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9) + \u2191(nndist f g) \u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9 ** rwa [coe_nnreal_ennreal_nndist] at le ** \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 le_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g) \u03c9 : \u03a9 \u22a2 \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g ** apply (NNReal.le_add_nndist (f \u03c9) (g \u03c9)).trans ** \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 le_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g) \u03c9 : \u03a9 \u22a2 \u2191g \u03c9 + nndist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191g \u03c9 + nndist f g ** rw [add_le_add_iff_left] ** \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 le_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g) \u03c9 : \u03a9 \u22a2 nndist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 nndist f g ** exact dist_le_coe.mp (le_dist \u03c9) ** \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 g : \u03a9 \u2192\u1d47 \u211d\u22650 le_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g) \u03c9 : \u03a9 le' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g \u22a2 \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9) + \u2191(nndist f g) ** rw [\u2190 ENNReal.coe_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 g : \u03a9 \u2192\u1d47 \u211d\u22650 le_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g) \u03c9 : \u03a9 le' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g \u22a2 \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9 + nndist f g) ** exact ENNReal.coe_mono le' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.hausdorffMeasure_smul\u2080 ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E \u22a2 \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s ** intro r _ s ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r\u271d : \ud835\udd5c hr : r\u271d \u2260 0 s\u271d : Set E r : \ud835\udd5c a\u271d : r \u2260 0 s : Set E \u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s ** simp only [NNReal.rpow_eq_pow, ENNReal.smul_def, \u2190 ENNReal.coe_rpow_of_nonneg _ hd, smul_eq_mul] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r\u271d : \ud835\udd5c hr : r\u271d \u2260 0 s\u271d : Set E r : \ud835\udd5c a\u271d : r \u2260 0 s : Set E \u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 \u2191\u2016r\u2016\u208a ^ d * \u2191\u2191\u03bcH[d] s ** exact (lipschitzWith_smul (\u03b2 := E) r).hausdorffMeasure_image_le hd s ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E this : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) = NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s ** refine' le_antisymm (this hr s) _ ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E this : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u2264 \u2191\u2191\u03bcH[d] (r \u2022 s) ** rw [\u2190 ENNReal.le_inv_smul_iff] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E this : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u22a2 \u2191\u2191\u03bcH[d] s \u2264 (NNReal.rpow \u2016r\u2016\u208a d)\u207b\u00b9 \u2022 \u2191\u2191\u03bcH[d] (r \u2022 s)  \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E this : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2260 0 ** dsimp ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E this : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u22a2 \u2191\u2191\u03bcH[d] s \u2264 \u2191(\u2016r\u2016\u208a ^ d)\u207b\u00b9 * \u2191\u2191\u03bcH[d] (r \u2022 s)  \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E this : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2260 0 ** rw [\u2190 NNReal.inv_rpow, \u2190 nnnorm_inv] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E this : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u22a2 \u2191\u2191\u03bcH[d] s \u2264 \u2191(\u2016r\u207b\u00b9\u2016\u208a ^ d) * \u2191\u2191\u03bcH[d] (r \u2022 s) ** refine' Eq.trans_le _ (this (inv_ne_zero hr) (r \u2022 s)) ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E this : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u22a2 \u2191\u2191\u03bcH[d] s = \u2191\u2191\u03bcH[d] (r\u207b\u00b9 \u2022 r \u2022 s) ** rw [inv_smul_smul\u2080 hr] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u2070 : EMetricSpace X inst\u271d\u2079 : EMetricSpace Y inst\u271d\u2078 : MeasurableSpace X inst\u271d\u2077 : BorelSpace X inst\u271d\u2076 : MeasurableSpace Y inst\u271d\u2075 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E d : \u211d hd : 0 \u2264 d r : \ud835\udd5c hr : r \u2260 0 s : Set E this : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2260 0 ** simp [hr] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ordConnected_range ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1 inst\u271d\u00b9 : Preorder \u03b2 s t : Set \u03b1 E : Type u_3 inst\u271d : OrderIsoClass E \u03b1 \u03b2 e : E \u22a2 OrdConnected (range \u2191e) ** simp_rw [\u2190 image_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1 inst\u271d\u00b9 : Preorder \u03b2 s t : Set \u03b1 E : Type u_3 inst\u271d : OrderIsoClass E \u03b1 \u03b2 e : E \u22a2 OrdConnected (\u2191e '' univ) ** exact ordConnected_image (e : \u03b1 \u2243o \u03b2) ** Qed",
        "informal": ""
    },
    {
        "formal": "Multiset.toFinset_dedup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t m : Multiset \u03b1 \u22a2 toFinset (dedup m) = toFinset m ** simp_rw [toFinset, dedup_idem] ** Qed",
        "informal": ""
    },
    {
        "formal": "Partrec.rfind ** \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 p : \u03b1 \u2192 \u2115 \u2192. Bool hp : Partrec\u2082 p n : \u2115 \u22a2 (Nat.rfind fun n_1 =>       (fun m => decide (m = 0)) <$>         Part.bind \u2191(decode (Nat.pair n n_1)) fun a =>           Part.map encode ((fun a => Part.map (fun b => bif (a, b).2 then 0 else 1) (p a.1 a.2)) a)) =     Part.bind \u2191(decode n) fun a => Part.map encode ((fun a => Nat.rfind (p a)) a) ** cases' e : decode (\u03b1 := \u03b1) n with a <;> simp [e, Nat.rfind_zero_none, map_id'] ** case some \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 p : \u03b1 \u2192 \u2115 \u2192. Bool hp : Partrec\u2082 p n : \u2115 a : \u03b1 e : decode n = Option.some a \u22a2 (Nat.rfind fun n => Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n))) =     Nat.rfind (p a) ** congr ** case some.e_p \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 p : \u03b1 \u2192 \u2115 \u2192. Bool hp : Partrec\u2082 p n : \u2115 a : \u03b1 e : decode n = Option.some a \u22a2 (fun n => Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n))) = p a ** funext n ** case some.e_p.h \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 p : \u03b1 \u2192 \u2115 \u2192. Bool hp : Partrec\u2082 p n\u271d : \u2115 a : \u03b1 e : decode n\u271d = Option.some a n : \u2115 \u22a2 Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n)) = p a n ** simp only [map_map, Function.comp] ** case some.e_p.h \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 p : \u03b1 \u2192 \u2115 \u2192. Bool hp : Partrec\u2082 p n\u271d : \u2115 a : \u03b1 e : decode n\u271d = Option.some a n : \u2115 \u22a2 Part.map (fun x => decide ((bif x then 0 else 1) = 0)) (p a n) = p a n ** refine map_id' (fun b => ?_) _ ** case some.e_p.h \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 p : \u03b1 \u2192 \u2115 \u2192. Bool hp : Partrec\u2082 p n\u271d : \u2115 a : \u03b1 e : decode n\u271d = Option.some a n : \u2115 b : Bool \u22a2 decide ((bif b then 0 else 1) = 0) = b ** cases b <;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.cardinal_generateMeasurable_le_continuum ** \u03b1 : Type u s : Set (Set \u03b1) hs : #\u2191s \u2264 \ud835\udd20 \u22a2 max (#\u2191s) 2 ^ \u2135\u2080 \u2264 \ud835\udd20 ** rw [\u2190 continuum_power_aleph0] ** \u03b1 : Type u s : Set (Set \u03b1) hs : #\u2191s \u2264 \ud835\udd20 \u22a2 max (#\u2191s) 2 ^ \u2135\u2080 \u2264 \ud835\udd20 ^ \u2135\u2080 ** exact_mod_cast power_le_power_right (max_le hs (nat_lt_continuum 2).le) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_integral_tendsto ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 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), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc)) ** refine' \u27e8_, tendsto_of_forall_integral_tendsto\u27e9 ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 \u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2192 \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc)) ** rw [tendsto_iff_forall_lintegral_tendsto] ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 \u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))) \u2192     \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc)) ** intro h f ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 h : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) f : \u03a9 \u2192\u1d47 \u211d \u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc)) ** simp_rw [BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub] ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 h : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) f : \u03a9 \u2192\u1d47 \u211d \u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc)) ** set f_pos := f.nnrealPart with _def_f_pos ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 h : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) f : \u03a9 \u2192\u1d47 \u211d f_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f _def_f_pos : f_pos = nnrealPart f \u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc)) ** set f_neg := (-f).nnrealPart with _def_f_neg ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 h : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) f : \u03a9 \u2192\u1d47 \u211d f_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f _def_f_pos : f_pos = nnrealPart f f_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f) _def_f_neg : f_neg = nnrealPart (-f) \u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc)) ** have tends_pos := (ENNReal.tendsto_toReal (f_pos.lintegral_lt_top_of_nnreal \u03bc).ne).comp (h f_pos) ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 h : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) f : \u03a9 \u2192\u1d47 \u211d f_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f _def_f_pos : f_pos = nnrealPart f f_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f) _def_f_neg : f_neg = nnrealPart (-f) tends_pos :   Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc))) \u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc)) ** have tends_neg := (ENNReal.tendsto_toReal (f_neg.lintegral_lt_top_of_nnreal \u03bc).ne).comp (h f_neg) ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 h : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) f : \u03a9 \u2192\u1d47 \u211d f_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f _def_f_pos : f_pos = nnrealPart f f_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f) _def_f_neg : f_neg = nnrealPart (-f) tends_pos :   Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc))) tends_neg :   Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))) \u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc)) ** have aux :\n  \u2200 g : \u03a9 \u2192\u1d47 \u211d\u22650,\n    (ENNReal.toReal \u2218 fun i : \u03b3 => \u222b\u207b x : \u03a9, \u2191(g x) \u2202(\u03bcs i : Measure \u03a9)) = fun i : \u03b3 =>\n      (\u222b\u207b x : \u03a9, \u2191(g x) \u2202(\u03bcs i : Measure \u03a9)).toReal :=\n  fun _ => rfl ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 h : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) f : \u03a9 \u2192\u1d47 \u211d f_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f _def_f_pos : f_pos = nnrealPart f f_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f) _def_f_neg : f_neg = nnrealPart (-f) tends_pos :   Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc))) tends_neg :   Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc))) aux :   \u2200 (g : \u03a9 \u2192\u1d47 \u211d\u22650),     (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) = fun i => ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) \u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc)) ** simp_rw [aux, BoundedContinuousFunction.toReal_lintegral_coe_eq_integral] at tends_pos tends_neg ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 h : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) f : \u03a9 \u2192\u1d47 \u211d f_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f _def_f_pos : f_pos = nnrealPart f f_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f) _def_f_neg : f_neg = nnrealPart (-f) aux :   \u2200 (g : \u03a9 \u2192\u1d47 \u211d\u22650),     (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) = fun i => ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) tends_pos : Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191\u03bc)) tends_neg :   Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc)) \u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i) - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F     (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191\u03bc - \u222b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191\u03bc)) ** exact Tendsto.sub tends_pos tends_neg ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.min_eq_right ** a b : Int h : b \u2264 a \u22a2 min a b = b ** rw [Int.min_comm a b] ** a b : Int h : b \u2264 a \u22a2 min b a = b ** exact Int.min_eq_left h ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.countable_iUnion ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x t : \u03b9 \u2192 Set \u03b1 inst\u271d : Countable \u03b9 ht : \u2200 (i : \u03b9), Set.Countable (t i) \u22a2 Set.Countable (\u22c3 i, t i) ** haveI := fun a => (ht a).to_subtype ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x t : \u03b9 \u2192 Set \u03b1 inst\u271d : Countable \u03b9 ht : \u2200 (i : \u03b9), Set.Countable (t i) this : \u2200 (a : \u03b9), Countable \u2191(t a) \u22a2 Set.Countable (\u22c3 i, t i) ** rw [iUnion_eq_range_psigma] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x t : \u03b9 \u2192 Set \u03b1 inst\u271d : Countable \u03b9 ht : \u2200 (i : \u03b9), Set.Countable (t i) this : \u2200 (a : \u03b9), Countable \u2191(t a) \u22a2 Set.Countable (range fun a => \u2191a.snd) ** apply countable_range ** Qed",
        "informal": ""
    },
    {
        "formal": "BddBelow.finite_of_bddAbove ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 s : Set \u03b1 h\u2080 : BddBelow s h\u2081 : BddAbove s \u22a2 Set.Finite s ** let \u27e8a, ha\u27e9 := h\u2080 ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 s : Set \u03b1 h\u2080 : BddBelow s h\u2081 : BddAbove s a : \u03b1 ha : a \u2208 lowerBounds s \u22a2 Set.Finite s ** let \u27e8b, hb\u27e9 := h\u2081 ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c x : \u03b1 s : Set \u03b1 h\u2080 : BddBelow s h\u2081 : BddAbove s a : \u03b1 ha : a \u2208 lowerBounds s b : \u03b1 hb : b \u2208 upperBounds s \u22a2 Set.Finite s ** classical exact \u27e8Set.fintypeOfMemBounds ha hb\u27e9 ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c x : \u03b1 s : Set \u03b1 h\u2080 : BddBelow s h\u2081 : BddAbove s a : \u03b1 ha : a \u2208 lowerBounds s b : \u03b1 hb : b \u2208 upperBounds s \u22a2 Set.Finite s ** exact \u27e8Set.fintypeOfMemBounds ha hb\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub ** X : Type u_1 inst\u271d\u00b3 : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : OpensMeasurableSpace X \u03bc : Measure X inst\u271d : IsFiniteMeasure \u03bc f : X \u2192\u1d47 \u211d \u22a2 \u222b (x : X), \u2191f x \u2202\u03bc = \u222b (x : X), \u2191(\u2191(nnrealPart f) x) \u2202\u03bc - \u222b (x : X), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u03bc ** simp only [f.self_eq_nnrealPart_sub_nnrealPart_neg, Pi.sub_apply, integral_sub,\n           integrable_of_nnreal] ** X : Type u_1 inst\u271d\u00b3 : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : OpensMeasurableSpace X \u03bc : Measure X inst\u271d : IsFiniteMeasure \u03bc f : X \u2192\u1d47 \u211d \u22a2 \u222b (a : X), (NNReal.toReal \u2218 \u2191(nnrealPart f)) a \u2202\u03bc - \u222b (a : X), (NNReal.toReal \u2218 \u2191(nnrealPart (-f))) a \u2202\u03bc =     \u222b (x : X), \u2191(\u2191(nnrealPart f) x) \u2202\u03bc - \u222b (x : X), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u03bc ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.mod_zero ** n : Num \u22a2 mod n 0 = n ** cases n ** case zero  \u22a2 mod zero 0 = zero ** rfl ** case pos a\u271d : PosNum \u22a2 mod (pos a\u271d) 0 = pos a\u271d ** simp [Num.mod] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite ** \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 hf : FinStronglyMeasurable f \u03bc \u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t) ** rcases hf with \u27e8fs, hT_lt_top, h_approx\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 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) \u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t) ** let T n := support (fs n) ** case intro.intro \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) \u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t) ** have hT_meas : \u2200 n, MeasurableSet (T n) := fun n => SimpleFunc.measurableSet_support (fs n) ** case intro.intro \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) \u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t) ** let t := \u22c3 n, T n ** case intro.intro \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n \u22a2 \u2203 t, MeasurableSet t \u2227 (\u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0) \u2227 SigmaFinite (Measure.restrict \u03bc t) ** refine' \u27e8t, MeasurableSet.iUnion hT_meas, _, _\u27e9 ** case intro.intro.refine'_1 \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n \u22a2 \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0 ** have h_fs_zero : \u2200 n, \u2200 x \u2208 t\u1d9c, fs n x = 0 := by\n  intro n x hxt\n  rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt\n  simpa using hxt n ** case intro.intro.refine'_1 \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n h_fs_zero : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0 \u22a2 \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0 ** refine' fun x hxt => tendsto_nhds_unique (h_approx x) _ ** case intro.intro.refine'_1 \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n h_fs_zero : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0 x : \u03b1 hxt : x \u2208 t\u1d9c \u22a2 Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd 0) ** rw [funext fun n => h_fs_zero n x hxt] ** case intro.intro.refine'_1 \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n h_fs_zero : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0 x : \u03b1 hxt : x \u2208 t\u1d9c \u22a2 Tendsto (fun n => 0) atTop (\ud835\udcdd 0) ** exact tendsto_const_nhds ** \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n \u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 t\u1d9c \u2192 \u2191(fs n) x = 0 ** intro n x hxt ** \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n n : \u2115 x : \u03b1 hxt : x \u2208 t\u1d9c \u22a2 \u2191(fs n) x = 0 ** rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt ** \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n n : \u2115 x : \u03b1 hxt : \u2200 (x_1 : \u2115), \u00acx \u2208 T x_1 \u22a2 \u2191(fs n) x = 0 ** simpa using hxt n ** case intro.intro.refine'_2 \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n \u22a2 SigmaFinite (Measure.restrict \u03bc t) ** refine' \u27e8\u27e8\u27e8fun n => t\u1d9c \u222a T n, fun _ => trivial, fun n => _, _\u27e9\u27e9\u27e9 ** case intro.intro.refine'_2.refine'_1 \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n n : \u2115 \u22a2 \u2191\u2191(Measure.restrict \u03bc t) ((fun n => t\u1d9c \u222a T n) n) < \u22a4 ** rw [Measure.restrict_apply' (MeasurableSet.iUnion hT_meas), Set.union_inter_distrib_right,\n  Set.compl_inter_self t, Set.empty_union] ** case intro.intro.refine'_2.refine'_1 \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n n : \u2115 \u22a2 \u2191\u2191\u03bc (T n \u2229 \u22c3 b, T b) < \u22a4 ** exact (measure_mono (Set.inter_subset_left _ _)).trans_lt (hT_lt_top n) ** case intro.intro.refine'_2.refine'_2 \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n \u22a2 \u22c3 i, (fun n => t\u1d9c \u222a T n) i = univ ** rw [\u2190 Set.union_iUnion t\u1d9c T] ** case intro.intro.refine'_2.refine'_2 \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 : Zero \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : T2Space \u03b2 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 hT_lt_top : \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 h_approx : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) T : \u2115 \u2192 Set \u03b1 := fun n => support \u2191(fs n) hT_meas : \u2200 (n : \u2115), MeasurableSet (T n) t : Set \u03b1 := \u22c3 n, T n \u22a2 t\u1d9c \u222a \u22c3 i, T i = univ ** exact Set.compl_union_self _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.le.intro_sub ** a b : Int n : Nat h : b - a = \u2191n \u22a2 a \u2264 b ** simp [le_def, h] ** a b : Int n : Nat h : b - a = \u2191n \u22a2 NonNeg \u2191n ** constructor ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.piecewise_empty ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 f g : \u03b1 \u2192\u209b \u03b2 \u22a2 \u2191(piecewise \u2205 (_ : MeasurableSet \u2205) f g) = \u2191g ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 f g : \u03b1 \u2192\u209b \u03b2 \u22a2 Set.piecewise \u2205 \u2191f \u2191g = \u2191g ** convert Set.piecewise_empty f g ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snormEssSup_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 hc : c \u2260 0 \u22a2 snormEssSup f (c \u2022 \u03bc) = snormEssSup f \u03bc ** simp_rw [snormEssSup] ** \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 hc : c \u2260 0 \u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a) (c \u2022 \u03bc) = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ** exact essSup_smul_measure hc ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_liminf' ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) \u22a2 Measurable fun x => liminf (fun i => f i x) v ** have : Countable (Subtype p) := Encodable.nonempty_encodable.1 hv.countable ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) \u22a2 Measurable fun x => liminf (fun i => f i x) v ** rcases isEmpty_or_nonempty (Subtype p) with hp|hp ** 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\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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : Nonempty (Subtype p) \u22a2 Measurable fun x => liminf (fun i => f i x) v ** by_cases H : \u2203 (j : Subtype p), s j = \u2205 ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 \u22a2 Measurable fun x => liminf (fun i => f i x) v ** simp_rw [hv.liminf_eq_ite, if_neg 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** have : \u2200 i, Countable (s i) := fun i \u21a6 countable_coe_iff.2 (hs i) ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** let m : Subtype p \u2192 Set \u03b4 := fun j \u21a6 {x | BddBelow (range (fun (i : s j) \u21a6 f i x))} ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** have m_meas : \u2200 j, MeasurableSet (m j) :=\n  fun j \u21a6 measurableSet_bddBelow_range (fun (i : s j) \u21a6 hf i) ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** have mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), x \u2209 m j} := by\n  rw [setOf_forall]\n  exact MeasurableSet.iInter (fun j \u21a6 (m_meas j).compl) ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** apply Measurable.piecewise mc_meas measurable_const ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} \u22a2 Measurable fun x => \u2a06 j, \u2a05 i, f (\u2191i) x ** apply measurable_iSup (fun j \u21a6 ?_) ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p \u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b ** let reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x \u21a6 liminf_reparam (fun i \u21a6 f i x) s p ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p \u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b ** let F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x \u21a6 \u2a05 (i : s j), f i x ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x \u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b ** have F0_meas : \u2200 j, Measurable (F0 j) := fun j \u21a6 measurable_iInf (fun (i : s j) \u21a6 hf 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\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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) \u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b ** set F1 : \u03b4 \u2192 \u03b1 := fun x \u21a6 F0 (reparam x j) x with hF1 ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x \u22a2 Measurable F1 ** let g : \u2115 \u2192 Subtype p := choose (exists_surjective_nat (Subtype p)) ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k this :   F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x \u22a2 Measurable F1 ** rw [this] ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k this :   F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x \u22a2 Measurable fun x =>     if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** apply Measurable.piecewise (m_meas j) (F0_meas j) ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k this :   F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x \u22a2 Measurable fun x => F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** apply Measurable.find (fun n \u21a6 F0_meas (g n)) (fun n \u21a6 ?_) ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k this :   F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x n : \u2115 \u22a2 MeasurableSet {x | x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k} ** exact (m_meas (g n)).union mc_meas ** 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\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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : IsEmpty (Subtype p) \u22a2 Measurable fun x => liminf (fun i => f i x) v ** simp [hv.liminf_eq_sSup_iUnion_iInter] ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u2203 j, s \u2191j = \u2205 \u22a2 Measurable fun x => liminf (fun i => f i x) v ** simp_rw [hv.liminf_eq_ite, if_pos H, 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) \u22a2 MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} ** rw [setOf_forall] ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) \u22a2 MeasurableSet (\u22c2 i, {x | \u00acx \u2208 m i}) ** exact MeasurableSet.iInter (fun j \u21a6 (m_meas j).compl) ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) \u22a2 \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** intro x ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** by_cases H : \u2203 k, x \u2208 m k ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H\u271d : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 H : \u2203 k, x \u2208 m k \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** rcases H with \u27e8k, hk\u27e9 ** case pos.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\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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 k : Subtype p hk : x \u2208 m k \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** rcases choose_spec (exists_surjective_nat (Subtype p)) k with \u27e8n, rfl\u27e9 ** case pos.intro.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\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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 n : \u2115 hk : x \u2208 m (choose (_ : \u2203 f, Function.Surjective f) n) \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** exact \u27e8n, Or.inl hk\u27e9 ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H\u271d : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 H : \u00ac\u2203 k, x \u2208 m k \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** push_neg at 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H\u271d : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 H : \u2200 (k : Subtype p), \u00acx \u2208 m k \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** exact \u27e80, Or.inr H\u27e9 ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k \u22a2 F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** ext x ** case 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\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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 \u22a2 F1 x = if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** have A : reparam x j = if x \u2208 m j then j else g (Nat.find (Z x)) := rfl ** case 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\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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) \u22a2 F1 x = if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** split_ifs with hjx ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : x \u2208 m j \u22a2 F1 x = F0 j x ** have : reparam x j = j := by rw [A, if_pos hjx] ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : x \u2208 m j this : reparam x j = j \u22a2 F1 x = F0 j x ** simp only [hF1, this] ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : x \u2208 m j \u22a2 reparam x j = j ** rw [A, if_pos hjx] ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : \u00acx \u2208 m j \u22a2 F1 x = F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** have : reparam x j = g (Nat.find (Z x)) := by rw [A, if_neg hjx] ** 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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : \u00acx \u2208 m j this : reparam x j = g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) \u22a2 F1 x = F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** simp only [hF1, this] ** \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 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : \u00acx \u2208 m j \u22a2 reparam x j = g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) ** rw [A, if_neg hjx] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Finite.cast_ncard_eq ** \u03b1 : Type u_1 s t : Set \u03b1 hs : Set.Finite s \u22a2 \u2191(ncard s) = encard s ** rwa [ncard, ENat.coe_toNat_eq_self, ne_eq, encard_eq_top_iff, Set.Infinite, not_not] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.card_dvd_card_image\u2082_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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : 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 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s) id \u22a2 card t \u2223 card (image\u2082 f s t) ** induction' s using Finset.induction with a s _ ih ** case insert \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d\u00b9 a' : \u03b1 b b' : \u03b2 c : \u03b3 hf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id a : \u03b1 s : Finset \u03b1 a\u271d : \u00aca \u2208 s ih :   (\u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)) \u2192     PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s) id \u2192 card t \u2223 card (image\u2082 f s t) hf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id \u22a2 card t \u2223 card (image\u2082 f (insert a s) t) ** specialize ih (forall_of_forall_insert hf)\n  (hs.subset <| Set.image_subset _ <| coe_subset.2 <| subset_insert _ _) ** case insert \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d\u00b9 a' : \u03b1 b b' : \u03b2 c : \u03b3 hf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id a : \u03b1 s : Finset \u03b1 a\u271d : \u00aca \u2208 s hf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id ih : card t \u2223 card (image\u2082 f s t) \u22a2 card t \u2223 card (image\u2082 f (insert a s) t) ** rw [image\u2082_insert_left] ** case insert \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d\u00b9 a' : \u03b1 b b' : \u03b2 c : \u03b3 hf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id a : \u03b1 s : Finset \u03b1 a\u271d : \u00aca \u2208 s hf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id ih : card t \u2223 card (image\u2082 f s t) \u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t) ** by_cases h : Disjoint (image (f a) t) (image\u2082 f s t) ** case neg \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d\u00b9 a' : \u03b1 b b' : \u03b2 c : \u03b3 hf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id a : \u03b1 s : Finset \u03b1 a\u271d : \u00aca \u2208 s hf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id ih : card t \u2223 card (image\u2082 f s t) h : \u00acDisjoint (image (f a) t) (image\u2082 f s t) \u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t) ** simp_rw [\u2190 biUnion_image_left, disjoint_biUnion_right, not_forall] at h ** case neg \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d\u00b9 a' : \u03b1 b b' : \u03b2 c : \u03b3 hf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id a : \u03b1 s : Finset \u03b1 a\u271d : \u00aca \u2208 s hf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id ih : card t \u2223 card (image\u2082 f s t) h : \u2203 x x_1, \u00acDisjoint (image (f a) t) (image (f x) t) \u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t) ** obtain \u27e8b, hb, h\u27e9 := h ** case neg.intro.intro \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d\u00b9 a' : \u03b1 b\u271d b' : \u03b2 c : \u03b3 hf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id a : \u03b1 s : Finset \u03b1 a\u271d : \u00aca \u2208 s hf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id ih : card t \u2223 card (image\u2082 f s t) b : \u03b1 hb : b \u2208 s h : \u00acDisjoint (image (f a) t) (image (f b) t) \u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t) ** rwa [union_eq_right.2] ** case neg.intro.intro \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d\u00b9 a' : \u03b1 b\u271d b' : \u03b2 c : \u03b3 hf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id a : \u03b1 s : Finset \u03b1 a\u271d : \u00aca \u2208 s hf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id ih : card t \u2223 card (image\u2082 f s t) b : \u03b1 hb : b \u2208 s h : \u00acDisjoint (image (f a) t) (image (f b) t) \u22a2 image (fun b => f a b) t \u2286 image\u2082 f s t ** exact (hs.eq (Set.mem_image_of_mem _ <| mem_insert_self _ _)\n    (Set.mem_image_of_mem _ <| mem_insert_of_mem hb) h).trans_subset\n  (image_subset_image\u2082_right hb) ** case empty \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : 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 hf\u271d : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s) id hf : \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 Injective (f a) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191\u2205) id \u22a2 card t \u2223 card (image\u2082 f \u2205 t) ** simp ** case pos \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d\u00b9 a' : \u03b1 b b' : \u03b2 c : \u03b3 hf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id a : \u03b1 s : Finset \u03b1 a\u271d : \u00aca \u2208 s hf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id ih : card t \u2223 card (image\u2082 f s t) h : Disjoint (image (f a) t) (image\u2082 f s t) \u22a2 card t \u2223 card (image (fun b => f a b) t \u222a image\u2082 f s t) ** rw [card_union_eq h] ** case pos \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d\u00b9 a' : \u03b1 b b' : \u03b2 c : \u03b3 hf\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Injective (f a) hs\u271d : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191s\u271d) id a : \u03b1 s : Finset \u03b1 a\u271d : \u00aca \u2208 s hf : \u2200 (a_1 : \u03b1), a_1 \u2208 insert a s \u2192 Injective (f a_1) hs : PairwiseDisjoint ((fun a => image (f a) t) '' \u2191(insert a s)) id ih : card t \u2223 card (image\u2082 f s t) h : Disjoint (image (f a) t) (image\u2082 f s t) \u22a2 card t \u2223 card (image (f a) t) + card (image\u2082 f s t) ** exact (card_image_of_injective _ <| hf _ <| mem_insert_self _ _).symm.dvd.add ih ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_countable ** \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 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hs : Set.Countable s \u22a2 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in \u22c3 x \u2208 s, {x}, f a \u2202\u03bc ** rw [biUnion_of_singleton] ** \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 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hs : Set.Countable s \u22a2 \u2211' (a : \u2191s), \u222b\u207b (x : \u03b1) in {\u2191a}, f x \u2202\u03bc = \u2211' (a : \u2191s), f \u2191a * \u2191\u2191\u03bc {\u2191a} ** simp only [lintegral_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "convex_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 \u22a2 Convex \u211d (parallelepiped v) ** rw [parallelepiped_eq_sum_segment] ** \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 Convex \u211d (\u2211 i : \u03b9, segment \u211d 0 (v i)) ** exact convex_sum _ fun _i _hi => convex_segment _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "AddCircle.volume_of_add_preimage_eq ** T : \u211d hT : Fact (0 < T) s I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u hs : u +\u1d65 s =\u1da0[ae volume] s hI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u))) \u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I) ** let G := AddSubgroup.zmultiples u ** T : \u211d hT : Fact (0 < T) s I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u hs : u +\u1d65 s =\u1da0[ae volume] s hI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u))) G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u \u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I) ** haveI : Fintype G := @Fintype.ofFinite _ hu.finite_zmultiples ** T : \u211d hT : Fact (0 < T) s I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u hs : u +\u1d65 s =\u1da0[ae volume] s hI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u))) G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u this : Fintype { x // x \u2208 G } \u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I) ** have hsG : \u2200 g : G, (g +\u1d65 s : Set <| AddCircle T) =\u1d50[volume] s := by\n  rintro \u27e8y, hy\u27e9; exact (vadd_ae_eq_self_of_mem_zmultiples hs hy : _) ** T : \u211d hT : Fact (0 < T) s I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u hs : u +\u1d65 s =\u1da0[ae volume] s hI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u))) G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u this : Fintype { x // x \u2208 G } hsG : \u2200 (g : { x // x \u2208 G }), g +\u1d65 s =\u1da0[ae volume] s \u22a2 \u2191\u2191volume s = addOrderOf u \u2022 \u2191\u2191volume (s \u2229 I) ** rw [(isAddFundamentalDomain_of_ae_ball I u x hu hI).measure_eq_card_smul_of_vadd_ae_eq_self s hsG,\n  add_order_eq_card_zmultiples' u, Nat.card_eq_fintype_card] ** T : \u211d hT : Fact (0 < T) s I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u hs : u +\u1d65 s =\u1da0[ae volume] s hI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u))) G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u this : Fintype { x // x \u2208 G } \u22a2 \u2200 (g : { x // x \u2208 G }), g +\u1d65 s =\u1da0[ae volume] s ** rintro \u27e8y, hy\u27e9 ** case mk T : \u211d hT : Fact (0 < T) s I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u hs : u +\u1d65 s =\u1da0[ae volume] s hI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u))) G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u this : Fintype { x // x \u2208 G } y : AddCircle T hy : y \u2208 G \u22a2 { val := y, property := hy } +\u1d65 s =\u1da0[ae volume] s ** exact (vadd_ae_eq_self_of_mem_zmultiples hs hy : _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.gcd_gcd_self_left_left ** m n : Nat \u22a2 gcd (gcd m n) m = gcd m n ** rw [gcd_comm m n, gcd_gcd_self_left_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.generateFrom_insert_univ ** \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 S : Set (Set \u03b1) \u22a2 generateFrom (insert univ S) = generateFrom S ** rw [insert_eq, \u2190 generateFrom_sup_generateFrom, generateFrom_singleton_univ, bot_sup_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.sum_smul_dirac ** \u03b1 : Type u_1 \u03b2 : Type ?u.17474 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 s : Set \u03b1 inst\u271d\u00b9 : Countable \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 \u03bc : Measure \u03b1 \u22a2 (sum fun a => \u2191\u2191\u03bc {a} \u2022 dirac a) = \u03bc ** simpa using (map_eq_sum \u03bc id measurable_id).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Setoid.classes_mkClasses ** \u03b1 : Type u_1 c : Set (Set \u03b1) hc : IsPartition c s : Set \u03b1 x\u271d : s \u2208 classes (mkClasses c (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b)) y : \u03b1 hs : s = {x | Rel (mkClasses c (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b)) x y} b : Set \u03b1 hm : b \u2208 c hb : y \u2208 b _hy : \u2200 (y_1 : Set \u03b1), y_1 \u2208 c \u2192 y \u2208 y_1 \u2192 y_1 = b \u22a2 s \u2208 c ** rwa [show s = b from hs.symm \u25b8 Set.ext fun x =>\n  \u27e8fun hx => symm' (mkClasses c hc.2) hx b hm hb, fun hx b' hc' hx' =>\n      eq_of_mem_eqv_class hc.2 hm hx hc' hx' \u25b8 hb\u27e9] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_mul_Ioo_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' Ioo a b = Ioo (b / c) (a / c) ** simpa only [mul_comm] using preimage_mul_const_Ioo_of_neg a b h ** Qed",
        "informal": ""
    },
    {
        "formal": "List.get?_set_ne ** \u03b1 : Type u_1 a : \u03b1 m n : Nat l : List \u03b1 h : m \u2260 n \u22a2 get? (set l m a) n = get? l n ** simp only [set_eq_modifyNth, get?_modifyNth_ne _ _ h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.ae_eq_of_forall_set_integral_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 E hf : Integrable f hg : Integrable g hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc \u22a2 f =\u1d50[\u03bc] g ** rw [\u2190 sub_ae_eq_zero] ** \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 E hf : Integrable f hg : Integrable g hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc \u22a2 f - g =\u1d50[\u03bc] 0 ** have hfg' : \u2200 s : Set \u03b1, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 := by\n  intro s hs h\u03bcs\n  rw [integral_sub' hf.integrableOn hg.integrableOn]\n  exact sub_eq_zero.mpr (hfg s hs h\u03bcs) ** \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 E hf : Integrable f hg : Integrable g hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0 \u22a2 f - g =\u1d50[\u03bc] 0 ** exact Integrable.ae_eq_zero_of_forall_set_integral_eq_zero (hf.sub hg) hfg' ** \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 E hf : Integrable f hg : Integrable g hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0 ** intro s hs h\u03bcs ** \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 E hf : Integrable f hg : Integrable g hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0 ** rw [integral_sub' hf.integrableOn hg.integrableOn] ** \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 E hf : Integrable f hg : Integrable g hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc - \u222b (a : \u03b1) in s, g a \u2202\u03bc = 0 ** exact sub_eq_zero.mpr (hfg s hs h\u03bcs) ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.one_mul ** n : Nat k : Fin (n + 1) \u22a2 1 * k = k ** rw [Fin.mul_comm, Fin.mul_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.surjOn_iff_exists_map_subtype ** \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\u271d g\u2081 g\u2082 : \u03b2 \u2192 \u03b3 f' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1 g' : \u03b3 \u2192 \u03b2 a : \u03b1 b : \u03b2 x\u271d : \u2203 t' g, t \u2286 t' \u2227 Surjective g \u2227 \u2200 (x : \u2191s), f \u2191x = \u2191(g x) y : \u03b2 hy : y \u2208 t t' : Set \u03b2 g : \u2191s \u2192 \u2191t' htt' : t \u2286 t' hg : Surjective g hfg : \u2200 (x : \u2191s), f \u2191x = \u2191(g x) x : \u2191s hx : g x = { val := y, property := (_ : y \u2208 t') } \u22a2 f \u2191x = y ** rw [hfg, hx, Subtype.coe_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux1 ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 this :   \u2200 (A : E \u2192L[\u211d] E),     \u2203 \u03b4,       0 < \u03b4 \u2227         (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227           \u2200 (t : Set E) (g : E \u2192 E),             ApproximatesLinearOn g A t \u03b4 \u2192               ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t \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 ** choose \u03b4 h\u03b4 using 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t \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 ** obtain \u27e8t, A, t_disj, t_meas, t_cover, ht, -\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 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 : 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) \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 ** have s_eq : s = \u22c3 n, s \u2229 t n := by\n  rw [\u2190 inter_iUnion]\n  exact Subset.antisymm (subset_inter Subset.rfl t_cover) (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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u22a2 \u2200 (A : E \u2192L[\u211d] E),     \u2203 \u03b4,       0 < \u03b4 \u2227         (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227           \u2200 (t : Set E) (g : E \u2192 E),             ApproximatesLinearOn g A t \u03b4 \u2192               ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** obtain \u27e8\u03b4', \u03b4'pos, h\u03b4'\u27e9 : \u2203 (\u03b4' : \u211d), 0 < \u03b4' \u2227 \u2200 B, dist B A < \u03b4' \u2192 dist B.det A.det < \u2191\u03b5 :=\n  continuousAt_iff.1 ContinuousLinearMap.continuous_det.continuousAt \u03b5 \u03b5pos ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** let \u03b4'' : \u211d\u22650 := \u27e8\u03b4' / 2, (half_pos \u03b4'pos).le\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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** have I'' : \u2200 B : E \u2192L[\u211d] E, \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |B.det - A.det| \u2264 \u2191\u03b5 := by\n  intro B hB\n  rw [\u2190 Real.dist_eq]\n  apply (h\u03b4' B _).le\n  rw [dist_eq_norm]\n  exact hB.trans_lt (half_lt_self \u03b4'pos) ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** rcases eq_or_ne A.det 0 with (hA | hA) ** case intro.intro.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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** let m : \u211d\u22650 := Real.toNNReal |A.det| - \u03b5 ** case intro.intro.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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** rcases ((mul_le_addHaar_image_of_lt_det \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, \u03b4pos\u27e9 ** case intro.intro.inr.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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** refine' \u27e8min \u03b4 \u03b4'', lt_min \u03b4pos (half_pos \u03b4'pos), _, _\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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } \u22a2 \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 ** intro B hB ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } B : E \u2192L[\u211d] E hB : \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u22a2 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 ** rw [\u2190 Real.dist_eq] ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } B : E \u2192L[\u211d] E hB : \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u22a2 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) \u2264 \u2191\u03b5 ** apply (h\u03b4' B _).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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } B : E \u2192L[\u211d] E hB : \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u22a2 dist B A < \u03b4' ** rw [dist_eq_norm] ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } B : E \u2192L[\u211d] E hB : \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u22a2 \u2016B - A\u2016 < \u03b4' ** exact hB.trans_lt (half_lt_self \u03b4'pos) ** case intro.intro.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 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A = 0 \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** refine' \u27e8\u03b4'', half_pos \u03b4'pos, I'', _\u27e9 ** case intro.intro.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 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A = 0 \u22a2 \u2200 (t : Set E) (g : E \u2192 E),     ApproximatesLinearOn g A t \u03b4'' \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** simp only [hA, forall_const, zero_mul, ENNReal.ofReal_zero, imp_true_iff,\n  zero_le, abs_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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 \u22a2 \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| ** simp only [ENNReal.ofReal, ENNReal.coe_sub] ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 \u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det A|) - \u2191\u03b5 < \u2191(Real.toNNReal |ContinuousLinearMap.det A|) ** apply ENNReal.sub_lt_self ENNReal.coe_ne_top ** case ha\u2080 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 \u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det A|) \u2260 0 ** simpa only [abs_nonpos_iff, Real.toNNReal_eq_zero, ENNReal.coe_eq_zero, Ne.def] using hA ** case hb 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 \u22a2 \u2191\u03b5 \u2260 0 ** simp only [\u03b5pos.ne', ENNReal.coe_eq_zero, Ne.def, not_false_iff] ** case intro.intro.inr.intro.intro.refine'_1 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 \u22a2 \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'') \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 ** intro B hB ** case intro.intro.inr.intro.intro.refine'_1 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 B : E \u2192L[\u211d] E hB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'') \u22a2 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 ** apply I'' _ (hB.trans _) ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 B : E \u2192L[\u211d] E hB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'') \u22a2 \u2191(min \u03b4 \u03b4'') \u2264 \u2191\u03b4'' ** simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff] ** case intro.intro.inr.intro.intro.refine'_2 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 \u22a2 \u2200 (t : Set E) (g : E \u2192 E),     ApproximatesLinearOn g A t (min \u03b4 \u03b4'') \u2192       ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** intro t g htg ** case intro.intro.inr.intro.intro.refine'_2 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 t : Set E g : E \u2192 E htg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'') \u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** rcases eq_or_ne (\u03bc t) \u221e with (ht | ht) ** case intro.intro.inr.intro.intro.refine'_2.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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 t : Set E g : E \u2192 E htg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'') ht : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** have := h t g (htg.mono_num (min_le_left _ _)) ** case intro.intro.inr.intro.intro.refine'_2.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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 t : Set E g : E \u2192 E htg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'') ht : \u2191\u2191\u03bc t \u2260 \u22a4 this : \u2191m * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) \u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** rwa [ENNReal.coe_sub, ENNReal.sub_mul, tsub_le_iff_right] at this ** case intro.intro.inr.intro.intro.refine'_2.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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 t : Set E g : E \u2192 E htg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'') ht : \u2191\u2191\u03bc t \u2260 \u22a4 this : (\u2191(Real.toNNReal |ContinuousLinearMap.det A|) - \u2191\u03b5) * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) \u22a2 0 < \u2191\u03b5 \u2192 \u2191\u03b5 < \u2191(Real.toNNReal |ContinuousLinearMap.det A|) \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ** simp only [ht, imp_true_iff, Ne.def, not_false_iff] ** case intro.intro.inr.intro.intro.refine'_2.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 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 A : E \u2192L[\u211d] E \u03b4' : \u211d \u03b4'pos : 0 < \u03b4' h\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5 \u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) } I'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5 hA : ContinuousLinearMap.det A \u2260 0 m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5 I : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) \u03b4pos : 0 < \u03b4 t : Set E g : E \u2192 E htg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'') ht : \u2191\u2191\u03bc t = \u22a4 \u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t ** simp only [ht, \u03b5pos.ne', ENNReal.mul_top, ENNReal.coe_eq_zero, le_top, Ne.def,\n  not_false_iff, _root_.add_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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) \u22a2 s = \u22c3 n, s \u2229 t n ** 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) \u22a2 s = s \u2229 \u22c3 i, t i ** exact Subset.antisymm (subset_inter Subset.rfl t_cover) (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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =     \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** conv_lhs => rw [s_eq] ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u222b\u207b (x : E) in \u22c3 n, s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =     \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t 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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i) ** exact fun n => hs.inter (t_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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 Pairwise (Disjoint on fun n => s \u2229 t n) ** exact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _ ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264     \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n n : \u2115 \u22a2 \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264     \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc ** apply lintegral_mono_ae ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n n : \u2115 \u22a2 \u2200\u1d50 (a : E) \u2202Measure.restrict \u03bc (s \u2229 t n),     ENNReal.ofReal |ContinuousLinearMap.det (f' a)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 ** filter_upwards [(ht n).norm_fderiv_sub_le \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n    (hf' x hx.1).mono (inter_subset_left _ _)] ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n n : \u2115 \u22a2 \u2200 (a : E),     \u2016f' a - A n\u2016\u208a \u2264 \u03b4 (A n) \u2192       ENNReal.ofReal |ContinuousLinearMap.det (f' a)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 ** intro x hx ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n n : \u2115 x : E hx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n) \u22a2 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 ** have I : |(f' x).det| \u2264 |(A n).det| + \u03b5 :=\n  calc\n    |(f' x).det| = |(A n).det + ((f' x).det - (A n).det)| := by congr 1; abel\n    _ \u2264 |(A n).det| + |(f' x).det - (A n).det| := (abs_add _ _)\n    _ \u2264 |(A n).det| + \u03b5 := add_le_add le_rfl ((h\u03b4 (A n)).2.1 _ hx) ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n n : \u2115 x : E hx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n) I : |ContinuousLinearMap.det (f' x)| \u2264 |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u22a2 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 ** calc\n  ENNReal.ofReal |(f' x).det| \u2264 ENNReal.ofReal (|(A n).det| + \u03b5) :=\n    ENNReal.ofReal_le_ofReal I\n  _ = ENNReal.ofReal |(A n).det| + \u03b5 := by\n    simp only [ENNReal.ofReal_add, abs_nonneg, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal] ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n n : \u2115 x : E hx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n) \u22a2 |ContinuousLinearMap.det (f' x)| =     |ContinuousLinearMap.det (A n) + (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))| ** congr 1 ** case e_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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n n : \u2115 x : E hx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n) \u22a2 ContinuousLinearMap.det (f' x) =     ContinuousLinearMap.det (A n) + (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n)) ** 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 : 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n n : \u2115 x : E hx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n) I : |ContinuousLinearMap.det (f' x)| \u2264 |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u22a2 ENNReal.ofReal (|ContinuousLinearMap.det (A n)| + \u2191\u03b5) = ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 ** simp only [ENNReal.ofReal_add, abs_nonneg, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal] ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc =     \u2211' (n : \u2115), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) ** simp only [set_lintegral_const, lintegral_add_right _ measurable_const] ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) \u2264     \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) ** refine' ENNReal.tsum_le_tsum fun n => add_le_add_right _ _ ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n n : \u2115 \u22a2 ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) \u2264 \u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) ** exact (h\u03b4 (A n)).2.2 _ _ (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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) = \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s ** conv_rhs => rw [s_eq] ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =     \u2191\u2191\u03bc (f '' \u22c3 n, s \u2229 t n) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, s \u2229 t n) ** rw [image_iUnion, measure_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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =     \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, s \u2229 t n)  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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 Pairwise (Disjoint on fun i => f '' (s \u2229 t i))  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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2200 (i : \u2115), MeasurableSet (f '' (s \u2229 t i)) ** rotate_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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =     \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, s \u2229 t n) ** rw [measure_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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =     \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i)  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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 Pairwise (Disjoint on fun n => s \u2229 t 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i) ** rotate_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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =     \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i) ** rw [\u2190 ENNReal.tsum_mul_left, \u2190 ENNReal.tsum_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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =     \u2211' (a : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t a)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t a)) ** congr 1 ** case e_f 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 (fun n => \u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) = fun a =>     \u2191\u2191\u03bc (f '' (s \u2229 t a)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t a) ** ext1 i ** 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 \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n i : \u2115 \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t i)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i) = \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i) ** rw [mul_assoc, two_mul, add_assoc] ** 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 Pairwise (Disjoint on fun i => f '' (s \u2229 t i)) ** intro i j hij ** 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n i j : \u2115 hij : i \u2260 j \u22a2 (Disjoint on fun i => f '' (s \u2229 t i)) i j ** apply Disjoint.image _ hf (inter_subset_left _ _) (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 hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n i j : \u2115 hij : i \u2260 j \u22a2 Disjoint (s \u2229 t i) (s \u2229 t j) ** exact Disjoint.mono (inter_subset_right _ _) (inter_subset_right _ _) (t_disj hij) ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2200 (i : \u2115), MeasurableSet (f '' (s \u2229 t i)) ** intro i ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n i : \u2115 \u22a2 MeasurableSet (f '' (s \u2229 t i)) ** exact\n  measurable_image_of_fderivWithin (hs.inter (t_meas i))\n    (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _))\n    (hf.mono (inter_subset_left _ _)) ** 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 Pairwise (Disjoint on fun n => s \u2229 t n) ** exact pairwise_disjoint_mono t_disj fun i => 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227         \u2200 (t : Set E) (g : E \u2192 E),           ApproximatesLinearOn g A t (\u03b4 A) \u2192             ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\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)) s_eq : s = \u22c3 n, s \u2229 t n \u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i) ** exact fun i => hs.inter (t_meas i) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one ** 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 this : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) \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 1) ** refine this.congr 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 this : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) r : \u211d \u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) ** congr 1 ** case e_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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 this : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) r : \u211d \u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) ** apply measure_toMeasurable_inter_of_sigmaFinite ** case e_a.hs 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 this : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) r : \u211d \u22a2 MeasurableSet ({x} + r \u2022 t) ** simp only [image_add_left, singleton_add] ** case e_a.hs 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 this : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) r : \u211d \u22a2 MeasurableSet ((fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 t)) ** apply (continuous_add_left (-x)).measurable (ht.const_smul\u2080 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) ** apply\n  tendsto_addHaar_inter_smul_one_of_density_one_aux \u03bc _ (measurableSet_toMeasurable _ _) _ _\n    t ht h't 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) ** apply tendsto_of_tendsto_of_tendsto_of_le_of_le' h tendsto_const_nhds ** case hgf 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0,     \u2191\u2191\u03bc (s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) \u2264 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) ** refine' eventually_of_forall fun r => mul_le_mul_right' _ _ ** case hgf 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 r : \u211d \u22a2 \u2191\u2191\u03bc (s \u2229 closedBall x r) \u2264 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) ** exact measure_mono (inter_subset_inter_left _ (subset_toMeasurable _ _)) ** case hfh 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) \u2264 1 ** 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a) \u2264 1 ** rintro 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 r : \u211d \u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) \u2264 1 ** apply ENNReal.div_le_of_le_mul ** case h.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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 r : \u211d \u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) \u2264 1 * \u2191\u2191\u03bc (closedBall x r) ** rw [one_mul] ** case h.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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 r : \u211d \u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) \u2264 \u2191\u2191\u03bc (closedBall x r) ** exact measure_mono (inter_subset_right _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.coeFn_negPart ** \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 g : E \u2192 F c : \u211d\u22650 f : { x // x \u2208 Lp \u211d p } a : \u03b1 h : \u2191\u2191(negPart f) a = max (-\u2191\u2191f a) 0 \u22a2 \u2191\u2191(negPart f) a = -min (\u2191\u2191f a) 0 ** rw [h, \u2190 max_neg_neg, neg_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.lintegral_swapLeft ** \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\u271d : \u03b3 \u2192 \u03b1 \u03ba : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b2 \u00d7 \u03b1 g : \u03b3 \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (c : \u03b3), g c \u2202\u2191(swapLeft \u03ba) a = \u222b\u207b (c : \u03b3), g c \u2202\u2191\u03ba (Prod.swap a) ** rw [swapLeft, lintegral_comap _ measurable_swap a] ** Qed",
        "informal": ""
    },
    {
        "formal": "StieltjesFunction.measure_Ici ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atTop (\ud835\udcdd l) x : \u211d \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ici x) = ofReal (l - leftLim (\u2191f) x) ** refine' tendsto_nhds_unique (tendsto_measure_Ico_atTop _ _) _ ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atTop (\ud835\udcdd l) x : \u211d \u22a2 Tendsto (fun x_1 => \u2191\u2191(StieltjesFunction.measure f) (Ico x x_1)) atTop (\ud835\udcdd (ofReal (l - leftLim (\u2191f) x))) ** simp_rw [measure_Ico] ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atTop (\ud835\udcdd l) x : \u211d \u22a2 Tendsto (fun x_1 => ofReal (leftLim (\u2191f) x_1 - leftLim (\u2191f) x)) atTop (\ud835\udcdd (ofReal (l - leftLim (\u2191f) x))) ** refine' ENNReal.tendsto_ofReal (Tendsto.sub_const _ _) ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atTop (\ud835\udcdd l) x : \u211d \u22a2 Tendsto (fun x => leftLim (\u2191f) x) atTop (\ud835\udcdd l) ** have h_le1 : \u2200 x, f (x - 1) \u2264 leftLim f x := fun x => Monotone.le_leftLim f.mono (sub_one_lt x) ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atTop (\ud835\udcdd l) x : \u211d h_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x \u22a2 Tendsto (fun x => leftLim (\u2191f) x) atTop (\ud835\udcdd l) ** have h_le2 : \u2200 x, leftLim f x \u2264 f x := fun x => Monotone.leftLim_le f.mono le_rfl ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atTop (\ud835\udcdd l) x : \u211d h_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x h_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x \u22a2 Tendsto (fun x => leftLim (\u2191f) x) atTop (\ud835\udcdd l) ** refine' tendsto_of_tendsto_of_tendsto_of_le_of_le (hf.comp _) hf h_le1 h_le2 ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atTop (\ud835\udcdd l) x : \u211d h_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x h_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x \u22a2 Tendsto (fun i => i - 1) atTop atTop ** rw [tendsto_atTop_atTop] ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atTop (\ud835\udcdd l) x : \u211d h_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x h_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x \u22a2 \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 b \u2264 a - 1 ** exact fun y => \u27e8y + 1, fun z hyz => by rwa [le_sub_iff_add_le]\u27e9 ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atTop (\ud835\udcdd l) x : \u211d h_le1 : \u2200 (x : \u211d), \u2191f (x - 1) \u2264 leftLim (\u2191f) x h_le2 : \u2200 (x : \u211d), leftLim (\u2191f) x \u2264 \u2191f x y z : \u211d hyz : y + 1 \u2264 z \u22a2 y \u2264 z - 1 ** rwa [le_sub_iff_add_le] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.lipschitzWith_pos_part ** \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 g : E \u2192 F c : \u211d\u22650 x y : \u211d \u22a2 dist (max x 0) (max y 0) \u2264 \u21911 * dist x y ** simp [Real.dist_eq, abs_max_sub_max_le_abs] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Nontrivial.ne_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : Finset \u03b1 a b : \u03b1 hs : Finset.Nontrivial s \u22a2 s \u2260 {a} ** rintro rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b : \u03b1 hs : Finset.Nontrivial {a} \u22a2 False ** exact not_nontrivial_singleton hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.haarContent_self ** 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 \u22a2 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K\u2080.toCompacts = 1 ** simp_rw [\u2190 ENNReal.coe_one, haarContent_apply, ENNReal.coe_eq_coe, chaar_self] ** 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 \u22a2 { val := 1, property := (_ : (fun r => 0 \u2264 r) 1) } = 1 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_eq_stieltjes_id ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 \u22a2 volume = StieltjesFunction.measure StieltjesFunction.id ** haveI : IsAddLeftInvariant StieltjesFunction.id.measure :=\n  \u27e8fun a =>\n    Eq.symm <|\n      Real.measure_ext_Ioo_rat fun p q => by\n        simp only [Measure.map_apply (measurable_const_add a) measurableSet_Ioo,\n          sub_sub_sub_cancel_right, StieltjesFunction.measure_Ioo, StieltjesFunction.id_leftLim,\n          StieltjesFunction.id_apply, id.def, preimage_const_add_Ioo]\u27e9 ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 this : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id) \u22a2 volume = StieltjesFunction.measure StieltjesFunction.id ** have A : StieltjesFunction.id.measure (stdOrthonormalBasis \u211d \u211d).toBasis.parallelepiped = 1 := by\n  change StieltjesFunction.id.measure (parallelepiped (stdOrthonormalBasis \u211d \u211d)) = 1\n  rcases parallelepiped_orthonormalBasis_one_dim (stdOrthonormalBasis \u211d \u211d) with (H | H) <;>\n    simp only [H, StieltjesFunction.measure_Icc, StieltjesFunction.id_apply, id.def, tsub_zero,\n      StieltjesFunction.id_leftLim, sub_neg_eq_add, zero_add, ENNReal.ofReal_one] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 this : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id) A :   \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)       \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =     1 \u22a2 volume = StieltjesFunction.measure StieltjesFunction.id ** conv_rhs =>\n  rw [addHaarMeasure_unique StieltjesFunction.id.measure\n      (stdOrthonormalBasis \u211d \u211d).toBasis.parallelepiped, A] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 this : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id) A :   \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)       \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =     1 \u22a2 volume = 1 \u2022 addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) ** simp only [volume, Basis.addHaar, one_smul] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d p q : \u211a \u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) (Ioo \u2191p \u2191q) =     \u2191\u2191(Measure.map (fun x => a + x) (StieltjesFunction.measure StieltjesFunction.id)) (Ioo \u2191p \u2191q) ** simp only [Measure.map_apply (measurable_const_add a) measurableSet_Ioo,\n  sub_sub_sub_cancel_right, StieltjesFunction.measure_Ioo, StieltjesFunction.id_leftLim,\n  StieltjesFunction.id_apply, id.def, preimage_const_add_Ioo] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 this : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id) \u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)       \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =     1 ** change StieltjesFunction.id.measure (parallelepiped (stdOrthonormalBasis \u211d \u211d)) = 1 ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 this : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id) \u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) (parallelepiped \u2191(stdOrthonormalBasis \u211d \u211d)) = 1 ** rcases parallelepiped_orthonormalBasis_one_dim (stdOrthonormalBasis \u211d \u211d) with (H | H) <;>\n  simp only [H, StieltjesFunction.measure_Icc, StieltjesFunction.id_apply, id.def, tsub_zero,\n    StieltjesFunction.id_leftLim, sub_neg_eq_add, zero_add, ENNReal.ofReal_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurePreserving_sumPiEquivProdPi_symm ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u2074 : Fintype \u03b9 m\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) m : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b3 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b2 : Fintype \u03b9' \u03c0 : \u03b9 \u2295 \u03b9' \u2192 Type u_4 inst\u271d\u00b9 : (i : \u03b9 \u2295 \u03b9') \u2192 MeasurableSpace (\u03c0 i) \u03bc : (i : \u03b9 \u2295 \u03b9') \u2192 Measure (\u03c0 i) inst\u271d : \u2200 (i : \u03b9 \u2295 \u03b9'), SigmaFinite (\u03bc i) \u22a2 Measure.map (\u2191(MeasurableEquiv.symm (MeasurableEquiv.sumPiEquivProdPi \u03c0)))       (Measure.prod (Measure.pi fun i => \u03bc (Sum.inl i)) (Measure.pi fun i => \u03bc (Sum.inr 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\u2074 : Fintype \u03b9 m\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) m : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b3 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b2 : Fintype \u03b9' \u03c0 : \u03b9 \u2295 \u03b9' \u2192 Type u_4 inst\u271d\u00b9 : (i : \u03b9 \u2295 \u03b9') \u2192 MeasurableSpace (\u03c0 i) \u03bc : (i : \u03b9 \u2295 \u03b9') \u2192 Measure (\u03c0 i) inst\u271d : \u2200 (i : \u03b9 \u2295 \u03b9'), SigmaFinite (\u03bc i) s : (i : \u03b9 \u2295 \u03b9') \u2192 Set (\u03c0 i) x\u271d : \u2200 (i : \u03b9 \u2295 \u03b9'), MeasurableSet (s i) \u22a2 \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm (MeasurableEquiv.sumPiEquivProdPi \u03c0)))             (Measure.prod (Measure.pi fun i => \u03bc (Sum.inl i)) (Measure.pi fun i => \u03bc (Sum.inr i))))       (Set.pi univ s) =     \u220f i : \u03b9 \u2295 \u03b9', \u2191\u2191(\u03bc i) (s i) ** simp_rw [MeasurableEquiv.map_apply, MeasurableEquiv.coe_sumPiEquivProdPi_symm,\n  Equiv.sumPiEquivProdPi_symm_preimage_univ_pi, Measure.prod_prod, Measure.pi_pi,\n  Fintype.prod_sum_type] ** 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 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 r \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * ENNReal.ofReal r ** refine' snorm_one_le_of_le hfint hfint' _ ** \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 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 r \u22a2 \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191(Real.toNNReal r) ** simp only [Real.coe_toNNReal', le_max_iff] ** \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 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 r \u22a2 \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 r \u2228 f \u03c9 \u2264 0 ** filter_upwards [hf] with \u03c9 h\u03c9 using Or.inl h\u03c9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.normalize_eq_iff ** d\u2081 d\u2082 : Nat n\u2081 n\u2082 : Int z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 \u22a2 normalize n\u2081 d\u2081 = normalize n\u2082 d\u2082 \u2194 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081 ** constructor <;> intro h ** case mp d\u2081 d\u2082 : Nat n\u2081 n\u2082 : Int z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 h : normalize n\u2081 d\u2081 = normalize n\u2082 d\u2082 \u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081 ** simp only [normalize_eq, mk'.injEq] at h ** case mp d\u2081 d\u2082 : Nat n\u2081 n\u2082 : Int z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 h :   n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227     d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082 \u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081 ** have' hn\u2081 := Int.ofNat_dvd_left.2 <| Nat.gcd_dvd_left n\u2081.natAbs d\u2081 ** case mp d\u2081 d\u2082 : Nat n\u2081 n\u2082 : Int z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 h :   n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227     d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082 hn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081 \u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081 ** have' hn\u2082 := Int.ofNat_dvd_left.2 <| Nat.gcd_dvd_left n\u2082.natAbs d\u2082 ** case mp d\u2081 d\u2082 : Nat n\u2081 n\u2082 : Int z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 h :   n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227     d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082 hn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081 hn\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 n\u2082 \u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081 ** have' hd\u2081 := Int.ofNat_dvd.2 <| Nat.gcd_dvd_right n\u2081.natAbs d\u2081 ** case mp d\u2081 d\u2082 : Nat n\u2081 n\u2082 : Int z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 h :   n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227     d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082 hn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081 hn\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 n\u2082 hd\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 \u2191d\u2081 \u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081 ** have' hd\u2082 := Int.ofNat_dvd.2 <| Nat.gcd_dvd_right n\u2082.natAbs d\u2082 ** case mp d\u2081 d\u2082 : Nat n\u2081 n\u2082 : Int z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 h :   n\u2081 / \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) = n\u2082 / \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2227     d\u2081 / Nat.gcd (Int.natAbs n\u2081) d\u2081 = d\u2082 / Nat.gcd (Int.natAbs n\u2082) d\u2082 hn\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 n\u2081 hn\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 n\u2082 hd\u2081 : \u2191(Nat.gcd (Int.natAbs n\u2081) d\u2081) \u2223 \u2191d\u2081 hd\u2082 : \u2191(Nat.gcd (Int.natAbs n\u2082) d\u2082) \u2223 \u2191d\u2082 \u22a2 n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081 ** rw [\u2190 Int.ediv_mul_cancel (Int.dvd_trans hd\u2082 (Int.dvd_mul_left ..)),\n  Int.mul_ediv_assoc _ hd\u2082, \u2190 Int.ofNat_ediv, \u2190 h.2, Int.ofNat_ediv,\n  \u2190 Int.mul_ediv_assoc _ hd\u2081, Int.mul_ediv_assoc' _ hn\u2081,\n  Int.mul_right_comm, h.1, Int.ediv_mul_cancel hn\u2082] ** case mpr d\u2081 d\u2082 : Nat n\u2081 n\u2082 : Int z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 h : n\u2081 * \u2191d\u2082 = n\u2082 * \u2191d\u2081 \u22a2 normalize n\u2081 d\u2081 = normalize n\u2082 d\u2082 ** rw [\u2190 normalize_mul_right _ z\u2082, \u2190 normalize_mul_left z\u2082 z\u2081, Int.mul_comm d\u2081, h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrableOn_iUnion_of_summable_norm_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : Countable \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc f : C(\u03b1, E) s : \u03b2 \u2192 Compacts \u03b1 hf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i)) \u22a2 IntegrableOn (\u2191f) (\u22c3 i, \u2191(s i)) ** refine'\n  integrableOn_iUnion_of_summable_integral_norm (fun i => (s i).isCompact.isClosed.measurableSet)\n    (fun i => (map_continuous f).continuousOn.integrableOn_compact (s i).isCompact)\n    (summable_of_nonneg_of_le (fun \u03b9 => integral_nonneg fun x => norm_nonneg _) (fun i => _) hf) ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : Countable \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc f : C(\u03b1, E) s : \u03b2 \u2192 Compacts \u03b1 hf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i)) i : \u03b2 \u22a2 \u222b (a : \u03b1) in \u2191(s i), \u2016\u2191f a\u2016 \u2202\u03bc \u2264 \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i)) ** rw [\u2190 (Real.norm_of_nonneg (integral_nonneg fun a => norm_nonneg _) : \u2016_\u2016 = \u222b x in s i, \u2016f x\u2016 \u2202\u03bc)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : Countable \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc f : C(\u03b1, E) s : \u03b2 \u2192 Compacts \u03b1 hf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i)) i : \u03b2 \u22a2 \u2016\u222b (x : \u03b1) in \u2191(s i), \u2016\u2191f x\u2016 \u2202\u03bc\u2016 \u2264 \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i)) ** exact\n  norm_set_integral_le_of_norm_le_const' (s i).isCompact.measure_lt_top\n    (s i).isCompact.isClosed.measurableSet fun x hx =>\n    (norm_norm (f x)).symm \u25b8 (f.restrict (s i : Set \u03b1)).norm_coe_le_norm \u27e8x, hx\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "PosNum.bit1_of_bit1 ** \u03b1 : Type u_1 n : PosNum \u22a2 _root_.bit0 n + 1 = bit1 n ** rw [add_one, bit0_of_bit0] ** \u03b1 : Type u_1 n : PosNum \u22a2 succ (bit0 n) = bit1 n ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.smul_map_volume_mul_left ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d h : a \u2260 0 \u22a2 ofReal |a| \u2022 Measure.map (fun x => a * x) volume = volume ** refine' (Real.measure_ext_Ioo_rat fun p q => _).symm ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d h : a \u2260 0 p q : \u211a \u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q) ** cases' lt_or_gt_of_ne h with h h ** case inl \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d h\u271d : a \u2260 0 p q : \u211a h : a < 0 \u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q) ** simp only [Real.volume_Ioo, Measure.smul_apply, \u2190 ENNReal.ofReal_mul (le_of_lt <| neg_pos.2 h),\n  Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, neg_sub_neg, neg_mul,\n  preimage_const_mul_Ioo_of_neg _ _ h, abs_of_neg h, mul_sub, smul_eq_mul,\n  mul_div_cancel' _ (ne_of_lt h)] ** case inr \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d h\u271d : a \u2260 0 p q : \u211a h : a > 0 \u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q) ** simp only [Real.volume_Ioo, Measure.smul_apply, \u2190 ENNReal.ofReal_mul (le_of_lt h),\n  Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, preimage_const_mul_Ioo _ _ h,\n  abs_of_pos h, mul_sub, mul_div_cancel' _ (ne_of_gt h), smul_eq_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.smul_finite ** \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 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc c : \u211d\u22650\u221e hc : c \u2260 \u22a4 \u22a2 IsFiniteMeasure (c \u2022 \u03bc) ** lift c to \u211d\u22650 using hc ** 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 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 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc c : \u211d\u22650 \u22a2 IsFiniteMeasure (\u2191c \u2022 \u03bc) ** exact MeasureTheory.isFiniteMeasureSMulNNReal ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.vars_rename ** 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\u00b2 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d\u00b9 : CommSemiring S inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R \u22a2 vars (\u2191(rename f) \u03c6) \u2286 Finset.image f (vars \u03c6) ** classical\nintro i hi\nsimp only [vars_def, exists_prop, Multiset.mem_toFinset, Finset.mem_image] at hi \u22a2\nsimpa only [Multiset.mem_map] using degrees_rename _ _ hi ** 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\u00b2 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d\u00b9 : CommSemiring S inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R \u22a2 vars (\u2191(rename f) \u03c6) \u2286 Finset.image f (vars \u03c6) ** intro i hi ** 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\u00b2 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d\u00b9 : CommSemiring S inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 hi : i \u2208 vars (\u2191(rename f) \u03c6) \u22a2 i \u2208 Finset.image f (vars \u03c6) ** simp only [vars_def, exists_prop, Multiset.mem_toFinset, Finset.mem_image] at hi \u22a2 ** 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\u00b2 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d\u00b9 : CommSemiring S inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 hi : i \u2208 degrees (\u2191(rename f) \u03c6) \u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i ** simpa only [Multiset.mem_map] using degrees_rename _ _ hi ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : 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\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc \u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) ** refine' \u27e8fun x => hf.mk f x.2, hf.stronglyMeasurable_mk.comp_measurable measurable_snd, _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : 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\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc \u22a2 (fun x => f x.2) =\u1d50[Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc] fun x => AEStronglyMeasurable.mk f hf x.2 ** suffices h : Measure.QuasiMeasurePreserving Prod.snd (\u03bc.map fun \u03c9 => (X \u03c9, \u03c9)) \u03bc ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : 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\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc \u22a2 Measure.QuasiMeasurePreserving Prod.snd ** refine' \u27e8measurable_snd, Measure.AbsolutelyContinuous.mk fun s hs h\u03bcs => _\u27e9 ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191(Measure.map Prod.snd (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc)) s = 0 ** rw [Measure.map_apply _ hs] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0  \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 Measurable Prod.snd ** swap ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0 ** by_cases hX : AEMeasurable X \u03bc ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : 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\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc h : Measure.QuasiMeasurePreserving Prod.snd \u22a2 (fun x => f x.2) =\u1d50[Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc] fun x => AEStronglyMeasurable.mk f hf x.2 ** exact Measure.QuasiMeasurePreserving.ae_eq h hf.ae_eq_mk ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 Measurable Prod.snd ** exact measurable_snd ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hX : AEMeasurable X \u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0 ** rw [Measure.map_apply_of_aemeasurable] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hX : AEMeasurable X \u22a2 \u2191\u2191\u03bc ((fun \u03c9 => (X \u03c9, \u03c9)) \u207b\u00b9' (Prod.snd \u207b\u00b9' s)) = 0 ** rw [\u2190 univ_prod, mk_preimage_prod, preimage_univ, univ_inter, preimage_id'] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hX : AEMeasurable X \u22a2 \u2191\u2191\u03bc s = 0 ** exact h\u03bcs ** case pos.hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hX : AEMeasurable X \u22a2 AEMeasurable fun \u03c9 => (X \u03c9, \u03c9) ** exact hX.prod_mk aemeasurable_id ** case pos.hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hX : AEMeasurable X \u22a2 MeasurableSet (Prod.snd \u207b\u00b9' s) ** exact measurable_snd hs ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hX : \u00acAEMeasurable X \u22a2 \u2191\u2191(Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) (Prod.snd \u207b\u00b9' s) = 0 ** rw [Measure.map_of_not_aemeasurable] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hX : \u00acAEMeasurable X \u22a2 \u2191\u21910 (Prod.snd \u207b\u00b9' s) = 0 ** simp ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 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\u271d : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9\u271d inst\u271d\u2076 : MeasurableSpace \u03a9\u271d inst\u271d\u2075 : PolishSpace \u03a9\u271d inst\u271d\u2074 : BorelSpace \u03a9\u271d inst\u271d\u00b3 : Nonempty \u03a9\u271d inst\u271d\u00b2 : NormedAddCommGroup F\u271d m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F\u271d \u03a9 : Type u_5 F : Type u_6 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 inst\u271d : TopologicalSpace F f : \u03a9 \u2192 F hf : AEStronglyMeasurable f \u03bc s : Set \u03a9 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hX : AEMeasurable fun \u03c9 => (X \u03c9, \u03c9) \u22a2 AEMeasurable X ** exact measurable_fst.comp_aemeasurable hX ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_nnnorm_condexpL2_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 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } \u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc ** let h_meas := lpMeas.aeStronglyMeasurable' (condexpL2 \u211d \u211d 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 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) \u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc ** let g := h_meas.choose ** \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 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas \u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc ** have hg_meas : StronglyMeasurable[m] g := h_meas.choose_spec.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 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g \u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc ** have hg_eq : g =\u1d50[\u03bc] condexpL2 \u211d \u211d hm f := h_meas.choose_spec.2.symm ** \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 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) \u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc ** have hg_eq_restrict : g =\u1d50[\u03bc.restrict s] condexpL2 \u211d \u211d hm f := ae_restrict_of_ae hg_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 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) \u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc ** have hg_nnnorm_eq : (fun x => (\u2016g x\u2016\u208a : \u211d\u22650\u221e)) =\u1d50[\u03bc.restrict s] fun x =>\n    (\u2016(condexpL2 \u211d \u211d hm f : \u03b1 \u2192 \u211d) x\u2016\u208a : \u211d\u22650\u221e) := by\n  refine' hg_eq_restrict.mono fun x hx => _\n  dsimp only\n  simp_rw [hx] ** \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 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc ** rw [lintegral_congr_ae hg_nnnorm_eq.symm] ** \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 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016g a\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc ** refine'\n  lintegral_nnnorm_le_of_forall_fin_meas_integral_eq hm (Lp.stronglyMeasurable f) _ _ _ _ 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 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) \u22a2 (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a ** refine' hg_eq_restrict.mono fun x hx => _ ** \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 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x : \u03b1 hx : g x = \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x \u22a2 (fun x => \u2191\u2016g x\u2016\u208a) x = (fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a) x ** dsimp only ** \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 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x : \u03b1 hx : g x = \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x \u22a2 \u2191\u2016Exists.choose (_ : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc) x\u2016\u208a = \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a ** simp_rw [hx] ** 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 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u22a2 IntegrableOn (fun x => \u2191\u2191f x) s ** exact integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim 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 t : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u22a2 StronglyMeasurable fun a => g a ** exact hg_meas ** 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 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u22a2 IntegrableOn (fun a => g a) s ** rw [IntegrableOn, integrable_congr hg_eq_restrict] ** 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 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u22a2 Integrable \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) ** exact integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs f ** 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 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u22a2 \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, \u2191\u2191f x \u2202\u03bc ** intro t ht h\u03bct ** 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\u271d : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191f x \u2202\u03bc ** rw [\u2190 integral_condexpL2_eq_of_fin_meas_real f ht h\u03bct.ne] ** 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\u271d : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d ?m.623979) f) x \u2202\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\u271d : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \u211d 2 } h_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f) g : \u03b1 \u2192 \u211d := Exists.choose h_meas hg_meas : StronglyMeasurable g hg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) hg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t < \u22a4 \u22a2 m \u2264 m0 ** exact set_integral_congr_ae (hm t ht) (hg_eq.mono fun x hx _ => hx) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.norm_constL_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\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : IsFiniteMeasure \u03bc c : E \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : Fact (1 \u2264 p) \u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc Set.univ) ^ (1 / ENNReal.toReal p) ** positivity ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.mul_pow ** a b n : Nat \u22a2 (a * b) ^ n = a ^ n * b ^ n ** induction n with\n| zero => rw [Nat.pow_zero, Nat.pow_zero, Nat.pow_zero, Nat.mul_one]\n| succ _ ih => rw [Nat.pow_succ, Nat.pow_succ, Nat.pow_succ, Nat.mul_mul_mul_comm, ih] ** case zero a b : Nat \u22a2 (a * b) ^ zero = a ^ zero * b ^ zero ** rw [Nat.pow_zero, Nat.pow_zero, Nat.pow_zero, Nat.mul_one] ** case succ a b n\u271d : Nat ih : (a * b) ^ n\u271d = a ^ n\u271d * b ^ n\u271d \u22a2 (a * b) ^ succ n\u271d = a ^ succ n\u271d * b ^ succ n\u271d ** rw [Nat.pow_succ, Nat.pow_succ, Nat.pow_succ, Nat.mul_mul_mul_comm, ih] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.simpleFunc.denseEmbedding ** \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 inst\u271d : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 \u22a2 DenseEmbedding Subtype.val ** borelize E ** \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 inst\u271d : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E \u22a2 DenseEmbedding Subtype.val ** apply simpleFunc.uniformEmbedding.denseEmbedding ** \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 inst\u271d : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E \u22a2 DenseRange Subtype.val ** intro f ** \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 inst\u271d : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f : { x // x \u2208 Lp E p } \u22a2 f \u2208 closure (Set.range Subtype.val) ** 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 inst\u271d : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f : { x // x \u2208 Lp E p } \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f) ** have hfi' : Mem\u2112p f p \u03bc := Lp.mem\u2112p f ** \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 inst\u271d : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f : { x // x \u2208 Lp E p } hfi' : Mem\u2112p (\u2191\u2191f) p \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f) ** haveI : SeparableSpace (range f \u222a {0} : Set E) :=\n  (Lp.stronglyMeasurable f).separableSpace_range_union_singleton ** \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 inst\u271d : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f : { x // x \u2208 Lp E p } hfi' : Mem\u2112p (\u2191\u2191f) p this : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0}) \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f) ** refine'\n  \u27e8fun n =>\n    toLp\n      (SimpleFunc.approxOn f (Lp.stronglyMeasurable f).measurable (range f \u222a {0}) 0 _ n)\n      (SimpleFunc.mem\u2112p_approxOn_range (Lp.stronglyMeasurable f).measurable hfi' n),\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 inst\u271d : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f : { x // x \u2208 Lp E p } hfi' : Mem\u2112p (\u2191\u2191f) p this : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0}) \u22a2 Tendsto     (fun n =>       \u2191(toLp (SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n)           (_ :             Mem\u2112p               (\u2191(SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n))               p)))     atTop (\ud835\udcdd f) ** convert SimpleFunc.tendsto_approxOn_range_Lp hp_ne_top (Lp.stronglyMeasurable f).measurable hfi' ** case h.e'_5.h.e'_3 \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 inst\u271d : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f : { x // x \u2208 Lp E p } hfi' : Mem\u2112p (\u2191\u2191f) p this : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0}) \u22a2 f = Mem\u2112p.toLp (\u2191\u2191f) hfi' ** rw [toLp_coeFn f (Lp.mem\u2112p f)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.scanl_get ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 v : Vector \u03b1 n i : Fin n \u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) ** cases' n with n ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 n : \u2115 v : Vector \u03b1 (Nat.succ n) i : Fin (Nat.succ n) \u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) ** induction' n with n hn generalizing b ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 v : Vector \u03b1 Nat.zero i : Fin Nat.zero \u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) ** exact i.elim0 ** case succ.zero \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b\u271d : \u03b2 n : \u2115 v\u271d : Vector \u03b1 (Nat.succ n) i\u271d : Fin (Nat.succ n) b : \u03b2 v : Vector \u03b1 (Nat.succ Nat.zero) i : Fin (Nat.succ Nat.zero) \u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) ** have i0 : i = 0 := Fin.eq_zero _ ** case succ.zero \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b\u271d : \u03b2 n : \u2115 v\u271d : Vector \u03b1 (Nat.succ n) i\u271d : Fin (Nat.succ n) b : \u03b2 v : Vector \u03b1 (Nat.succ Nat.zero) i : Fin (Nat.succ Nat.zero) i0 : i = 0 \u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) ** simp [scanl_singleton, i0, get_zero] ** case succ.zero \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b\u271d : \u03b2 n : \u2115 v\u271d : Vector \u03b1 (Nat.succ n) i\u271d : Fin (Nat.succ n) b : \u03b2 v : Vector \u03b1 (Nat.succ Nat.zero) i : Fin (Nat.succ Nat.zero) i0 : i = 0 \u22a2 get (b ::\u1d65 f b (head v) ::\u1d65 nil) 1 = f b (head v) ** simp [get_eq_get, List.get] ** case succ.succ \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b\u271d : \u03b2 n\u271d : \u2115 v\u271d : Vector \u03b1 (Nat.succ n\u271d) i\u271d : Fin (Nat.succ n\u271d) n : \u2115 hn :   \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),     get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) b : \u03b2 v : Vector \u03b1 (Nat.succ (Nat.succ n)) i : Fin (Nat.succ (Nat.succ n)) \u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) ** rw [\u2190 cons_head_tail v, scanl_cons, get_cons_succ] ** case succ.succ \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b\u271d : \u03b2 n\u271d : \u2115 v\u271d : Vector \u03b1 (Nat.succ n\u271d) i\u271d : Fin (Nat.succ n\u271d) n : \u2115 hn :   \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),     get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) b : \u03b2 v : Vector \u03b1 (Nat.succ (Nat.succ n)) i : Fin (Nat.succ (Nat.succ n)) \u22a2 get (scanl f (f b (head v)) (tail v)) i =     f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc i)) (get (head v ::\u1d65 tail v) i) ** refine' Fin.cases _ _ i ** case succ.succ.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b\u271d : \u03b2 n\u271d : \u2115 v\u271d : Vector \u03b1 (Nat.succ n\u271d) i\u271d : Fin (Nat.succ n\u271d) n : \u2115 hn :   \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),     get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) b : \u03b2 v : Vector \u03b1 (Nat.succ (Nat.succ n)) i : Fin (Nat.succ (Nat.succ n)) \u22a2 get (scanl f (f b (head v)) (tail v)) 0 =     f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc 0)) (get (head v ::\u1d65 tail v) 0) ** simp only [get_zero, scanl_head, Fin.castSucc_zero, head_cons] ** case succ.succ.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b\u271d : \u03b2 n\u271d : \u2115 v\u271d : Vector \u03b1 (Nat.succ n\u271d) i\u271d : Fin (Nat.succ n\u271d) n : \u2115 hn :   \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),     get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) b : \u03b2 v : Vector \u03b1 (Nat.succ (Nat.succ n)) i : Fin (Nat.succ (Nat.succ n)) \u22a2 \u2200 (i : Fin (n + 1)),     get (scanl f (f b (head v)) (tail v)) (Fin.succ i) =       f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc (Fin.succ i))) (get (head v ::\u1d65 tail v) (Fin.succ i)) ** intro i' ** case succ.succ.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b\u271d : \u03b2 n\u271d : \u2115 v\u271d : Vector \u03b1 (Nat.succ n\u271d) i\u271d : Fin (Nat.succ n\u271d) n : \u2115 hn :   \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),     get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i) b : \u03b2 v : Vector \u03b1 (Nat.succ (Nat.succ n)) i : Fin (Nat.succ (Nat.succ n)) i' : Fin (n + 1) \u22a2 get (scanl f (f b (head v)) (tail v)) (Fin.succ i') =     f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc (Fin.succ i'))) (get (head v ::\u1d65 tail v) (Fin.succ i')) ** simp only [hn, Fin.castSucc_fin_succ, get_cons_succ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.reverse_snoc ** \u03b1 : Type u_1 n : \u2115 xs : Vector \u03b1 n x : \u03b1 \u22a2 reverse (snoc xs x) = x ::\u1d65 reverse xs ** cases xs ** case mk \u03b1 : Type u_1 n : \u2115 x : \u03b1 val\u271d : List \u03b1 property\u271d : List.length val\u271d = n \u22a2 reverse (snoc { val := val\u271d, property := property\u271d } x) = x ::\u1d65 reverse { val := val\u271d, property := property\u271d } ** simp only [reverse, snoc, cons, toList_mk] ** case mk \u03b1 : Type u_1 n : \u2115 x : \u03b1 val\u271d : List \u03b1 property\u271d : List.length val\u271d = n \u22a2 {       val :=         List.reverse           (toList             (append { val := val\u271d, property := property\u271d }               { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })),       property :=         (_ :           (fun l => List.length l = n + 1)             (List.reverse               (toList                 (append { val := val\u271d, property := property\u271d }                   { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })))) } =     { val := x :: List.reverse val\u271d, property := (_ : Nat.succ (List.length (List.reverse val\u271d)) = Nat.succ n) } ** congr ** case mk.e_val \u03b1 : Type u_1 n : \u2115 x : \u03b1 val\u271d : List \u03b1 property\u271d : List.length val\u271d = n \u22a2 [] ++ [x] ++ List.reverse val\u271d = x :: List.reverse val\u271d ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.not_nontrivial_singleton ** \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 : \u03b1 H : Set.Nontrivial {x} \u22a2 False ** rw [nontrivial_iff_exists_ne (mem_singleton x)] at H ** \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 : \u03b1 H : \u2203 y, y \u2208 {x} \u2227 y \u2260 x \u22a2 False ** let \u27e8y, hy, hya\u27e9 := H ** \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 : \u03b1 H : \u2203 y, y \u2208 {x} \u2227 y \u2260 x y : \u03b1 hy : y \u2208 {x} hya : y \u2260 x \u22a2 False ** exact hya (mem_singleton_iff.1 hy) ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.cast_zmod_eq_zero_iff_of_lt ** m n : \u2115 inst\u271d : NeZero m h : m < n a : ZMod m \u22a2 \u2191a = 0 \u2194 a = 0 ** rw [\u2190 ZMod.cast_zero (n := m)] ** m n : \u2115 inst\u271d : NeZero m h : m < n a : ZMod m \u22a2 \u2191a = \u21910 \u2194 a = 0 ** exact Injective.eq_iff' (cast_injective_of_lt h) rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.le_min ** a b c : Int x\u271d : a \u2264 b \u2227 a \u2264 c h\u2081 : a \u2264 b h\u2082 : a \u2264 c \u22a2 a \u2264 min b c ** rw [Int.min_def] ** a b c : Int x\u271d : a \u2264 b \u2227 a \u2264 c h\u2081 : a \u2264 b h\u2082 : a \u2264 c \u22a2 a \u2264 if b \u2264 c then b else c ** split <;> assumption ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.measurable_floor ** \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 : OpensMeasurableSpace R x : R \u22a2 MeasurableSet (floor \u207b\u00b9' {\u230ax\u230b}) ** simpa only [Int.preimage_floor_singleton] using measurableSet_Ico ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.cast_bitm1 ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 n : ZNum \u22a2 \u2191(ZNum.bitm1 n) = _root_.bit0 \u2191n - 1 ** conv =>\n  lhs\n  rw [\u2190 zneg_zneg n] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 n : ZNum \u22a2 \u2191(ZNum.bitm1 (- -n)) = _root_.bit0 \u2191n - 1 ** rw [\u2190 zneg_bit1, cast_zneg, cast_bit1] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 n : ZNum \u22a2 -_root_.bit1 \u2191(-n) = _root_.bit0 \u2191n - 1 ** have : ((-1 + n + n : \u2124) : \u03b1) = (n + n + -1 : \u2124) := by simp [add_comm, add_left_comm] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 n : ZNum this : \u2191(-1 + \u2191n + \u2191n) = \u2191(\u2191n + \u2191n + -1) \u22a2 -_root_.bit1 \u2191(-n) = _root_.bit0 \u2191n - 1 ** simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 n : ZNum \u22a2 \u2191(-1 + \u2191n + \u2191n) = \u2191(\u2191n + \u2191n + -1) ** simp [add_comm, add_left_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.liftRel_iff_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 r : \u03b2 \u2192 \u03b3 \u2192 Prop f : \u03b1 \u2192\u2098[\u03bc] \u03b2 g : \u03b1 \u2192\u2098[\u03bc] \u03b3 \u22a2 LiftRel r f g \u2194 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, r (\u2191f a) (\u2191g a) ** rw [\u2190 liftRel_mk_mk, mk_coeFn, mk_coeFn] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.mem_condCdfSet_ae ** \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, a \u2208 condCdfSet \u03c1 ** simp_rw [ae_iff, condCdfSet, not_mem_compl_iff, setOf_mem_eq, measure_toMeasurable] ** \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 \u2191\u2191(Measure.fst \u03c1) {b | \u00acHasCondCdf \u03c1 b} = 0 ** exact hasCondCdf_ae \u03c1 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Finite.fin_embedding ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 h : Set.Finite s \u22a2 range \u2191(Equiv.asEmbedding (Fintype.equivFin \u2191\u2191(Finite.toFinset h)).symm) = s ** simp only [Finset.coe_sort_coe, Equiv.asEmbedding_range, Finite.coe_toFinset, setOf_mem_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "borel_eq_generateFrom_Ico ** \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\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 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : OrderTopology \u03b1 \u22a2 borel \u03b1 = MeasurableSpace.generateFrom {S | \u2203 l u, l < u \u2227 Ico l u = S} ** simpa only [exists_prop, mem_univ, true_and_iff] using\n  (@dense_univ \u03b1 _).borel_eq_generateFrom_Ico_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>\n    mem_univ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.integral_norm_condexpKernel ** \u03a9 : Type u_1 F : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03a9 m m\u03a9 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : NormedAddCommGroup F f : \u03a9 \u2192 F hf_int : Integrable f \u22a2 Integrable fun \u03c9 => \u222b (y : \u03a9), \u2016f y\u2016 \u2202\u2191(condexpKernel \u03bc m) \u03c9 ** rw [condexpKernel] ** \u03a9 : Type u_1 F : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03a9 m m\u03a9 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : NormedAddCommGroup F f : \u03a9 \u2192 F hf_int : Integrable f \u22a2 Integrable fun \u03c9 => \u222b (y : \u03a9), \u2016f y\u2016 \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9 ** exact Integrable.integral_norm_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": "MeasurableSpace.cardinal_generateMeasurable_le ** \u03b1 : Type u s : Set (Set \u03b1) \u22a2 #\u2191{t | GenerateMeasurable s t} \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** rw [generateMeasurable_eq_rec] ** \u03b1 : Type u s : Set (Set \u03b1) \u22a2 #\u2191(\u22c3 i, generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** apply (mk_iUnion_le _).trans ** \u03b1 : Type u s : Set (Set \u03b1) \u22a2 #(Quotient.out (ord (aleph 1))).\u03b1 * \u2a06 i, #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** rw [(aleph 1).mk_ord_out] ** \u03b1 : Type u s : Set (Set \u03b1) \u22a2 aleph 1 * \u2a06 i, #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** refine le_trans (mul_le_mul' aleph_one_le_continuum\n    (ciSup_le' fun i => cardinal_generateMeasurableRec_le s i)) ?_ ** \u03b1 : Type u s : Set (Set \u03b1) \u22a2 \ud835\udd20 * max (#\u2191s) 2 ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** refine (mul_le_max_of_aleph0_le_left aleph0_le_continuum).trans (max_le ?_ le_rfl) ** \u03b1 : Type u s : Set (Set \u03b1) \u22a2 \ud835\udd20 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** exact power_le_power_right (le_max_right _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.Tape.move_right_nth ** \u0393 : Type u_1 inst\u271d : Inhabited \u0393 T : Tape \u0393 i : \u2124 \u22a2 nth (move Dir.right T) i = nth T (i + 1) ** conv => rhs; rw [\u2190 T.move_right_left] ** \u0393 : Type u_1 inst\u271d : Inhabited \u0393 T : Tape \u0393 i : \u2124 \u22a2 nth (move Dir.right T) i = nth (move Dir.left (move Dir.right T)) (i + 1) ** rw [Tape.move_left_nth, add_sub_cancel] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_injective ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 \u22a2 Injective (preimage f) \u2194 Surjective f ** refine' \u27e8fun h y => _, Surjective.preimage_injective\u27e9 ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Injective (preimage f) y : \u03b2 \u22a2 \u2203 a, f a = y ** obtain \u27e8x, hx\u27e9 : (f \u207b\u00b9' {y}).Nonempty := by\n  rw [h.nonempty_apply_iff preimage_empty]\n  apply singleton_nonempty ** case intro \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Injective (preimage f) y : \u03b2 x : \u03b1 hx : x \u2208 f \u207b\u00b9' {y} \u22a2 \u2203 a, f a = y ** exact \u27e8x, hx\u27e9 ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Injective (preimage f) y : \u03b2 \u22a2 Set.Nonempty (f \u207b\u00b9' {y}) ** rw [h.nonempty_apply_iff preimage_empty] ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Injective (preimage f) y : \u03b2 \u22a2 Set.Nonempty {y} ** apply singleton_nonempty ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_lintegral_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\u22650\u221e inst\u271d : IsProbabilityMeasure \u03bc hint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4 \u22a2 \u2203 x, \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x ** simpa only [laverage_eq_lintegral] using\n  exists_laverage_le (IsProbabilityMeasure.ne_zero \u03bc) hint ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableEquiv.measurableSet_image ** \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 \u03b1 \u22a2 MeasurableSet (\u2191e '' s) \u2194 MeasurableSet s ** rw [image_eq_preimage, measurableSet_preimage] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.degreeOf_sub_lt ** R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q : MvPolynomial \u03c3 R x : \u03c3 f g : MvPolynomial \u03c3 R k : \u2115 h : 0 < k hf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g hg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g \u22a2 degreeOf x (f - g) < k ** rw [degreeOf_lt_iff h] ** R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q : MvPolynomial \u03c3 R x : \u03c3 f g : MvPolynomial \u03c3 R k : \u2115 h : 0 < k hf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g hg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g \u22a2 \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support (f - g) \u2192 \u2191m x < k ** intro m hm ** R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q : MvPolynomial \u03c3 R x : \u03c3 f g : MvPolynomial \u03c3 R k : \u2115 h : 0 < k hf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g hg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g m : \u03c3 \u2192\u2080 \u2115 hm : m \u2208 support (f - g) \u22a2 \u2191m x < k ** by_contra' hc ** R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q : MvPolynomial \u03c3 R x : \u03c3 f g : MvPolynomial \u03c3 R k : \u2115 h : 0 < k hf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g hg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g m : \u03c3 \u2192\u2080 \u2115 hm : m \u2208 support (f - g) hc : k \u2264 \u2191m x \u22a2 False ** have h := support_sub \u03c3 f g hm ** R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q : MvPolynomial \u03c3 R x : \u03c3 f g : MvPolynomial \u03c3 R k : \u2115 h\u271d : 0 < k hf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g hg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g m : \u03c3 \u2192\u2080 \u2115 hm : m \u2208 support (f - g) hc : k \u2264 \u2191m x h : m \u2208 support f \u222a support g \u22a2 False ** simp only [mem_support_iff, Ne.def, coeff_sub, sub_eq_zero] at hm ** R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q : MvPolynomial \u03c3 R x : \u03c3 f g : MvPolynomial \u03c3 R k : \u2115 h\u271d : 0 < k hf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g hg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g m : \u03c3 \u2192\u2080 \u2115 hc : k \u2264 \u2191m x h : m \u2208 support f \u222a support g hm : \u00accoeff m f = coeff m g \u22a2 False ** cases' Finset.mem_union.1 h with cf cg ** case inl R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q : MvPolynomial \u03c3 R x : \u03c3 f g : MvPolynomial \u03c3 R k : \u2115 h\u271d : 0 < k hf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g hg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g m : \u03c3 \u2192\u2080 \u2115 hc : k \u2264 \u2191m x h : m \u2208 support f \u222a support g hm : \u00accoeff m f = coeff m g cf : m \u2208 support f \u22a2 False ** exact hm (hf m cf hc) ** case inr R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q : MvPolynomial \u03c3 R x : \u03c3 f g : MvPolynomial \u03c3 R k : \u2115 h\u271d : 0 < k hf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g hg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g m : \u03c3 \u2192\u2080 \u2115 hc : k \u2264 \u2191m x h : m \u2208 support f \u222a support g hm : \u00accoeff m f = coeff m g cg : m \u2208 support g \u22a2 False ** exact hm (hg m cg hc) ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Path.insert_toList ** \u03b1 : Type u_1 t : RBNode \u03b1 v : \u03b1 p : Path \u03b1 \u22a2 toList (insert p t v) = withList p (toList (setRoot v t)) ** simp [insert] ** \u03b1 : Type u_1 t : RBNode \u03b1 v : \u03b1 p : Path \u03b1 \u22a2 toList       (match t with       | nil => insertNew p v       | node c a v_1 b => fill p (node c a v b)) =     listL p ++ (toList (setRoot v t) ++ listR p) ** split <;> simp [setRoot] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Infinite.exists_subset_ncard_eq ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 hs : Set.Infinite s k : \u2115 \u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 Set.ncard t = k ** have := hs.to_subtype ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 hs : Set.Infinite s k : \u2115 this : Infinite \u2191s \u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 Set.ncard t = k ** obtain \u27e8t', -, rfl\u27e9 := @Infinite.exists_subset_card_eq s univ infinite_univ k ** case intro.intro \u03b1 : Type u_1 s\u271d t s : Set \u03b1 hs : Set.Infinite s this : Infinite \u2191s t' : Finset \u2191s \u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 Set.ncard t = Finset.card t' ** refine' \u27e8Subtype.val '' (t' : Set s), by simp, Finite.image _ (by simp), _\u27e9 ** case intro.intro \u03b1 : Type u_1 s\u271d t s : Set \u03b1 hs : Set.Infinite s this : Infinite \u2191s t' : Finset \u2191s \u22a2 Set.ncard (Subtype.val '' \u2191t') = Finset.card t' ** rw [ncard_image_of_injective _ Subtype.coe_injective] ** case intro.intro \u03b1 : Type u_1 s\u271d t s : Set \u03b1 hs : Set.Infinite s this : Infinite \u2191s t' : Finset \u2191s \u22a2 Set.ncard \u2191t' = Finset.card t' ** simp ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 hs : Set.Infinite s this : Infinite \u2191s t' : Finset \u2191s \u22a2 Subtype.val '' \u2191t' \u2286 s ** simp ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 hs : Set.Infinite s this : Infinite \u2191s t' : Finset \u2191s \u22a2 Set.Finite \u2191t' ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "blimsup_cthickening_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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p \u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** let r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / ((i : \u211d) + 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 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) \u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** have h\u2080 : \u2200 i, p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) := by\n  rintro i \u27e8-, hi\u27e9; congr! 1; change r i = ite (0 < r i) (r i) _; simp [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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) \u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** have h\u2081 : \u2200 i, p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) := by\n  rintro i \u27e8-, hi\u27e9; simp only [hi, mul_ite, if_true] ** \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) \u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** have h\u2082 : \u2200 i, p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) := by\n  rintro i \u27e8-, hi\u27e9\n  have hi' : M * r i \u2264 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi\n  rw [cthickening_of_nonpos hi, cthickening_of_nonpos 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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) h\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) \u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** have hp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0 := by\n  ext i; simp [\u2190 and_or_left, lt_or_le 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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) h\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) hp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0 \u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** rw [hp, blimsup_or_eq_sup, blimsup_or_eq_sup] ** \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) h\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) hp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0 \u22a2 ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u2294       blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0) =\u1d50[\u03bc]     (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u2294       blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0 ** simp only [sup_eq_union] ** \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) h\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) hp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0 \u22a2 ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u222a       blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0) =\u1d50[\u03bc]     (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 0 < r i) \u222a       blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0 ** rw [blimsup_congr (eventually_of_forall h\u2080), blimsup_congr (eventually_of_forall h\u2081),\n  blimsup_congr (eventually_of_forall 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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) h\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) hp : p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0 \u22a2 ((blimsup (fun x => cthickening (M * r' x) (s x)) atTop fun x => p x \u2227 0 < r x) \u222a       blimsup (fun x => cthickening (r x) (s x)) atTop fun x => p x \u2227 r x \u2264 0) =\u1d50[\u03bc]     (blimsup (fun x => cthickening (r' x) (s x)) atTop fun x => p x \u2227 0 < r x) \u222a       blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i \u2227 r i \u2264 0 ** exact ae_eq_set_union (this (fun i => p i \u2227 0 < r i) hr') (ae_eq_refl _) ** \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 \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** clear p hr r ** \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 s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M \u22a2 \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** intro p r hr ** \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 s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M p : \u2115 \u2192 Prop r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** have hr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[>] 0) := by\n  convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [mul_zero] ** \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 s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M p : \u2115 \u2192 Prop r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) hr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0) \u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** refine' eventuallyLE_antisymm_iff.mpr \u27e8_, _\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 s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M p : \u2115 \u2192 Prop r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u22a2 Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0) ** convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [mul_zero] ** 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 s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M p : \u2115 \u2192 Prop r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) hr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0) \u22a2 blimsup (fun i => cthickening (M * r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p ** exact blimsup_cthickening_ae_le_of_eventually_mul_le \u03bc p (inv_pos.mpr hM) hr'\n  (eventually_of_forall fun i => by rw [inv_mul_cancel_left\u2080 hM.ne' (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 s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M p : \u2115 \u2192 Prop r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) hr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0) i : \u2115 \u22a2 M\u207b\u00b9 * (M * r i) \u2264 r i ** rw [inv_mul_cancel_left\u2080 hM.ne' (r i)] ** 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 s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M p : \u2115 \u2192 Prop r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) hr' : Tendsto (fun i => M * r i) atTop (\ud835\udcdd[Ioi 0] 0) \u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (M * r i) (s i)) atTop p ** exact blimsup_cthickening_ae_le_of_eventually_mul_le \u03bc p hM hr\n  (eventually_of_forall fun i => le_refl _) ** \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) \u22a2 Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) ** refine' tendsto_nhdsWithin_iff.mpr\n  \u27e8Tendsto.if' hr tendsto_one_div_add_atTop_nhds_0_nat, eventually_of_forall fun i => _\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 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) i : \u2115 \u22a2 r' i \u2208 Ioi 0 ** by_cases hi : 0 < r i ** case pos \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) i : \u2115 hi : 0 < r i \u22a2 r' i \u2208 Ioi 0 ** simp [hi] ** case neg \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) i : \u2115 hi : \u00ac0 < r i \u22a2 r' i \u2208 Ioi 0 ** simp only [hi, one_div, mem_Ioi, if_false, inv_pos] ** case neg \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) i : \u2115 hi : \u00ac0 < r i \u22a2 0 < \u2191i + 1 ** 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 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) \u22a2 \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) ** rintro i \u27e8-, hi\u27e9 ** case 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 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) i : \u2115 hi : 0 < r i \u22a2 cthickening (r i) (s i) = cthickening (r' i) (s i) ** congr! 1 ** case intro.h.e'_3 \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) i : \u2115 hi : 0 < r i \u22a2 r i = r' i ** change r i = ite (0 < r i) (r i) _ ** case intro.h.e'_3 \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) i : \u2115 hi : 0 < r i \u22a2 r i = if 0 < r i then r i else 1 / (\u2191i + 1) ** simp [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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) \u22a2 \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) ** rintro i \u27e8-, hi\u27e9 ** case 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 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) i : \u2115 hi : 0 < r i \u22a2 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) ** simp only [hi, mul_ite, if_true] ** \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) \u22a2 \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) ** rintro i \u27e8-, hi\u27e9 ** case 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 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) i : \u2115 hi : r i \u2264 0 \u22a2 cthickening (M * r i) (s i) = cthickening (r i) (s i) ** have hi' : M * r i \u2264 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi ** case 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 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) i : \u2115 hi : r i \u2264 0 hi' : M * r i \u2264 0 \u22a2 cthickening (M * r i) (s i) = cthickening (r i) (s i) ** rw [cthickening_of_nonpos hi, cthickening_of_nonpos 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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) h\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) \u22a2 p = fun i => p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0 ** ext i ** case h.a \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) this :   \u2200 (p : \u2115 \u2192 Prop) {r : \u2115 \u2192 \u211d},     Tendsto r atTop (\ud835\udcdd[Ioi 0] 0) \u2192       blimsup (fun i => cthickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => cthickening (r i) (s i)) atTop p r' : \u2115 \u2192 \u211d := fun i => if 0 < r i then r i else 1 / (\u2191i + 1) hr' : Tendsto r' atTop (\ud835\udcdd[Ioi 0] 0) h\u2080 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (r i) (s i) = cthickening (r' i) (s i) h\u2081 : \u2200 (i : \u2115), p i \u2227 0 < r i \u2192 cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) h\u2082 : \u2200 (i : \u2115), p i \u2227 r i \u2264 0 \u2192 cthickening (M * r i) (s i) = cthickening (r i) (s i) i : \u2115 \u22a2 p i \u2194 p i \u2227 0 < r i \u2228 p i \u2227 r i \u2264 0 ** simp [\u2190 and_or_left, lt_or_le 0 (r i)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.insert_union ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b : \u03b2 a s\u2081 s\u2082 : Finmap \u03b2 a\u2081 a\u2082 : AList \u03b2 \u22a2 insert a b (\u27e6a\u2081\u27e7 \u222a \u27e6a\u2082\u27e7) = insert a b \u27e6a\u2081\u27e7 \u222a \u27e6a\u2082\u27e7 ** simp [AList.insert_union] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.fixed_point\u2082 ** f : Code \u2192 \u2115 \u2192. \u2115 hf : Partrec\u2082 f cf : Code ef : eval cf = fun n => Part.bind \u2191(decode n) fun a => Part.map encode ((fun p => f p.1 p.2) a) c : Code e : eval (curry cf (encode c)) = eval c n : \u2115 \u22a2 eval c n = f c n ** simp [e.symm, ef, Part.map_id'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Holor.cprankMax_1 ** \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 ds h : CPRankMax1 x \u22a2 CPRankMax 1 x ** have h' := CPRankMax.succ 0 x 0 h 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 ds h : CPRankMax1 x h' : CPRankMax (0 + 1) (x + 0) \u22a2 CPRankMax 1 x ** rwa [zero_add, add_zero] at h' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.norm_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 hp_pos : p \u2260 0 h\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0 \u22a2 \u2016indicatorConstLp p hs h\u03bcs c\u2016 = \u2016c\u2016 * ENNReal.toReal (\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 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E hp_pos : p \u2260 0 h\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0 hp_top : p = \u22a4 \u22a2 \u2016indicatorConstLp p hs h\u03bcs c\u2016 = \u2016c\u2016 * ENNReal.toReal (\u2191\u2191\u03bc s) ^ (1 / ENNReal.toReal p) ** rw [hp_top, ENNReal.top_toReal, _root_.div_zero, Real.rpow_zero, mul_one] ** 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 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E hp_pos : p \u2260 0 h\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0 hp_top : p = \u22a4 \u22a2 \u2016indicatorConstLp \u22a4 hs h\u03bcs c\u2016 = \u2016c\u2016 ** exact norm_indicatorConstLp_top h\u03bcs_pos ** 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 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E hp_pos : p \u2260 0 h\u03bcs_pos : \u2191\u2191\u03bc s \u2260 0 hp_top : \u00acp = \u22a4 \u22a2 \u2016indicatorConstLp p hs h\u03bcs c\u2016 = \u2016c\u2016 * ENNReal.toReal (\u2191\u2191\u03bc s) ^ (1 / ENNReal.toReal p) ** exact norm_indicatorConstLp hp_pos hp_top ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.map_bind\u2082 ** \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 g : S \u2192+* T \u03c6 : MvPolynomial \u03c3 R \u22a2 \u2191(map g) (\u2191(bind\u2082 f) \u03c6) = \u2191(bind\u2082 (RingHom.comp (map g) f)) \u03c6 ** simp only [bind\u2082, eval\u2082_comp_right, coe_eval\u2082Hom, eval\u2082_map] ** \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 g : S \u2192+* T \u03c6 : MvPolynomial \u03c3 R \u22a2 eval\u2082 (RingHom.comp (map g) f) (\u2191(map g) \u2218 X) \u03c6 = eval\u2082 (RingHom.comp (map g) f) X \u03c6 ** congr 1 with : 1 ** case e_g.h \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 g : S \u2192+* T \u03c6 : MvPolynomial \u03c3 R x\u271d : \u03c3 \u22a2 (\u2191(map g) \u2218 X) x\u271d = X x\u271d ** simp only [Function.comp_apply, map_X] ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.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 \u22a2 core (res f s) t = s\u1d9c \u222a f \u207b\u00b9' t ** ext x ** case h \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\u1d9c \u222a f \u207b\u00b9' t ** rw [mem_core_res] ** case h \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 s \u2192 f x \u2208 t \u2194 x \u2208 s\u1d9c \u222a f \u207b\u00b9' t ** by_cases h : x \u2208 s <;> simp [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.sym_nonempty ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n : \u2115 m : Sym \u03b1 n \u22a2 Finset.Nonempty (Finset.sym s n) \u2194 n = 0 \u2228 Finset.Nonempty s ** simp_rw [nonempty_iff_ne_empty, Ne.def] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n : \u2115 m : Sym \u03b1 n \u22a2 \u00acFinset.sym s n = \u2205 \u2194 n = 0 \u2228 \u00acs = \u2205 ** rwa [sym_eq_empty, not_and_or, not_ne_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.append_data ** \u03b1 : Type u_1 arr arr' : Array \u03b1 \u22a2 (arr ++ arr').data = arr.data ++ arr'.data ** rw [\u2190 append_eq_append] ** \u03b1 : Type u_1 arr arr' : Array \u03b1 \u22a2 (Array.append arr arr').data = arr.data ++ arr'.data ** unfold Array.append ** \u03b1 : Type u_1 arr arr' : Array \u03b1 \u22a2 (foldl (fun r v => push r v) arr arr' 0 (size arr')).data = arr.data ++ arr'.data ** rw [foldl_eq_foldl_data] ** \u03b1 : Type u_1 arr arr' : Array \u03b1 \u22a2 (List.foldl (fun r v => push r v) arr arr'.data).data = arr.data ++ arr'.data ** induction arr'.data generalizing arr <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pairwise_disjoint_Ico_add_int_cast ** \u03b1 : Type u_1 inst\u271d : OrderedRing \u03b1 a : \u03b1 \u22a2 Pairwise (Disjoint on fun n => Ico (a + \u2191n) (a + \u2191n + 1)) ** simpa only [zsmul_one, Int.cast_add, Int.cast_one, \u2190 add_assoc] using\n  pairwise_disjoint_Ico_add_zsmul a (1 : \u03b1) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_strict_mono ** \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 h\u03bc : \u03bc \u2260 0 hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < g x \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc < \u222b\u207b (x : \u03b1), g x \u2202\u03bc ** rw [Ne.def, \u2190 Measure.measure_univ_eq_zero] at h\u03bc ** \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 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < g x \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc < \u222b\u207b (x : \u03b1), g x \u2202\u03bc ** refine' lintegral_strict_mono_of_ae_le_of_ae_lt_on hg hfi (ae_le_of_ae_lt h) h\u03bc _ ** \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 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < g x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 univ \u2192 f x < g x ** simpa using h ** Qed",
        "informal": ""
    },
    {
        "formal": "meas_essSup_lt ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2 x : \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : FirstCountableTopology \u03b2 inst\u271d : OrderTopology \u03b2 hf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f) _auto\u271d \u22a2 \u2191\u2191\u03bc {y | essSup f \u03bc < f y} = 0 ** simp_rw [\u2190 not_le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2 x : \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : FirstCountableTopology \u03b2 inst\u271d : OrderTopology \u03b2 hf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f) _auto\u271d \u22a2 \u2191\u2191\u03bc {y | \u00acf y \u2264 essSup f \u03bc} = 0 ** exact ae_le_essSup hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.normalize.reduced' ** num : Int den g : Nat den_nz : den \u2260 0 e : g = Nat.gcd (Int.natAbs num) den \u22a2 Nat.Coprime (Int.natAbs (num / \u2191g)) (den / g) ** rw [\u2190 Int.div_eq_ediv_of_dvd (e \u25b8 Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))] ** num : Int den g : Nat den_nz : den \u2260 0 e : g = Nat.gcd (Int.natAbs num) den \u22a2 Nat.Coprime (Int.natAbs (Int.div num \u2191g)) (den / g) ** exact normalize.reduced den_nz e ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_tsum ** \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) \u22a2 \u222b\u207b (a : \u03b1), \u2211' (i : \u03b2), f i a \u2202\u03bc = \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1), f i a \u2202\u03bc ** simp only [ENNReal.tsum_eq_iSup_sum] ** \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) \u22a2 \u222b\u207b (a : \u03b1), \u2a06 s, \u2211 i in s, f i a \u2202\u03bc = \u2a06 s, \u2211 i in s, \u222b\u207b (a : \u03b1), f i a \u2202\u03bc ** rw [lintegral_iSup_directed] ** \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) \u22a2 \u2a06 b, \u222b\u207b (a : \u03b1), \u2211 i in b, f i a \u2202\u03bc = \u2a06 s, \u2211 i in s, \u222b\u207b (a : \u03b1), f i a \u2202\u03bc ** simp [lintegral_finset_sum' _ fun i _ => hf i] ** 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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) \u22a2 \u2200 (b : Finset \u03b2), AEMeasurable fun a => \u2211 i in b, f i a ** intro b ** 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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) b : Finset \u03b2 \u22a2 AEMeasurable fun a => \u2211 i in b, f i a ** exact Finset.aemeasurable_sum _ fun i _ => hf i ** case h_directed \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) \u22a2 Directed (fun x x_1 => x \u2264 x_1) fun s a => \u2211 i in s, f i a ** intro s t ** case h_directed \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) s t : Finset \u03b2 \u22a2 \u2203 z,     (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) z) \u2227       (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) z) ** use s \u222a t ** 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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) s t : Finset \u03b2 \u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) (s \u222a t)) \u2227     (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) (s \u222a t)) ** constructor ** case h.left \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) s t : Finset \u03b2 \u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) (s \u222a t)) ** exact fun a => Finset.sum_le_sum_of_subset (Finset.subset_union_left _ _) ** case h.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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b2), AEMeasurable (f i) s t : Finset \u03b2 \u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) (s \u222a t)) ** exact fun a => Finset.sum_le_sum_of_subset (Finset.subset_union_right _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.hasSum_intervalIntegral_of_summable_norm ** \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 \u22a2 HasSum (fun i => \u222b (x : \u211d) in a..b, \u2191(f i) x) (\u222b (x : \u211d) in a..b, \u2211' (i : \u03b9), \u2191(f i) x) ** apply hasSum_integral_of_dominated_convergence\n  (fun i (x : \u211d) => \u2016(f i).restrict \u2191(\u27e8uIcc a b, isCompact_uIcc\u27e9 : Compacts \u211d)\u2016)\n  (fun i => (map_continuous <| f i).aestronglyMeasurable) ** case h_bound \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 \u22a2 \u2200 (n : \u03b9),     \u2200\u1d50 (t : \u211d),       t \u2208 \u0399 a b \u2192         \u2016\u2191(f n) t\u2016 \u2264 \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f n)\u2016 ** refine fun i => ae_of_all _ fun x hx => ?_ ** case h_bound \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 i : \u03b9 x : \u211d hx : x \u2208 \u0399 a b \u22a2 \u2016\u2191(f i) x\u2016 \u2264 \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 ** apply ContinuousMap.norm_coe_le_norm ((f i).restrict _) \u27e8x, _\u27e9 ** \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 i : \u03b9 x : \u211d hx : x \u2208 \u0399 a b \u22a2 x \u2208 \u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } ** exact \u27e8hx.1.le, hx.2\u27e9 ** case bound_summable \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 \u22a2 \u2200\u1d50 (t : \u211d),     t \u2208 \u0399 a b \u2192       Summable fun n =>         \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f n)\u2016 ** exact ae_of_all _ fun x _ => hf_sum ** case bound_integrable \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 \u22a2 IntervalIntegrable     (fun t =>       \u2211' (n : \u03b9), \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f n)\u2016)     volume a b ** exact intervalIntegrable_const ** case h_lim \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 \u22a2 \u2200\u1d50 (t : \u211d), t \u2208 \u0399 a b \u2192 HasSum (fun n => \u2191(f n) t) (\u2211' (i : \u03b9), \u2191(f i) t) ** refine ae_of_all _ fun x hx => Summable.hasSum ?_ ** case h_lim \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 x : \u211d hx : x \u2208 \u0399 a b \u22a2 Summable fun n => \u2191(f n) x ** let x : (\u27e8uIcc a b, isCompact_uIcc\u27e9 : Compacts \u211d) := \u27e8x, ?_\u27e9 ** case h_lim.refine_2 \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 x\u271d : \u211d hx : x\u271d \u2208 \u0399 a b x : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=   { val := x\u271d, property := ?h_lim.refine_1 } \u22a2 Summable fun n => \u2191(f n) x\u271d  case h_lim.refine_1 \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 x : \u211d hx : x \u2208 \u0399 a b \u22a2 x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } ** swap ** case h_lim.refine_1 \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 x : \u211d hx : x \u2208 \u0399 a b \u22a2 x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }  case h_lim.refine_2 \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 x\u271d : \u211d hx : x\u271d \u2208 \u0399 a b x : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=   { val := x\u271d, property := ?h_lim.refine_1 } \u22a2 Summable fun n => \u2191(f n) x\u271d ** exact \u27e8hx.1.le, hx.2\u27e9 ** case h_lim.refine_2 \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 x\u271d : \u211d hx : x\u271d \u2208 \u0399 a b x : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=   { val := x\u271d, property := (_ : a \u2293 b \u2264 x\u271d \u2227 x\u271d \u2264 a \u2294 b) } \u22a2 Summable fun n => \u2191(f n) x\u271d ** have := summable_of_summable_norm hf_sum ** case h_lim.refine_2 \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 d : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : Countable \u03b9 f : \u03b9 \u2192 C(\u211d, E) hf_sum :   Summable fun i => \u2016ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f i)\u2016 x\u271d : \u211d hx : x\u271d \u2208 \u0399 a b x : { x // x \u2208 { carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) } } :=   { val := x\u271d, property := (_ : a \u2293 b \u2264 x\u271d \u2227 x\u271d \u2264 a \u2294 b) } this :   Summable fun a_1 => ContinuousMap.restrict (\u2191{ carrier := [[a, b]], isCompact' := (_ : IsCompact [[a, b]]) }) (f a_1) \u22a2 Summable fun n => \u2191(f n) x\u271d ** simpa only [Compacts.coe_mk, ContinuousMap.restrict_apply]\n  using ContinuousMap.summable_apply this x ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.ext ** \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 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc\u2081 s = \u2191\u2191\u03bc\u2082 s \u22a2 \u2191\u03bc\u2081 = \u2191\u03bc\u2082 ** rw [\u2190 trimmed, OuterMeasure.trim_congr (h _), trimmed] ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.preimage_asSubtype ** \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 asSubtype f \u207b\u00b9' s = Subtype.val \u207b\u00b9' preimage f s ** ext x ** case h \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 x : \u2191(Dom f) \u22a2 x \u2208 asSubtype f \u207b\u00b9' s \u2194 x \u2208 Subtype.val \u207b\u00b9' preimage f s ** simp only [Set.mem_preimage, Set.mem_setOf_eq, PFun.asSubtype, PFun.mem_preimage] ** case h \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 x : \u2191(Dom f) \u22a2 fn f \u2191x (_ : \u2191x \u2208 Dom f) \u2208 s \u2194 \u2203 y, y \u2208 s \u2227 y \u2208 f \u2191x ** exact\n  Iff.intro (fun h => \u27e8_, h, Part.get_mem _\u27e9) fun \u27e8y, ys, fxy\u27e9 =>\n    have : f.fn x.val x.property \u2208 f x.val := Part.get_mem _\n    Part.mem_unique fxy this \u25b8 ys ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.not_monotoneOn_not_antitoneOn_iff_exists_le_le ** \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 \u2264 b \u2227 b \u2264 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_le_le, @and_left_comm (_ \u2208 s)] ** Qed",
        "informal": ""
    },
    {
        "formal": "circleIntegral.integral_eq_zero_of_hasDerivWithinAt' ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f f' : \u2102 \u2192 E c : \u2102 R : \u211d h : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z \u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = 0 ** by_cases hi : CircleIntegrable f' c R ** case pos E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f f' : \u2102 \u2192 E c : \u2102 R : \u211d h : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z hi : CircleIntegrable f' c R \u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = 0 ** rw [\u2190 sub_eq_zero.2 ((periodic_circleMap c R).comp f).eq] ** case pos E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f f' : \u2102 \u2192 E c : \u2102 R : \u211d h : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z hi : CircleIntegrable f' c R \u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = (f \u2218 circleMap c R) (2 * \u03c0) - (f \u2218 circleMap c R) 0 ** refine' intervalIntegral.integral_eq_sub_of_hasDerivAt (fun \u03b8 _ => _) hi.out ** case pos E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f f' : \u2102 \u2192 E c : \u2102 R : \u211d h : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z hi : CircleIntegrable f' c R \u03b8 : \u211d x\u271d : \u03b8 \u2208 [[0, 2 * \u03c0]] \u22a2 HasDerivAt (f \u2218 circleMap c R) (deriv (circleMap c R) \u03b8 \u2022 (fun z => f' z) (circleMap c R \u03b8)) \u03b8 ** exact (h _ (circleMap_mem_sphere' _ _ _)).scomp_hasDerivAt \u03b8\n  (differentiable_circleMap _ _ _).hasDerivAt (circleMap_mem_sphere' _ _) ** case neg E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f f' : \u2102 \u2192 E c : \u2102 R : \u211d h : \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 HasDerivWithinAt f (f' z) (sphere c |R|) z hi : \u00acCircleIntegrable f' c R \u22a2 (\u222e (z : \u2102) in C(c, R), f' z) = 0 ** exact integral_undef hi ** Qed",
        "informal": ""
    },
    {
        "formal": "Holor.cprankMax_upper_bound ** \u03b1 : Type d\u271d : \u2115 ds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 d : \u2115 ds : List \u2115 x : Holor \u03b1 (d :: ds) \u22a2 CPRankMax (prod (d :: ds)) x ** have h_summands :\n  \u2200 i : { x // x \u2208 Finset.range d },\n    CPRankMax ds.prod (unitVec d i.1 \u2297 slice x i.1 (mem_range.1 i.2)) :=\n  fun i => cprankMax_mul _ _ _ (cprankMax_upper_bound (slice x i.1 (mem_range.1 i.2))) ** \u03b1 : Type d\u271d : \u2115 ds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 d : \u2115 ds : List \u2115 x : Holor \u03b1 (d :: ds) h_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) \u22a2 CPRankMax (prod (d :: ds)) x ** have h_dds_prod : (List.cons d ds).prod = Finset.card (Finset.range d) * prod ds := by\n  simp [Finset.card_range] ** \u03b1 : Type d\u271d : \u2115 ds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 d : \u2115 ds : List \u2115 x : Holor \u03b1 (d :: ds) h_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) h_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds \u22a2 CPRankMax (prod (d :: ds)) x ** have :\n  CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d),\n      unitVec d i.val \u2297 slice x i.val (mem_range.1 i.2)) :=\n  cprankMax_sum (Finset.range d).attach _ fun i _ => h_summands i ** \u03b1 : Type d\u271d : \u2115 ds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 d : \u2115 ds : List \u2115 x : Holor \u03b1 (d :: ds) h_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) h_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds this :   CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) \u22a2 CPRankMax (prod (d :: ds)) x ** have h_cprankMax_sum :\n  CPRankMax (Finset.card (Finset.range d) * prod ds)\n    (\u2211 i in Finset.attach (Finset.range d),\n      unitVec d i.val \u2297 slice x i.val (mem_range.1 i.2)) := by rwa [Finset.card_attach] at this ** \u03b1 : Type d\u271d : \u2115 ds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 d : \u2115 ds : List \u2115 x : Holor \u03b1 (d :: ds) h_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) h_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds this :   CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) h_cprankMax_sum :   CPRankMax (Finset.card (Finset.range d) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) \u22a2 CPRankMax (prod (d :: ds)) x ** rw [\u2190 sum_unitVec_mul_slice x] ** \u03b1 : Type d\u271d : \u2115 ds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 d : \u2115 ds : List \u2115 x : Holor \u03b1 (d :: ds) h_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) h_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds this :   CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) h_cprankMax_sum :   CPRankMax (Finset.card (Finset.range d) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) \u22a2 CPRankMax (prod (d :: ds)) (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d)) ** rw [h_dds_prod] ** \u03b1 : Type d\u271d : \u2115 ds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 d : \u2115 ds : List \u2115 x : Holor \u03b1 (d :: ds) h_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) h_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds this :   CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) h_cprankMax_sum :   CPRankMax (Finset.card (Finset.range d) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) \u22a2 CPRankMax (Finset.card (Finset.range d) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d)) ** exact h_cprankMax_sum ** \u03b1 : Type d\u271d : \u2115 ds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 d : \u2115 ds : List \u2115 x : Holor \u03b1 (d :: ds) h_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) \u22a2 prod (d :: ds) = Finset.card (Finset.range d) * prod ds ** simp [Finset.card_range] ** \u03b1 : Type d\u271d : \u2115 ds\u271d ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 d : \u2115 ds : List \u2115 x : Holor \u03b1 (d :: ds) h_summands : \u2200 (i : { x // x \u2208 Finset.range d }), CPRankMax (prod ds) (unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) h_dds_prod : prod (d :: ds) = Finset.card (Finset.range d) * prod ds this :   CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) \u22a2 CPRankMax (Finset.card (Finset.range d) * prod ds)     (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : \u2191i < d)) ** rwa [Finset.card_attach] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L2.inner_indicatorConstLp_eq_set_integral_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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 inner (indicatorConstLp 2 hs h\u03bcs c) f = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc ** rw [inner_def, \u2190 integral_add_compl hs (L2.integrable_inner _ f)] ** \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc +       \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc =     \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc ** have h_left : (\u222b x in s, \u27ea(indicatorConstLp 2 hs h\u03bcs c) x, f x\u27eb \u2202\u03bc) = \u222b x in s, \u27eac, f x\u27eb \u2202\u03bc := by\n  suffices h_ae_eq : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 \u27eaindicatorConstLp 2 hs h\u03bcs c x, f x\u27eb = \u27eac, f x\u27eb\n  exact set_integral_congr_ae hs h_ae_eq\n  have h_indicator : \u2200\u1d50 x : \u03b1 \u2202\u03bc, x \u2208 s \u2192 indicatorConstLp 2 hs h\u03bcs c x = c :=\n    indicatorConstLp_coeFn_mem\n  refine' h_indicator.mono fun x hx hxs => _\n  congr\n  exact hx hxs ** \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc h_right : \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = 0 \u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc +       \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc =     \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc ** rw [h_left, h_right, add_zero] ** \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc ** suffices h_ae_eq : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 \u27eaindicatorConstLp 2 hs h\u03bcs c x, f x\u27eb = \u27eac, f x\u27eb ** \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x) \u22a2 \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc  case h_ae_eq \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x) ** exact set_integral_congr_ae hs h_ae_eq ** case h_ae_eq \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x) ** have h_indicator : \u2200\u1d50 x : \u03b1 \u2202\u03bc, x \u2208 s \u2192 indicatorConstLp 2 hs h\u03bcs c x = c :=\n  indicatorConstLp_coeFn_mem ** case h_ae_eq \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x) ** refine' h_indicator.mono fun x hx hxs => _ ** case h_ae_eq \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c x : \u03b1 hx : x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c hxs : x \u2208 s \u22a2 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = inner c (\u2191\u2191f x) ** congr ** case h_ae_eq.e_a \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c x : \u03b1 hx : x \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c hxs : x \u2208 s \u22a2 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = c ** exact hx hxs ** \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = 0 ** suffices h_ae_eq : \u2200\u1d50 x \u2202\u03bc, x \u2209 s \u2192 \u27eaindicatorConstLp 2 hs h\u03bcs c x, f x\u27eb = 0 ** case h_ae_eq \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0 ** have h_indicator : \u2200\u1d50 x : \u03b1 \u2202\u03bc, x \u2209 s \u2192 indicatorConstLp 2 hs h\u03bcs c x = 0 :=\n  indicatorConstLp_coeFn_nmem ** case h_ae_eq \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc h_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0 ** refine' h_indicator.mono fun x hx hxs => _ ** case h_ae_eq \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc h_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0 x : \u03b1 hx : \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0 hxs : \u00acx \u2208 s \u22a2 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0 ** rw [hx hxs] ** case h_ae_eq \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc h_indicator : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0 x : \u03b1 hx : \u00acx \u2208 s \u2192 \u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x = 0 hxs : \u00acx \u2208 s \u22a2 inner 0 (\u2191\u2191f x) = 0 ** exact inner_zero_left _ ** \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc h_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0 \u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = 0 ** simp_rw [\u2190 Set.mem_compl_iff] at h_ae_eq ** \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc h_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0 \u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = 0 ** suffices h_int_zero :\n  (\u222b x in s\u1d9c, inner (indicatorConstLp 2 hs h\u03bcs c x) (f x) \u2202\u03bc) = \u222b _ in s\u1d9c, (0 : \ud835\udd5c) \u2202\u03bc ** case h_int_zero \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc h_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0 \u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc ** exact set_integral_congr_ae hs.compl h_ae_eq ** \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc h_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0 h_int_zero : \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = 0 ** rw [h_int_zero] ** \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 s : Set \u03b1 f : { x // x \u2208 Lp E 2 } hs : MeasurableSet s c : E h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h_left : \u222b (x : \u03b1) in s, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s, inner c (\u2191\u2191f x) \u2202\u03bc h_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) = 0 h_int_zero : \u222b (x : \u03b1) in s\u1d9c, inner (\u2191\u2191(indicatorConstLp 2 hs h\u03bcs c) x) (\u2191\u2191f x) \u2202\u03bc = \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s\u1d9c, 0 \u2202\u03bc = 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "borel_eq_generateFrom_Ioc ** \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\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 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : OrderTopology \u03b1 \u22a2 borel \u03b1 = MeasurableSpace.generateFrom {S | \u2203 l u, l < u \u2227 Ioc l u = S} ** simpa only [exists_prop, mem_univ, true_and_iff] using\n  (@dense_univ \u03b1 _).borel_eq_generateFrom_Ioc_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>\n    mem_univ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "PosNum.minFacAux_to_nat ** fuel : \u2115 n k : PosNum h : Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u22a2 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k) ** induction' fuel with fuel ih generalizing k <;> rw [minFacAux, Nat.minFacAux] ** case succ fuel\u271d : \u2115 n k\u271d : PosNum h\u271d : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d) fuel : \u2115 ih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k) k : PosNum h : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k) \u22a2 \u2191(if n < bit1 k * bit1 k then n else if bit1 k \u2223 n then bit1 k else minFacAux n fuel (succ k)) =     if h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) then \u2191n     else       if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k)       else         let_fun this := (_ : Nat.sqrt \u2191n - \u2191(bit1 k) < Nat.sqrt \u2191n + 2 - \u2191(bit1 k));         Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2) ** simp_rw [\u2190 mul_to_nat] ** case succ fuel\u271d : \u2115 n k\u271d : PosNum h\u271d : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d) fuel : \u2115 ih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k) k : PosNum h : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k) \u22a2 \u2191(if n < bit1 k * bit1 k then n else if bit1 k \u2223 n then bit1 k else minFacAux n fuel (succ k)) =     if h : \u2191n < \u2191(bit1 k * bit1 k) then \u2191n else if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k) else Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2) ** simp only [cast_lt, dvd_to_nat] ** case succ fuel\u271d : \u2115 n k\u271d : PosNum h\u271d : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d) fuel : \u2115 ih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k) k : PosNum h : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k) \u22a2 \u2191(if n < bit1 k * bit1 k then n else if bit1 k \u2223 n then bit1 k else minFacAux n fuel (succ k)) =     if h : n < bit1 k * bit1 k then \u2191n else if bit1 k \u2223 n then \u2191(bit1 k) else Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2) ** split_ifs <;> try rfl ** case neg fuel\u271d : \u2115 n k\u271d : PosNum h\u271d\u00b2 : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d) fuel : \u2115 ih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k) k : PosNum h : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k) h\u271d\u00b9 : \u00acn < bit1 k * bit1 k h\u271d : \u00acbit1 k \u2223 n \u22a2 \u2191(minFacAux n fuel (succ k)) = Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2) ** rw [ih] <;> [congr; convert Nat.lt_succ_of_lt h using 1] <;>\n  simp only [_root_.bit1, _root_.bit0, cast_bit1, cast_succ, Nat.succ_eq_add_one, add_assoc,\n    add_left_comm, \u2190 one_add_one_eq_two] ** case zero fuel : \u2115 n k\u271d : PosNum h\u271d : Nat.sqrt \u2191n < fuel + \u2191(bit1 k\u271d) k : PosNum h : Nat.sqrt \u2191n < Nat.zero + \u2191(bit1 k) \u22a2 \u2191n =     if h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) then \u2191n     else       if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k)       else         let_fun this := (_ : Nat.sqrt \u2191n - \u2191(bit1 k) < Nat.sqrt \u2191n + 2 - \u2191(bit1 k));         Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2) ** rw [Nat.zero_add, Nat.sqrt_lt] at h ** case zero fuel : \u2115 n k\u271d : PosNum h\u271d : Nat.sqrt \u2191n < fuel + \u2191(bit1 k\u271d) k : PosNum h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) \u22a2 \u2191n =     if h : \u2191n < \u2191(bit1 k) * \u2191(bit1 k) then \u2191n     else       if \u2191(bit1 k) \u2223 \u2191n then \u2191(bit1 k)       else         let_fun this := (_ : Nat.sqrt \u2191n - \u2191(bit1 k) < Nat.sqrt \u2191n + 2 - \u2191(bit1 k));         Nat.minFacAux (\u2191n) (\u2191(bit1 k) + 2) ** simp only [h, dite_true] ** case pos fuel\u271d : \u2115 n k\u271d : PosNum h\u271d\u00b2 : Nat.sqrt \u2191n < fuel\u271d + \u2191(bit1 k\u271d) fuel : \u2115 ih : \u2200 {k : PosNum}, Nat.sqrt \u2191n < fuel + \u2191(bit1 k) \u2192 \u2191(minFacAux n fuel k) = Nat.minFacAux \u2191n \u2191(bit1 k) k : PosNum h : Nat.sqrt \u2191n < Nat.succ fuel + \u2191(bit1 k) h\u271d\u00b9 : \u00acn < bit1 k * bit1 k h\u271d : bit1 k \u2223 n \u22a2 \u2191(bit1 k) = \u2191(bit1 k) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_union_eq_left_of_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 ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 \u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** have ht : IntegrableOn f t \u03bc := by apply integrableOn_zero.congr_fun_ae; symm; exact ht_eq ** \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 ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t \u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** by_cases H : IntegrableOn f (s \u222a t) \u03bc ** case pos \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 ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t H : IntegrableOn f (s \u222a t) \u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc  case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t H : \u00acIntegrableOn f (s \u222a t) \u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** swap ** case pos \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 ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t H : IntegrableOn f (s \u222a t) \u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** let f' := H.1.mk f ** case pos \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 ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t H : IntegrableOn f (s \u222a t) f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) \u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** calc\n  \u222b x : \u03b1 in s \u222a t, f x \u2202\u03bc = \u222b x : \u03b1 in s \u222a t, f' x \u2202\u03bc := integral_congr_ae H.1.ae_eq_mk\n  _ = \u222b x in s, f' x \u2202\u03bc := by\n    apply\n      integral_union_eq_left_of_ae_aux _ H.1.stronglyMeasurable_mk (H.congr_fun_ae H.1.ae_eq_mk)\n    filter_upwards [ht_eq,\n      ae_mono (Measure.restrict_mono (subset_union_right s t) le_rfl) H.1.ae_eq_mk] with x hx h'x\n    rw [\u2190 h'x, hx]\n  _ = \u222b x in s, f x \u2202\u03bc :=\n    integral_congr_ae\n      (ae_mono (Measure.restrict_mono (subset_union_left s t) le_rfl) H.1.ae_eq_mk.symm) ** \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 ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 \u22a2 IntegrableOn f t ** apply integrableOn_zero.congr_fun_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 ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 \u22a2 (fun x => 0) =\u1d50[Measure.restrict \u03bc t] f ** symm ** \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 ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 \u22a2 f =\u1d50[Measure.restrict \u03bc t] fun x => 0 ** exact ht_eq ** case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t H : \u00acIntegrableOn f (s \u222a t) \u22a2 \u222b (x : \u03b1) in s \u222a t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** rw [integral_undef H, integral_undef] ** case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t H : \u00acIntegrableOn f (s \u222a t) \u22a2 \u00acIntegrable fun x => f x ** simpa [integrableOn_union, ht] using 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t H : IntegrableOn f (s \u222a t) f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) \u22a2 \u222b (x : \u03b1) in s \u222a t, f' x \u2202\u03bc = \u222b (x : \u03b1) in s, f' x \u2202\u03bc ** apply\n  integral_union_eq_left_of_ae_aux _ H.1.stronglyMeasurable_mk (H.congr_fun_ae H.1.ae_eq_mk) ** \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 ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t H : IntegrableOn f (s \u222a t) f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t,     AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x = 0 ** filter_upwards [ht_eq,\n  ae_mono (Measure.restrict_mono (subset_union_right s t) le_rfl) H.1.ae_eq_mk] with x hx h'x ** case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x = 0 ht : IntegrableOn f t H : IntegrableOn f (s \u222a t) f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x : \u03b1 hx : f x = 0 h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x \u22a2 AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc (s \u222a t))) x = 0 ** rw [\u2190 h'x, hx] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Nontrivial.ne_singleton ** \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 : \u03b1 hs : Set.Nontrivial s H : s = {x} \u22a2 False ** rw [H] at hs ** \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 : \u03b1 hs : Set.Nontrivial {x} H : s = {x} \u22a2 False ** exact not_nontrivial_singleton hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ProbabilityMeasure.apply_le_one ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 s : Set \u03a9 \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s \u2264 1 ** simpa using apply_mono \u03bc (subset_univ s) ** Qed",
        "informal": ""
    },
    {
        "formal": "Function.Periodic.intervalIntegral_add_zsmul_eq ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T n : \u2124 t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 \u22a2 \u222b (x : \u211d) in t..t + n \u2022 T, f x = n \u2022 \u222b (x : \u211d) in t..t + T, f x ** suffices (\u222b x in (0)..(n \u2022 T), f x) = n \u2022 \u222b x in (0)..T, f x by\n  simp only [hf.intervalIntegral_add_eq t 0, (hf.zsmul n).intervalIntegral_add_eq t 0, zero_add,\n    this] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T n : \u2124 t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 this : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x \u22a2 \u222b (x : \u211d) in 0 ..n \u2022 T, f x = n \u2022 \u222b (x : \u211d) in 0 ..T, f x ** cases' n with n n ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T n : \u2124 t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 this : \u222b (x : \u211d) in 0 ..n \u2022 T, f x = n \u2022 \u222b (x : \u211d) in 0 ..T, f x \u22a2 \u222b (x : \u211d) in t..t + n \u2022 T, f x = n \u2022 \u222b (x : \u211d) in t..t + T, f x ** simp only [hf.intervalIntegral_add_eq t 0, (hf.zsmul n).intervalIntegral_add_eq t 0, zero_add,\n  this] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T n : \u2124 t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 m : \u2115 \u22a2 \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x ** induction' m with m ih ** case zero E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T n : \u2124 t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 \u22a2 \u222b (x : \u211d) in 0 ..Nat.zero \u2022 T, f x = Nat.zero \u2022 \u222b (x : \u211d) in 0 ..T, f x ** simp ** case succ E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T n : \u2124 t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 m : \u2115 ih : \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x \u22a2 \u222b (x : \u211d) in 0 ..Nat.succ m \u2022 T, f x = Nat.succ m \u2022 \u222b (x : \u211d) in 0 ..T, f x ** simp only [succ_nsmul', hf.intervalIntegral_add_eq_add 0 (m \u2022 T) h_int, ih, zero_add] ** case ofNat E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 this : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x n : \u2115 \u22a2 \u222b (x : \u211d) in 0 ..Int.ofNat n \u2022 T, f x = Int.ofNat n \u2022 \u222b (x : \u211d) in 0 ..T, f x ** simp [\u2190 this n] ** case negSucc E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 this : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x n : \u2115 \u22a2 \u222b (x : \u211d) in 0 ..Int.negSucc n \u2022 T, f x = Int.negSucc n \u2022 \u222b (x : \u211d) in 0 ..T, f x ** conv_rhs => rw [negSucc_zsmul] ** case negSucc E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 this : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x n : \u2115 \u22a2 \u222b (x : \u211d) in 0 ..Int.negSucc n \u2022 T, f x = -((n + 1) \u2022 \u222b (x : \u211d) in 0 ..T, f x) ** have h\u2080 : Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0 := by simp; linarith ** case negSucc E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 this : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x n : \u2115 h\u2080 : Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0 \u22a2 \u222b (x : \u211d) in 0 ..Int.negSucc n \u2022 T, f x = -((n + 1) \u2022 \u222b (x : \u211d) in 0 ..T, f x) ** rw [integral_symm, \u2190 (hf.nsmul (n + 1)).funext, neg_inj] ** case negSucc E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 this : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x n : \u2115 h\u2080 : Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0 \u22a2 \u222b (x : \u211d) in Int.negSucc n \u2022 T..0, (fun x => f (x + (n + 1) \u2022 T)) x =     (n + 1) \u2022 \u222b (x : \u211d) in 0 ..T, (fun x => f (x + (n + 1) \u2022 T)) x ** simp_rw [integral_comp_add_right, h\u2080, zero_add, this (n + 1), add_comm T,\n  hf.intervalIntegral_add_eq ((n + 1) \u2022 T) 0, zero_add] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 this : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x n : \u2115 \u22a2 Int.negSucc n \u2022 T + (n + 1) \u2022 T = 0 ** simp ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T t : \u211d h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable f volume t\u2081 t\u2082 this : \u2200 (m : \u2115), \u222b (x : \u211d) in 0 ..m \u2022 T, f x = m \u2022 \u222b (x : \u211d) in 0 ..T, f x n : \u2115 \u22a2 (-1 + -\u2191n) * T + (\u2191n + 1) * T = 0 ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_findGreatest ** \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 p : \u03b1 \u2192 \u2115 \u2192 Prop inst\u271d : (x : \u03b1) \u2192 DecidablePred (p x) N : \u2115 hN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k} \u22a2 Measurable fun x => Nat.findGreatest (p x) N ** refine' measurable_findGreatest' fun k hk => _ ** \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 p : \u03b1 \u2192 \u2115 \u2192 Prop inst\u271d : (x : \u03b1) \u2192 DecidablePred (p x) N : \u2115 hN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k} k : \u2115 hk : k \u2264 N \u22a2 MeasurableSet {x | Nat.findGreatest (p x) N = k} ** simp only [Nat.findGreatest_eq_iff, setOf_and, setOf_forall, \u2190 compl_setOf] ** \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 p : \u03b1 \u2192 \u2115 \u2192 Prop inst\u271d : (x : \u03b1) \u2192 DecidablePred (p x) N : \u2115 hN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k} k : \u2115 hk : k \u2264 N \u22a2 MeasurableSet ({a | k \u2264 N} \u2229 ((\u22c2 (_ : k \u2260 0), {x | p x k}) \u2229 \u22c2 i, \u22c2 (_ : k < i), \u22c2 (_ : i \u2264 N), {a | p a i}\u1d9c)) ** repeat' apply_rules [MeasurableSet.inter, MeasurableSet.const, MeasurableSet.iInter,\n  MeasurableSet.compl, hN] <;> try intros ** case h\u2082.h\u2082 \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 p : \u03b1 \u2192 \u2115 \u2192 Prop inst\u271d : (x : \u03b1) \u2192 DecidablePred (p x) N : \u2115 hN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k} k : \u2115 hk : k \u2264 N b\u271d\u00b2 : \u2115 b\u271d\u00b9 : k < b\u271d\u00b2 b\u271d : b\u271d\u00b2 \u2264 N \u22a2 MeasurableSet {a | p a b\u271d\u00b2}\u1d9c ** apply_rules [MeasurableSet.inter, MeasurableSet.const, MeasurableSet.iInter,\nMeasurableSet.compl, hN] <;> try intros ** case h\u2082.h\u2082 \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 p : \u03b1 \u2192 \u2115 \u2192 Prop inst\u271d : (x : \u03b1) \u2192 DecidablePred (p x) N : \u2115 hN : \u2200 (k : \u2115), k \u2264 N \u2192 MeasurableSet {x | p x k} k : \u2115 hk : k \u2264 N b\u271d\u00b9 : \u2115 b\u271d : k < b\u271d\u00b9 \u22a2 b\u271d\u00b9 \u2264 N \u2192 MeasurableSet {a | p a b\u271d\u00b9}\u1d9c ** intros ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpInd_disjoint_union ** \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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 \u22a2 condexpInd G hm \u03bc (s \u222a t) = condexpInd G hm \u03bc s + condexpInd G hm \u03bc t ** ext1 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G \u22a2 \u2191(condexpInd G hm \u03bc (s \u222a t)) x = \u2191(condexpInd G hm \u03bc s + condexpInd G hm \u03bc t) x ** push_cast ** 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G \u22a2 \u2191(condexpInd G hm \u03bc (s \u222a t)) x = (\u2191(condexpInd G hm \u03bc s) + \u2191(condexpInd G hm \u03bc t)) x ** exact condexpInd_disjoint_union_apply hs ht h\u03bcs h\u03bct hst x ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condexp_ae_eq_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 hm : m \u2264 m\u03a9 hf_int : Integrable f \u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] \u03bc[f|m] ** rw [inf_of_le_left hm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.singularPart_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 \u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd ** by_cases hr : r = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 hr : \u00acr = 0 \u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd ** by_cases hl : HaveLebesgueDecomposition \u03bc \u03bd ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 hr : r = 0 \u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd ** rw [hr, zero_smul, zero_smul, singularPart_zero] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 hr : \u00acr = 0 hl : HaveLebesgueDecomposition \u03bc \u03bd \u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd ** haveI := hl ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 hr : \u00acr = 0 hl this : HaveLebesgueDecomposition \u03bc \u03bd \u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd ** refine'\n  (eq_singularPart ((measurable_rnDeriv \u03bc \u03bd).const_smul (r : \u211d\u22650\u221e))\n      (MutuallySingular.smul r (haveLebesgueDecomposition_spec _ _).2.1) _).symm ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 hr : \u00acr = 0 hl this : HaveLebesgueDecomposition \u03bc \u03bd \u22a2 r \u2022 \u03bc = \u2191r \u2022 singularPart \u03bc \u03bd + withDensity \u03bd (\u2191r \u2022 rnDeriv \u03bc \u03bd) ** rw [withDensity_smul _ (measurable_rnDeriv _ _), \u2190 smul_add,\n  \u2190 haveLebesgueDecomposition_add \u03bc \u03bd, ENNReal.smul_def] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 hr : \u00acr = 0 hl : \u00acHaveLebesgueDecomposition \u03bc \u03bd \u22a2 singularPart (r \u2022 \u03bc) \u03bd = r \u2022 singularPart \u03bc \u03bd ** rw [singularPart, singularPart, dif_neg hl, dif_neg, smul_zero] ** case neg.hnc \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 hr : \u00acr = 0 hl : \u00acHaveLebesgueDecomposition \u03bc \u03bd \u22a2 \u00acHaveLebesgueDecomposition (r \u2022 \u03bc) \u03bd ** refine' fun hl' => hl _ ** case neg.hnc \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 hr : \u00acr = 0 hl : \u00acHaveLebesgueDecomposition \u03bc \u03bd hl' : HaveLebesgueDecomposition (r \u2022 \u03bc) \u03bd \u22a2 HaveLebesgueDecomposition \u03bc \u03bd ** rw [\u2190 inv_smul_smul\u2080 hr \u03bc] ** case neg.hnc \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 r : \u211d\u22650 hr : \u00acr = 0 hl : \u00acHaveLebesgueDecomposition \u03bc \u03bd hl' : HaveLebesgueDecomposition (r \u2022 \u03bc) \u03bd \u22a2 HaveLebesgueDecomposition (r\u207b\u00b9 \u2022 r \u2022 \u03bc) \u03bd ** exact @Measure.haveLebesgueDecomposition_smul _ _ _ _ hl' _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.eraseNone_image_some ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq (Option \u03b1) s : Finset \u03b1 \u22a2 \u2191eraseNone (image some s) = s ** simpa only [map_eq_image] using eraseNone_map_some s ** Qed",
        "informal": ""
    },
    {
        "formal": "String.split_of_valid ** s : String p : Char \u2192 Bool \u22a2 split s p = List.map mk (List.splitOnP p s.data) ** simpa [split] using splitAux_of_valid p [] [] s.1 [] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.natAbs_sign_of_nonzero ** z : Int hz : z \u2260 0 \u22a2 natAbs (sign z) = 1 ** rw [Int.natAbs_sign, if_neg hz] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.indep_biSup_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 t : Set \u03b9 ht : p t \u22a2 Indep (\u2a06 n \u2208 t, s n) (limsup s f) ** refine' indep_of_indep_of_le_right (indep_biSup_compl h_le h_indep t) _ ** \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 t : Set \u03b9 ht : p t \u22a2 limsup s f \u2264 \u2a06 n \u2208 t\u1d9c, s n ** refine' limsSup_le_of_le (by isBoundedDefault) _ ** \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 t : Set \u03b9 ht : p t \u22a2 \u2200\u1da0 (n : MeasurableSpace \u03a9) in map s f, n \u2264 \u2a06 n \u2208 t\u1d9c, s n ** simp only [Set.mem_compl_iff, eventually_map] ** \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 t : Set \u03b9 ht : p t \u22a2 \u2200\u1da0 (a : \u03b9) in f, s a \u2264 \u2a06 n, \u2a06 (_ : \u00acn \u2208 t), s n ** exact eventually_of_mem (hf t ht) le_iSup\u2082 ** \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 t : Set \u03b9 ht : p t \u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map s f) ** isBoundedDefault ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.Tape.move_right_n_head ** \u0393 : Type u_1 inst\u271d : Inhabited \u0393 T : Tape \u0393 i : \u2115 \u22a2 ((move Dir.right)^[i] T).head = nth T \u2191i ** induction i generalizing T ** case zero \u0393 : Type u_1 inst\u271d : Inhabited \u0393 T : Tape \u0393 \u22a2 ((move Dir.right)^[Nat.zero] T).head = nth T \u2191Nat.zero ** rfl ** case succ \u0393 : Type u_1 inst\u271d : Inhabited \u0393 n\u271d : \u2115 n_ih\u271d : \u2200 (T : Tape \u0393), ((move Dir.right)^[n\u271d] T).head = nth T \u2191n\u271d T : Tape \u0393 \u22a2 ((move Dir.right)^[Nat.succ n\u271d] T).head = nth T \u2191(Nat.succ n\u271d) ** simp only [*, Tape.move_right_nth, Int.ofNat_succ, iterate_succ, Function.comp_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "parallelepiped_orthonormalBasis_one_dim ** \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 b : OrthonormalBasis \u03b9 \u211d \u211d \u22a2 parallelepiped \u2191b = Icc 0 1 \u2228 parallelepiped \u2191b = Icc (-1) 0 ** have e : \u03b9 \u2243 Fin 1 := by\n  apply Fintype.equivFinOfCardEq\n  simp only [\u2190 finrank_eq_card_basis b.toBasis, finrank_self] ** \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 b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 \u22a2 parallelepiped \u2191b = Icc 0 1 \u2228 parallelepiped \u2191b = Icc (-1) 0 ** have B : parallelepiped (b.reindex e) = parallelepiped b := by\n  convert parallelepiped_comp_equiv b e.symm\n  ext i\n  simp only [OrthonormalBasis.coe_reindex] ** \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 b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b \u22a2 parallelepiped \u2191b = Icc 0 1 \u2228 parallelepiped \u2191b = Icc (-1) 0 ** rw [\u2190 B] ** \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 b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b \u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0 ** let F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t => fun _i => t ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t A : Icc 0 1 = F '' Icc 0 1 \u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0 ** rcases orthonormalBasis_one_dim (b.reindex e) with (H | 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 b : OrthonormalBasis \u03b9 \u211d \u211d \u22a2 \u03b9 \u2243 Fin 1 ** apply Fintype.equivFinOfCardEq ** 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 b : OrthonormalBasis \u03b9 \u211d \u211d \u22a2 Fintype.card \u03b9 = 1 ** simp only [\u2190 finrank_eq_card_basis b.toBasis, finrank_self] ** \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 b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 \u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b ** convert parallelepiped_comp_equiv b e.symm ** case h.e'_2.h.e'_6 \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 b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 \u22a2 \u2191(OrthonormalBasis.reindex b e) = \u2191b \u2218 \u2191e.symm ** ext i ** case h.e'_2.h.e'_6.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 b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 i : Fin 1 \u22a2 \u2191(OrthonormalBasis.reindex b e) i = (\u2191b \u2218 \u2191e.symm) i ** simp only [OrthonormalBasis.coe_reindex] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t \u22a2 Icc 0 1 = F '' Icc 0 1 ** apply Subset.antisymm ** case h\u2081 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t \u22a2 Icc 0 1 \u2286 F '' Icc 0 1 ** intro x hx ** case h\u2081 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t x : Fin 1 \u2192 \u211d hx : x \u2208 Icc 0 1 \u22a2 x \u2208 F '' Icc 0 1 ** refine' \u27e8x 0, \u27e8hx.1 0, hx.2 0\u27e9, _\u27e9 ** case h\u2081 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t x : Fin 1 \u2192 \u211d hx : x \u2208 Icc 0 1 \u22a2 F (x 0) = x ** ext j ** case h\u2081.h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t x : Fin 1 \u2192 \u211d hx : x \u2208 Icc 0 1 j : Fin 1 \u22a2 F (x 0) j = x j ** simp only [Subsingleton.elim j 0] ** case h\u2082 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t \u22a2 F '' Icc 0 1 \u2286 Icc 0 1 ** rintro x \u27e8y, hy, rfl\u27e9 ** case h\u2082.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t y : \u211d hy : y \u2208 Icc 0 1 \u22a2 F y \u2208 Icc 0 1 ** exact \u27e8fun _j => hy.1, fun _j => hy.2\u27e9 ** case inl \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t A : Icc 0 1 = F '' Icc 0 1 H : \u2191(OrthonormalBasis.reindex b e) = fun x => 1 \u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0 ** left ** case inl.h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t A : Icc 0 1 = F '' Icc 0 1 H : \u2191(OrthonormalBasis.reindex b e) = fun x => 1 \u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 ** simp_rw [parallelepiped, H, A, Algebra.id.smul_eq_mul, mul_one] ** case inl.h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t A : Icc 0 1 = F '' Icc 0 1 H : \u2191(OrthonormalBasis.reindex b e) = fun x => 1 \u22a2 (fun a => \u2211 x : Fin 1, a x) '' ((fun a _i => a) '' Icc 0 1) = Icc 0 1 ** simp only [Finset.univ_unique, Fin.default_eq_zero, smul_eq_mul, mul_one, Finset.sum_singleton,\n  \u2190 image_comp, Function.comp_apply, image_id', ge_iff_le, zero_le_one, not_true, gt_iff_lt] ** case inr \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t A : Icc 0 1 = F '' Icc 0 1 H : \u2191(OrthonormalBasis.reindex b e) = fun x => -1 \u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc 0 1 \u2228 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0 ** right ** case inr.h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t A : Icc 0 1 = F '' Icc 0 1 H : \u2191(OrthonormalBasis.reindex b e) = fun x => -1 \u22a2 parallelepiped \u2191(OrthonormalBasis.reindex b e) = Icc (-1) 0 ** simp_rw [H, parallelepiped, Algebra.id.smul_eq_mul, A] ** case inr.h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F\u271d : 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\u271d inst\u271d : Module \u211d F\u271d b : OrthonormalBasis \u03b9 \u211d \u211d e : \u03b9 \u2243 Fin 1 B : parallelepiped \u2191(OrthonormalBasis.reindex b e) = parallelepiped \u2191b F : \u211d \u2192 Fin 1 \u2192 \u211d := fun t _i => t A : Icc 0 1 = F '' Icc 0 1 H : \u2191(OrthonormalBasis.reindex b e) = fun x => -1 \u22a2 (fun a => \u2211 x : Fin 1, a x * -1) '' ((fun a _i => a) '' Icc 0 1) = Icc (-1) 0 ** simp only [Finset.univ_unique, Fin.default_eq_zero, mul_neg, mul_one, Finset.sum_neg_distrib,\n  Finset.sum_singleton, \u2190 image_comp, Function.comp, image_neg, preimage_neg_Icc, neg_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_fintype ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E inst\u271d\u2079 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : CompleteSpace F G : Type u_5 inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : FirstCountableTopology X \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 inst\u271d : Fintype \u03b1 f : \u03b1 \u2192 E hf : Integrable f \u22a2 \u222b (x : \u03b1), f x \u2202\u03bc = \u2211 x : \u03b1, ENNReal.toReal (\u2191\u2191\u03bc {x}) \u2022 f x ** rw [\u2190 integral_finset .univ , Finset.coe_univ, Measure.restrict_univ] ** case hf \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E inst\u271d\u2079 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : CompleteSpace F G : Type u_5 inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : FirstCountableTopology X \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 inst\u271d : Fintype \u03b1 f : \u03b1 \u2192 E hf : Integrable f \u22a2 Integrable fun x => f x ** simp only [Finset.coe_univ, Measure.restrict_univ, hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.gcd_to_nat ** \u22a2 \u2200 (a b : Num), \u2191(gcd a b) = Nat.gcd \u2191a \u2191b ** have : \u2200 a b : Num, (a * b).natSize \u2264 a.natSize + b.natSize := by\n  intros\n  simp only [natSize_to_nat, cast_mul]\n  rw [Nat.size_le, pow_add]\n  exact mul_lt_mul'' (Nat.lt_size_self _) (Nat.lt_size_self _) (Nat.zero_le _) (Nat.zero_le _) ** this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b \u22a2 \u2200 (a b : Num), \u2191(gcd a b) = Nat.gcd \u2191a \u2191b ** intros ** this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b a\u271d b\u271d : Num \u22a2 \u2191(gcd a\u271d b\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d ** unfold gcd ** this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b a\u271d b\u271d : Num \u22a2 \u2191(if a\u271d \u2264 b\u271d then gcdAux (natSize a\u271d + natSize b\u271d) a\u271d b\u271d else gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) =     Nat.gcd \u2191a\u271d \u2191b\u271d ** split_ifs with h ** \u22a2 \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b ** intros ** a\u271d b\u271d : Num \u22a2 natSize (a\u271d * b\u271d) \u2264 natSize a\u271d + natSize b\u271d ** simp only [natSize_to_nat, cast_mul] ** a\u271d b\u271d : Num \u22a2 Nat.size (\u2191a\u271d * \u2191b\u271d) \u2264 Nat.size \u2191a\u271d + Nat.size \u2191b\u271d ** rw [Nat.size_le, pow_add] ** a\u271d b\u271d : Num \u22a2 \u2191a\u271d * \u2191b\u271d < 2 ^ Nat.size \u2191a\u271d * 2 ^ Nat.size \u2191b\u271d ** exact mul_lt_mul'' (Nat.lt_size_self _) (Nat.lt_size_self _) (Nat.zero_le _) (Nat.zero_le _) ** case pos this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b a\u271d b\u271d : Num h : a\u271d \u2264 b\u271d \u22a2 \u2191(gcdAux (natSize a\u271d + natSize b\u271d) a\u271d b\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d ** exact gcd_to_nat_aux h (this _ _) ** case neg this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b a\u271d b\u271d : Num h : \u00aca\u271d \u2264 b\u271d \u22a2 \u2191(gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d ** rw [Nat.gcd_comm] ** case neg this : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b a\u271d b\u271d : Num h : \u00aca\u271d \u2264 b\u271d \u22a2 \u2191(gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) = Nat.gcd \u2191b\u271d \u2191a\u271d ** exact gcd_to_nat_aux (le_of_not_le h) (this _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Iterator.ValidFor.setCurr' ** l r : List Char c : Char it : Iterator h : ValidFor l r it \u22a2 ValidFor l (List.modifyHead (fun x => c) r) (setCurr it c) ** cases h.out' ** case refl l r : List Char c : Char h : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } \u22a2 ValidFor l (List.modifyHead (fun x => c) r)     (setCurr { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } c) ** simp [Iterator.setCurr] ** case refl l r : List Char c : Char h : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } \u22a2 ValidFor l     (match r with     | [] => []     | a :: l => c :: l)     { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } } ** refine .of_eq _ ?_ (by simp) ** case refl l r : List Char c : Char h : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } \u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =     List.reverseAux l       (match r with       | [] => []       | a :: l => c :: l) ** have := set_of_valid l.reverse r c ** case refl l r : List Char c : Char h : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } this :   set { data := List.reverse l ++ r } { byteIdx := utf8Len (List.reverse l) } c =     { data := List.reverse l ++ List.modifyHead (fun x => c) r } \u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =     List.reverseAux l       (match r with       | [] => []       | a :: l => c :: l) ** simp at this ** case refl l r : List Char c : Char h : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } this :   set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c =     {       data :=         List.reverse l ++           match r with           | [] => []           | a :: l => c :: l } \u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =     List.reverseAux l       (match r with       | [] => []       | a :: l => c :: l) ** simp [List.reverseAux_eq, this] ** l r : List Char c : Char h : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } \u22a2 { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.i.byteIdx =     utf8Len l ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PartrecToTM2.head_stack_ok ** q : \u039b' s : Option \u0393' L\u2081 L\u2082 : List \u2115 L\u2083 : List \u0393' \u22a2 Reaches\u2081 (TM2.step tr)     { l := some (head stack q), var := s, stk := elim (trList L\u2081) [] [] (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083) }     { l := some q, var := none, stk := elim (trList (List.headI L\u2082 :: L\u2081)) [] [] L\u2083 } ** cases' L\u2082 with a L\u2082 ** case nil q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' \u22a2 Reaches\u2081 (TM2.step tr)     { l := some (head stack q), var := s, stk := elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083) }     { l := some q, var := none, stk := elim (trList (List.headI [] :: L\u2081)) [] [] L\u2083 } ** refine'\n  TransGen.trans\n    (move_ok (by decide)\n      (splitAtPred_eq _ _ [] (some \u0393'.cons\u2097) L\u2083 (by rintro _ \u27e8\u27e9) \u27e8rfl, rfl\u27e9))\n    (TransGen.head rfl (TransGen.head rfl _)) ** case nil q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     (TM2.stepAux       (tr         ((fun x =>             \u039b'.read fun s => ite (s = some \u0393'.cons\u2097) id (\u039b'.clear (fun x => decide (x = \u0393'.cons\u2097)) stack) (unrev q))           (some \u0393'.cons\u2097)))       (some \u0393'.cons\u2097)       (update         (update (update (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083)) stack L\u2083) rev           (List.reverseAux [] (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083) rev)))         rev         ((fun s => Option.iget ((fun x => some \u0393'.cons) s)) (some \u0393'.cons\u2097) ::           update (update (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083)) stack L\u2083) rev             (List.reverseAux [] (elim (trList L\u2081) [] [] (trList [] ++ \u0393'.cons\u2097 :: L\u2083) rev)) rev)))     { l := some q, var := none, stk := elim (trList (List.headI [] :: L\u2081)) [] [] L\u2083 } ** simp only [TM2.step, Option.mem_def, TM2.stepAux, ite_true, id_eq, trList, List.nil_append,\n  elim_update_stack, elim_rev, List.reverseAux_nil, elim_update_rev, Function.update_same,\n  List.headI_nil, trNat_default] ** case nil q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     { l := some (unrev q), var := some \u0393'.cons\u2097, stk := elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083 }     { l := some q, var := none, stk := elim (\u0393'.cons :: trList L\u2081) [] [] L\u2083 } ** convert unrev_ok using 2 ** case h.e'_2.h.e'_7 q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' \u22a2 elim (\u0393'.cons :: trList L\u2081) [] [] L\u2083 =     update (update (elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083) rev []) main       (List.reverseAux (elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083 rev)         (elim (trList L\u2081) [Option.iget (some \u0393'.cons)] [] L\u2083 main)) ** simp ** q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' \u22a2 stack \u2260 rev ** decide ** q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' \u22a2 \u2200 (x : \u0393'), x \u2208 [] \u2192 natEnd x = false ** rintro _ \u27e8\u27e9 ** case cons q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' a : \u2115 L\u2082 : List \u2115 \u22a2 Reaches\u2081 (TM2.step tr)     { l := some (head stack q), var := s, stk := elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) }     { l := some q, var := none, stk := elim (trList (List.headI (a :: L\u2082) :: L\u2081)) [] [] L\u2083 } ** refine'\n  TransGen.trans\n    (move_ok (by decide)\n      (splitAtPred_eq _ _ (trNat a) (some \u0393'.cons) (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083)\n        (trNat_natEnd _) \u27e8rfl, by simp\u27e9))\n    (TransGen.head rfl (TransGen.head rfl _)) ** case cons q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' a : \u2115 L\u2082 : List \u2115 \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     (TM2.stepAux       (tr         ((fun x =>             \u039b'.read fun s => ite (s = some \u0393'.cons\u2097) id (\u039b'.clear (fun x => decide (x = \u0393'.cons\u2097)) stack) (unrev q))           (some \u0393'.cons)))       (some \u0393'.cons)       (update         (update           (update (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083)) stack (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083)) rev           (List.reverseAux (trNat a) (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) rev)))         rev         ((fun s => Option.iget ((fun x => some \u0393'.cons) s)) (some \u0393'.cons) ::           update             (update (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083)) stack (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))             rev (List.reverseAux (trNat a) (elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) rev)) rev)))     { l := some q, var := none, stk := elim (trList (List.headI (a :: L\u2082) :: L\u2081)) [] [] L\u2083 } ** simp only [TM2.step, Option.mem_def, TM2.stepAux, ite_false, trList, List.append_assoc,\n  List.cons_append, elim_update_stack, elim_rev, elim_update_rev, Function.update_same,\n  List.headI_cons] ** case cons q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' a : \u2115 L\u2082 : List \u2115 \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     { l := some (\u039b'.clear (fun x => decide (x = \u0393'.cons\u2097)) stack (unrev q)), var := some \u0393'.cons,       stk :=         elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) [] (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083) }     { l := some q, var := none, stk := elim (trNat a ++ \u0393'.cons :: trList L\u2081) [] [] L\u2083 } ** refine'\n  TransGen.trans\n    (clear_ok\n      (splitAtPred_eq _ _ (trList L\u2082) (some \u0393'.cons\u2097) L\u2083\n        (fun x h => Bool.decide_false (trList_ne_cons\u2097 _ _ h)) \u27e8rfl, by simp\u27e9))\n    _ ** case cons q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' a : \u2115 L\u2082 : List \u2115 \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     { l := some (unrev q), var := some \u0393'.cons\u2097,       stk :=         update           (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []             (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))           stack L\u2083 }     { l := some q, var := none, stk := elim (trNat a ++ \u0393'.cons :: trList L\u2081) [] [] L\u2083 } ** convert unrev_ok using 2 ** case h.e'_2.h.e'_7 q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' a : \u2115 L\u2082 : List \u2115 \u22a2 elim (trNat a ++ \u0393'.cons :: trList L\u2081) [] [] L\u2083 =     update       (update         (update           (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []             (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))           stack L\u2083)         rev [])       main       (List.reverseAux         (update           (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []             (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))           stack L\u2083 rev)         (update           (elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) []             (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083))           stack L\u2083 main)) ** simp [List.reverseAux_eq] ** q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' a : \u2115 L\u2082 : List \u2115 \u22a2 stack \u2260 rev ** decide ** q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' a : \u2115 L\u2082 : List \u2115 \u22a2 elim (trList L\u2081) [] [] (trList (a :: L\u2082) ++ \u0393'.cons\u2097 :: L\u2083) stack =     trNat a ++ \u0393'.cons :: (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083) ** simp ** q : \u039b' s : Option \u0393' L\u2081 : List \u2115 L\u2083 : List \u0393' a : \u2115 L\u2082 : List \u2115 \u22a2 elim (trList L\u2081) (Option.iget (some \u0393'.cons) :: List.reverseAux (trNat a) []) [] (trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083) stack =     trList L\u2082 ++ \u0393'.cons\u2097 :: L\u2083 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image\u2082_left_identity ** \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t\u271d t' : Finset \u03b2 u u' : Finset \u03b3 a\u271d a' : \u03b1 b b' : \u03b2 c : \u03b3 f : \u03b1 \u2192 \u03b3 \u2192 \u03b3 a : \u03b1 h : \u2200 (b : \u03b3), f a b = b t : Finset \u03b3 \u22a2 \u2191(image\u2082 f {a} t) = \u2191t ** rw [coe_image\u2082, coe_singleton, Set.image2_left_identity h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_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 + 1) \u03c9 < N \u22a2 upperCrossingTime a b f M (n + 1) \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9 \u2227     lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 ** have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM\n  (lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).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 : \u03a9 \u2131 : Filtration \u2115 m0 M : \u2115 hNM : N \u2264 M h : upperCrossingTime a b f N (n + 1) \u03c9 < N this : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 \u22a2 upperCrossingTime a b f M (n + 1) \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9 \u2227     lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 ** refine' \u27e8_, this\u27e9 ** \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 + 1) \u03c9 < N this : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 \u22a2 upperCrossingTime a b f M (n + 1) \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9 ** rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this] ** \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 + 1) \u03c9 < N this : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 \u22a2 hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) N \u03c9 = hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) M \u03c9 ** refine' hitting_eq_hitting_of_exists hNM _ ** \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 + 1) \u03c9 < N this : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 \u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b ** rw [upperCrossingTime_succ_eq, hitting_lt_iff] at 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 M : \u2115 hNM : N \u2264 M h : \u2203 j, j \u2208 Set.Ico (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b this : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 \u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b  case 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 M : \u2115 hNM : N \u2264 M h : hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) N \u03c9 < N this : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 \u22a2 N \u2264 N ** obtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h ** case intro.intro \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 this : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 j : \u2115 hj\u2081 : j \u2208 Set.Ico (lowerCrossingTime a b f N n \u03c9) N hj\u2082 : f j \u03c9 \u2208 Set.Ici b \u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N n \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b  case 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 M : \u2115 hNM : N \u2264 M h : hitting f (Set.Ici b) (lowerCrossingTime a b f N n \u03c9) N \u03c9 < N this : lowerCrossingTime a b f M n \u03c9 = lowerCrossingTime a b f N n \u03c9 \u22a2 N \u2264 N ** exacts [\u27e8j, \u27e8hj\u2081.1, hj\u2081.2.le\u27e9, hj\u2082\u27e9, le_rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g \u22a2 \u2200 \u2983f : \u03b1 \u2192 F\u2984, Mem\u2112p f p \u2192 AEStronglyMeasurable' m f \u03bc \u2192 P f ** intro f hf hfm ** \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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc \u22a2 P f ** let f_Lp := hf.toLp f ** \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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f hf \u22a2 P f ** have hfm_Lp : AEStronglyMeasurable' m f_Lp \u03bc := hfm.congr hf.coeFn_toLp.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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc \u22a2 P f ** refine' h_ae hf.coeFn_toLp (Lp.mem\u2112p _) _ ** \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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc \u22a2 P \u2191\u2191(toLp f hf) ** change P f_Lp ** \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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc \u22a2 P \u2191\u2191f_Lp ** refine' Lp.induction_stronglyMeasurable hm hp_ne_top (P := fun f => P f) _ _ h_closed f_Lp hfm_Lp ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc \u22a2 \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     (fun f => P \u2191\u2191f) \u2191(Lp.simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) ** intro c s hs h\u03bcs ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc c : F s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 P \u2191\u2191\u2191(Lp.simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) ** rw [Lp.simpleFunc.coe_indicatorConst] ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc c : F s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 P \u2191\u2191(indicatorConstLp p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) ** refine' h_ae indicatorConstLp_coeFn.symm _ (h_ind c hs h\u03bcs) ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc c : F s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 Mem\u2112p (Set.indicator s fun x => c) p ** exact mem\u2112p_indicator_const p (hm s hs) c (Or.inr h\u03bcs.ne) ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f : \u03b1 \u2192 F hf : Mem\u2112p f p hfm : AEStronglyMeasurable' m f \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc \u22a2 \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           (fun f => P \u2191\u2191f) (toLp f hf) \u2192 (fun f => P \u2191\u2191f) (toLp g hg) \u2192 (fun f => P \u2191\u2191f) (toLp f hf + toLp g hg) ** intro f g hf_mem hg_mem hfm hgm h_disj hfP hgP ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f\u271d : \u03b1 \u2192 F hf : Mem\u2112p f\u271d p hfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc f g : \u03b1 \u2192 F hf_mem : Mem\u2112p f p hg_mem : Mem\u2112p g p hfm : StronglyMeasurable f hgm : StronglyMeasurable g h_disj : Disjoint (Function.support f) (Function.support g) hfP : P \u2191\u2191(toLp f hf_mem) hgP : P \u2191\u2191(toLp g hg_mem) \u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem) ** have hfP' : P f := h_ae hf_mem.coeFn_toLp (Lp.mem\u2112p _) hfP ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f\u271d : \u03b1 \u2192 F hf : Mem\u2112p f\u271d p hfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc f g : \u03b1 \u2192 F hf_mem : Mem\u2112p f p hg_mem : Mem\u2112p g p hfm : StronglyMeasurable f hgm : StronglyMeasurable g h_disj : Disjoint (Function.support f) (Function.support g) hfP : P \u2191\u2191(toLp f hf_mem) hgP : P \u2191\u2191(toLp g hg_mem) hfP' : P f \u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem) ** have hgP' : P g := h_ae hg_mem.coeFn_toLp (Lp.mem\u2112p _) hgP ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f\u271d : \u03b1 \u2192 F hf : Mem\u2112p f\u271d p hfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc f g : \u03b1 \u2192 F hf_mem : Mem\u2112p f p hg_mem : Mem\u2112p g p hfm : StronglyMeasurable f hgm : StronglyMeasurable g h_disj : Disjoint (Function.support f) (Function.support g) hfP : P \u2191\u2191(toLp f hf_mem) hgP : P \u2191\u2191(toLp g hg_mem) hfP' : P f hgP' : P g \u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem) ** specialize h_add h_disj hf_mem hg_mem hfm hgm hfP' hgP' ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f\u271d : \u03b1 \u2192 F hf : Mem\u2112p f\u271d p hfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc f g : \u03b1 \u2192 F hf_mem : Mem\u2112p f p hg_mem : Mem\u2112p g p hfm : StronglyMeasurable f hgm : StronglyMeasurable g h_disj : Disjoint (Function.support f) (Function.support g) hfP : P \u2191\u2191(toLp f hf_mem) hgP : P \u2191\u2191(toLp g hg_mem) hfP' : P f hgP' : P g h_add : P (f + g) \u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem) ** refine' h_ae _ (hf_mem.add hg_mem) h_add ** case 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\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 : (\u03b1 \u2192 F) \u2192 Prop h_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c) h_closed : IsClosed {f | P \u2191\u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g f\u271d : \u03b1 \u2192 F hf : Mem\u2112p f\u271d p hfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc f_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf hfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc f g : \u03b1 \u2192 F hf_mem : Mem\u2112p f p hg_mem : Mem\u2112p g p hfm : StronglyMeasurable f hgm : StronglyMeasurable g h_disj : Disjoint (Function.support f) (Function.support g) hfP : P \u2191\u2191(toLp f hf_mem) hgP : P \u2191\u2191(toLp g hg_mem) hfP' : P f hgP' : P g h_add : P (f + g) \u22a2 f + g =\u1d50[\u03bc] \u2191\u2191(toLp f hf_mem + toLp g hg_mem) ** exact (hf_mem.coeFn_toLp.symm.add hg_mem.coeFn_toLp.symm).trans (Lp.coeFn_add _ _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.restrict_const_lintegral ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : \u211d\u22650\u221e s : Set \u03b1 hs : MeasurableSet s \u22a2 lintegral (restrict (const \u03b1 c) s) \u03bc = c * \u2191\u2191\u03bc s ** rw [restrict_lintegral_eq_lintegral_restrict _ hs, const_lintegral_restrict] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.restrict_sInf_eq_sInf_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\u271d : 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 m0 : MeasurableSpace \u03b1 m : Set (Measure \u03b1) hm : Set.Nonempty m ht : MeasurableSet t \u22a2 restrict (sInf m) t = sInf ((fun \u03bc => restrict \u03bc t) '' m) ** ext1 s hs ** 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\u271d : 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 m0 : MeasurableSpace \u03b1 m : Set (Measure \u03b1) hm : Set.Nonempty m ht : MeasurableSet t s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(restrict (sInf m) t) s = \u2191\u2191(sInf ((fun \u03bc => restrict \u03bc t) '' m)) s ** simp_rw [sInf_apply hs, restrict_apply hs, sInf_apply (MeasurableSet.inter hs ht),\n  Set.image_image, restrict_toOuterMeasure_eq_toOuterMeasure_restrict ht, \u2190\n  Set.image_image _ toOuterMeasure, \u2190 OuterMeasure.restrict_sInf_eq_sInf_restrict _ (hm.image _),\n  OuterMeasure.restrict_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.exists_ne_of_one_lt_card ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 hs : 1 < card s a : \u03b1 \u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a ** obtain \u27e8x, hx, y, hy, hxy\u27e9 := Finset.one_lt_card.mp hs ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 hs : 1 < card s a x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x \u2260 y \u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a ** by_cases ha : y = a ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 hs : 1 < card s a x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x \u2260 y ha : y = a \u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a ** exact \u27e8x, hx, ne_of_ne_of_eq hxy ha\u27e9 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 hs : 1 < card s a x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x \u2260 y ha : \u00acy = a \u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a ** exact \u27e8y, hy, ha\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.laverage_mem_openSegment_compl_self ** \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 g : \u03b1 \u2192 \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hs : NullMeasurableSet s hs\u2080 : \u2191\u2191\u03bc s \u2260 0 hsc\u2080 : \u2191\u2191\u03bc s\u1d9c \u2260 0 \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc \u2208 openSegment \u211d\u22650\u221e (\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d\u207b (x : \u03b1) in s\u1d9c, f x \u2202\u03bc) ** simpa only [union_compl_self, restrict_univ] using\n  laverage_union_mem_openSegment aedisjoint_compl_right hs.compl hs\u2080 hsc\u2080 (measure_ne_top _ _)\n    (measure_ne_top _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "TypeVec.prod_map_id ** n : \u2115 \u03b1 \u03b2 : TypeVec.{u_1} n \u22a2 (id \u2297' id) = id ** ext i x : 2 ** case a.h n : \u2115 \u03b1 \u03b2 : TypeVec.{u_1} n i : Fin2 n x : (\u03b1 \u2297 \u03b2) i \u22a2 (id \u2297' id) i x = id i x ** induction i <;> simp only [TypeVec.prod.map, *, dropFun_id] ** case a.h.fz n n\u271d : \u2115 \u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d) x : (\u03b1 \u2297 \u03b2) Fin2.fz \u22a2 (id Fin2.fz x.fst, id Fin2.fz x.snd) = id Fin2.fz x  case a.h.fs n n\u271d : \u2115 a\u271d : Fin2 n\u271d a_ih\u271d : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (x : (\u03b1 \u2297 \u03b2) a\u271d), (id \u2297' id) a\u271d x = id a\u271d x \u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d) x : (\u03b1 \u2297 \u03b2) (Fin2.fs a\u271d) \u22a2 id a\u271d x = id (Fin2.fs a\u271d) x ** cases x ** case a.h.fz.mk n n\u271d : \u2115 \u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d) fst\u271d : last \u03b1 snd\u271d : last \u03b2 \u22a2 (id Fin2.fz (fst\u271d, snd\u271d).fst, id Fin2.fz (fst\u271d, snd\u271d).snd) = id Fin2.fz (fst\u271d, snd\u271d)  case a.h.fs n n\u271d : \u2115 a\u271d : Fin2 n\u271d a_ih\u271d : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (x : (\u03b1 \u2297 \u03b2) a\u271d), (id \u2297' id) a\u271d x = id a\u271d x \u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d) x : (\u03b1 \u2297 \u03b2) (Fin2.fs a\u271d) \u22a2 id a\u271d x = id (Fin2.fs a\u271d) x ** rfl ** case a.h.fs n n\u271d : \u2115 a\u271d : Fin2 n\u271d a_ih\u271d : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (x : (\u03b1 \u2297 \u03b2) a\u271d), (id \u2297' id) a\u271d x = id a\u271d x \u03b1 \u03b2 : TypeVec.{u_1} (Nat.succ n\u271d) x : (\u03b1 \u2297 \u03b2) (Fin2.fs a\u271d) \u22a2 id a\u271d x = id (Fin2.fs a\u271d) x ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.tr_respects ** 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 \u22a2 Respects (TM2.step M) (TM1.step (tr M)) TrCfg ** intro c\u2081 c\u2082 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 M : \u039b \u2192 Stmt\u2082 c\u2081 : Cfg\u2082 c\u2082 : TM1.Cfg \u0393' \u039b' \u03c3 h : TrCfg c\u2081 c\u2082 \u22a2 match TM2.step M c\u2081 with   | some b\u2081 => \u2203 b\u2082, TrCfg b\u2081 b\u2082 \u2227 Reaches\u2081 (TM1.step (tr M)) c\u2082 b\u2082   | none => TM1.step (tr M) c\u2082 = none ** cases' h with l v S L hT ** case mk 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 l : Option \u039b v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) \u22a2 match TM2.step M { l := l, var := v, stk := S } with   | some b\u2081 =>     \u2203 b\u2082,       TrCfg b\u2081 b\u2082 \u2227         Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal l, var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082   | none => TM1.step (tr M) { l := Option.map normal l, var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none ** cases' l with l ** case mk.some 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 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) l : \u039b \u22a2 match TM2.step M { l := some l, var := v, stk := S } with   | some b\u2081 =>     \u2203 b\u2082,       TrCfg b\u2081 b\u2082 \u2227         Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal (some l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082   | none => TM1.step (tr M) { l := Option.map normal (some l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none ** simp only [TM2.step, Respects, Option.map_some'] ** case mk.some 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 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) l : \u039b \u22a2 \u2203 b\u2082,     TrCfg (TM2.stepAux (M l) v S) b\u2082 \u2227       Reaches\u2081 (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082 ** rsuffices \u27e8b, c, r\u27e9 : \u2203 b, _ \u2227 Reaches (TM1.step (tr M)) _ _ ** 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 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) l : \u039b \u22a2 \u2203 b,     TrCfg (TM2.stepAux (M l) v S) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (tr M (normal l)) v (Tape.mk' \u2205 (addBottom L))) b ** simp only [tr] ** 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 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) l : \u039b \u22a2 \u2203 b,     TrCfg (TM2.stepAux (M l) v S) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (M l)) v (Tape.mk' \u2205 (addBottom L))) b ** generalize M l = N ** 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 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) l : \u039b N : Stmt\u2082 \u22a2 \u2203 b, TrCfg (TM2.stepAux N v S) b \u2227 Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal N) v (Tape.mk' \u2205 (addBottom L))) b ** induction N using stmtStRec generalizing v S L hT with\n| H\u2081 k s q IH => exact tr_respects_aux M hT s @IH\n| H\u2082 a _ IH => exact IH _ hT\n| H\u2083 p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n  unfold TM2.stepAux trNormal TM1.stepAux\n  simp only []\n  cases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT]\n| H\u2084 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n| H\u2085 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9 ** case mk.none 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 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) \u22a2 match TM2.step M { l := none, var := v, stk := S } with   | some b\u2081 =>     \u2203 b\u2082,       TrCfg b\u2081 b\u2082 \u2227         Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal none, var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082   | none => TM1.step (tr M) { l := Option.map normal none, var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none ** constructor ** case mk.some.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 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) l : \u039b b : ?m.732398 c : ?m.732726 b r : Reaches (TM1.step (tr M)) (?m.732727 b) (?m.732728 b) \u22a2 \u2203 b\u2082,     TrCfg (TM2.stepAux (M l) v S) b\u2082 \u2227       Reaches\u2081 (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082 ** exact \u27e8b, c, TransGen.head' rfl r\u27e9 ** case H\u2081 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 l : \u039b k : K s : StAct k q : Stmt\u2082 IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),     (\u2200 (k : K), ListBlank.map (proj k) L = 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 L))) b v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) \u22a2 \u2203 b,     TrCfg (TM2.stepAux (stRun s q) v S) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun s q)) v (Tape.mk' \u2205 (addBottom L))) b ** exact tr_respects_aux M hT s @IH ** case H\u2082 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 l : \u039b a : \u03c3 \u2192 \u03c3 q\u271d : Stmt\u2082 IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),     (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q\u271d v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u271d) v (Tape.mk' \u2205 (addBottom L))) b v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) \u22a2 \u2203 b,     TrCfg (TM2.stepAux (TM2.Stmt.load a q\u271d) v S) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.load a q\u271d)) v (Tape.mk' \u2205 (addBottom L))) b ** exact IH _ hT ** case H\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 l : \u039b p : \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2082 IH\u2081 :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),     (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q\u2081 v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b IH\u2082 :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),     (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q\u2082 v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) \u22a2 \u2203 b,     TrCfg (TM2.stepAux (TM2.Stmt.branch p q\u2081 q\u2082) v S) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.branch p q\u2081 q\u2082)) v (Tape.mk' \u2205 (addBottom L))) b ** unfold TM2.stepAux trNormal TM1.stepAux ** case H\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 l : \u039b p : \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2082 IH\u2081 :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),     (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q\u2081 v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b IH\u2082 :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),     (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q\u2082 v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) \u22a2 \u2203 b,     TrCfg (bif p v then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227       Reaches (TM1.step (tr M))         (bif (fun x => p) (Tape.mk' \u2205 (addBottom L)).head v then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))         else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))         b ** simp only [] ** case H\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 l : \u039b p : \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2082 IH\u2081 :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),     (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q\u2081 v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b IH\u2082 :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),     (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q\u2082 v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) \u22a2 \u2203 b,     TrCfg (bif p v then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227       Reaches (TM1.step (tr M))         (bif p v then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))         else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))         b ** cases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT] ** case H\u2084 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 l : \u039b l\u271d : \u03c3 \u2192 \u039b v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) \u22a2 \u2203 b,     TrCfg (TM2.stepAux (TM2.Stmt.goto l\u271d) v S) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.goto l\u271d)) v (Tape.mk' \u2205 (addBottom L))) b ** exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9 ** case H\u2085 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 l : \u039b v : \u03c3 S : (k : K) \u2192 List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) \u22a2 \u2203 b,     TrCfg (TM2.stepAux TM2.Stmt.halt v S) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal TM2.Stmt.halt) v (Tape.mk' \u2205 (addBottom L))) b ** exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.upperCrossingTime_eq_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 upperCrossingTime a b f N n \u03c9 = N ** refine' le_antisymm upperCrossingTime_le (not_lt.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 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hn : upcrossingsBefore a b f N \u03c9 < n \u22a2 \u00acupperCrossingTime a b f N n \u03c9 < N ** convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.mem\u2112p_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\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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 \u22a2 Mem\u2112p (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p ** refine' \u27e8(approxOn f fmeas s y\u2080 h\u2080 n).aestronglyMeasurable, _\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\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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 \u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \u22a4 ** suffices snorm (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc < \u22a4 by\n  have : Mem\u2112p (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc :=\n    \u27e8(approxOn f fmeas s y\u2080 h\u2080 n - const \u03b2 y\u2080).aestronglyMeasurable, this\u27e9\n  convert snorm_add_lt_top this hi\u2080\n  ext x\n  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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 \u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4 ** have hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p \u03bc := by\n  have h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 := by\n    simp only [\u2190 dist_eq_norm]\n    exact (continuous_id.dist continuous_const).measurable.comp fmeas\n  refine' \u27e8h_meas.aemeasurable.aestronglyMeasurable, _\u27e9\n  rw [snorm_norm]\n  convert snorm_add_lt_top hf hi\u2080.neg with x\n  simp [sub_eq_add_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\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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p \u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4 ** have : \u2200\u1d50 x \u2202\u03bc, \u2016approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 := by\n  refine' eventually_of_forall _\n  intro x\n  convert norm_approxOn_y\u2080_le fmeas h\u2080 x n using 1\n  rw [Real.norm_eq_abs, abs_of_nonneg]\n  exact add_nonneg (norm_nonneg _) (norm_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\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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p this : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 \u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4 ** calc\n  snorm (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc \u2264\n      snorm (fun x => \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016) p \u03bc :=\n    snorm_mono_ae this\n  _ < \u22a4 := snorm_add_lt_top hf' 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\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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 this : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4 \u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \u22a4 ** have : Mem\u2112p (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc :=\n  \u27e8(approxOn f fmeas s y\u2080 h\u2080 n - const \u03b2 y\u2080).aestronglyMeasurable, this\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\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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 this\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4 this : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \u22a4 ** convert snorm_add_lt_top this hi\u2080 ** case h.e'_3.h.e'_5 \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 this\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4 this : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u22a2 \u2191(approxOn f fmeas s y\u2080 h\u2080 n) = (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) + fun x => y\u2080 ** ext x ** case h.e'_3.h.e'_5.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 \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 this\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4 this : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p x : \u03b2 \u22a2 \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x = ((fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) + fun x => y\u2080) x ** 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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 \u22a2 Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p ** have h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 := by\n  simp only [\u2190 dist_eq_norm]\n  exact (continuous_id.dist continuous_const).measurable.comp fmeas ** \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 \u22a2 Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p ** refine' \u27e8h_meas.aemeasurable.aestronglyMeasurable, _\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\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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 \u22a2 snorm (fun x => \u2016f x - y\u2080\u2016) p \u03bc < \u22a4 ** rw [snorm_norm] ** \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 \u22a2 snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 ** convert snorm_add_lt_top hf hi\u2080.neg with x ** case h.e'_3.h.e'_5.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 \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 x : \u03b2 \u22a2 f x - y\u2080 = (f + -fun x => y\u2080) x ** simp [sub_eq_add_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\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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 \u22a2 Measurable fun x => \u2016f x - y\u2080\u2016 ** simp only [\u2190 dist_eq_norm] ** \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 \u22a2 Measurable fun x => dist (f x) y\u2080 ** exact (continuous_id.dist continuous_const).measurable.comp fmeas ** \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p \u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03bc, \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 ** refine' eventually_of_forall _ ** \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p \u22a2 \u2200 (x : \u03b2), \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 ** intro x ** \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p x : \u03b2 \u22a2 \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 ** convert norm_approxOn_y\u2080_le fmeas h\u2080 x n using 1 ** case h.e'_4 \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p x : \u03b2 \u22a2 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 = \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016 ** rw [Real.norm_eq_abs, abs_of_nonneg] ** case h.e'_4 \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 \u03bc : Measure \u03b2 fmeas : Measurable f hf : Mem\u2112p f p s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hi\u2080 : Mem\u2112p (fun x => y\u2080) p n : \u2115 hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p x : \u03b2 \u22a2 0 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016 ** exact add_nonneg (norm_nonneg _) (norm_nonneg _) ** Qed",
        "informal": ""
    },
    {
        "formal": "measurableSet_prod ** \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 s : Set \u03b1 t : Set \u03b2 \u22a2 MeasurableSet (s \u00d7\u02e2 t) \u2194 MeasurableSet s \u2227 MeasurableSet t \u2228 s = \u2205 \u2228 t = \u2205 ** cases' (s \u00d7\u02e2 t).eq_empty_or_nonempty with h h ** case inl \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 s : Set \u03b1 t : Set \u03b2 h : s \u00d7\u02e2 t = \u2205 \u22a2 MeasurableSet (s \u00d7\u02e2 t) \u2194 MeasurableSet s \u2227 MeasurableSet t \u2228 s = \u2205 \u2228 t = \u2205 ** simp [h, prod_eq_empty_iff.mp h] ** case inr \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 s : Set \u03b1 t : Set \u03b2 h : Set.Nonempty (s \u00d7\u02e2 t) \u22a2 MeasurableSet (s \u00d7\u02e2 t) \u2194 MeasurableSet s \u2227 MeasurableSet t \u2228 s = \u2205 \u2228 t = \u2205 ** simp [\u2190 not_nonempty_iff_eq_empty, prod_nonempty_iff.mp h, measurableSet_prod_of_nonempty h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Computable.map_decode_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 \u03b2 \u2192 \u03c3 \u22a2 (Computable\u2082 fun a n => Option.map (f a) (decode n)) \u2194 Computable\u2082 f ** convert (bind_decode_iff (f := fun a => Option.some \u2218 f a)).trans option_some_iff ** case h.e'_1.h.e'_7.h.h \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 \u2192 \u03c3 x\u271d\u00b9 : \u03b1 x\u271d : \u2115 \u22a2 Option.map (f x\u271d\u00b9) (decode x\u271d) = Option.bind (decode x\u271d) (Option.some \u2218 f x\u271d\u00b9) ** apply Option.map_eq_bind ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.inv_smul_integral_comp_sub_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 - b / c..d - a / c, f x ** by_cases hc : c = 0 <;> simp [hc, integral_comp_sub_div] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.bitwise_bit ** f : Bool \u2192 Bool \u2192 Bool a : Bool m : \u2124 b : Bool n : \u2124 \u22a2 bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n) ** cases' m with m m <;> cases' n with n n <;>\nsimp only [bitwise, ofNat_eq_coe, bit_coe_nat, natBitwise, Bool.not_false, Bool.not_eq_false',\n  bit_negSucc] ** case ofNat.ofNat f : Bool \u2192 Bool \u2192 Bool a b : Bool m n : \u2115 \u22a2 (bif f false false then -[Nat.bitwise (fun x y => !f x y) (Nat.bit a m) (Nat.bit b n)+1]     else \u2191(Nat.bitwise f (Nat.bit a m) (Nat.bit b n))) =     bit (f a b) (bif f false false then -[Nat.bitwise (fun x y => !f x y) m n+1] else \u2191(Nat.bitwise f m n)) ** by_cases h : f false false <;> simp [h] ** case ofNat.negSucc f : Bool \u2192 Bool \u2192 Bool a b : Bool m n : \u2115 \u22a2 (bif f false true then -[Nat.bitwise (fun x y => !f x !y) (Nat.bit a m) (Nat.bit (!b) n)+1]     else \u2191(Nat.bitwise (fun x y => f x !y) (Nat.bit a m) (Nat.bit (!b) n))) =     bit (f a b)       (bif f false true then -[Nat.bitwise (fun x y => !f x !y) m n+1] else \u2191(Nat.bitwise (fun x y => f x !y) m n)) ** by_cases h : f false true <;> simp [h] ** case negSucc.ofNat f : Bool \u2192 Bool \u2192 Bool a b : Bool m n : \u2115 \u22a2 (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) (Nat.bit (!a) m) (Nat.bit b n)+1]     else \u2191(Nat.bitwise (fun x y => f (!x) y) (Nat.bit (!a) m) (Nat.bit b n))) =     bit (f a b)       (bif f true false then -[Nat.bitwise (fun x y => !f (!x) y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) y) m n)) ** by_cases h : f true false <;> simp [h] ** case negSucc.negSucc f : Bool \u2192 Bool \u2192 Bool a b : Bool m n : \u2115 \u22a2 (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n)+1]     else \u2191(Nat.bitwise (fun x y => f (!x) !y) (Nat.bit (!a) m) (Nat.bit (!b) n))) =     bit (f a b)       (bif f true true then -[Nat.bitwise (fun x y => !f (!x) !y) m n+1] else \u2191(Nat.bitwise (fun x y => f (!x) !y) m n)) ** by_cases h : f true true <;> simp [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.restrict_iUnion_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\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 \u22a2 restrict \u03bc (\u22c3 i, s i) = restrict \u03bd (\u22c3 i, s i) \u2194 \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i) ** refine' \u27e8fun h i => restrict_congr_mono (subset_iUnion _ _) h, fun h => _\u27e9 ** \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\u271d s' t : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 h : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i) \u22a2 restrict \u03bc (\u22c3 i, s i) = restrict \u03bd (\u22c3 i, s i) ** ext1 t ht ** 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\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 h : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i) t : Set \u03b1 ht : MeasurableSet t D : Directed (fun x x_1 => x \u2286 x_1) fun t => \u22c3 i \u2208 t, s i \u22a2 \u2191\u2191(restrict \u03bc (\u22c3 i, s i)) t = \u2191\u2191(restrict \u03bd (\u22c3 i, s i)) t ** rw [iUnion_eq_iUnion_finset] ** 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\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 h : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i) t : Set \u03b1 ht : MeasurableSet t D : Directed (fun x x_1 => x \u2286 x_1) fun t => \u22c3 i \u2208 t, s i \u22a2 \u2191\u2191(restrict \u03bc (\u22c3 t, \u22c3 i \u2208 t, s i)) t = \u2191\u2191(restrict \u03bd (\u22c3 t, \u22c3 i \u2208 t, s i)) t ** simp only [restrict_iUnion_apply_eq_iSup D ht, restrict_finset_biUnion_congr.2 fun i _ => h i] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_univ ** \u03b1\u271d : Type ?u.175153 s t : Set \u03b1\u271d \u03b1 : Type u_1 \u22a2 ncard univ = Nat.card \u03b1 ** cases' finite_or_infinite \u03b1 with h h ** case inr \u03b1\u271d : Type ?u.175153 s t : Set \u03b1\u271d \u03b1 : Type u_1 h : Infinite \u03b1 \u22a2 ncard univ = Nat.card \u03b1 ** rw [Nat.card_eq_zero_of_infinite, Infinite.ncard] ** case inr \u03b1\u271d : Type ?u.175153 s t : Set \u03b1\u271d \u03b1 : Type u_1 h : Infinite \u03b1 \u22a2 Set.Infinite univ ** exact infinite_univ ** case inl \u03b1\u271d : Type ?u.175153 s t : Set \u03b1\u271d \u03b1 : Type u_1 h : Finite \u03b1 \u22a2 ncard univ = Nat.card \u03b1 ** have hft := Fintype.ofFinite \u03b1 ** case inl \u03b1\u271d : Type ?u.175153 s t : Set \u03b1\u271d \u03b1 : Type u_1 h : Finite \u03b1 hft : Fintype \u03b1 \u22a2 ncard univ = Nat.card \u03b1 ** rw [ncard_eq_toFinset_card, Finite.toFinset_univ, Finset.card_univ, Nat.card_eq_fintype_card] ** Qed",
        "informal": ""
    },
    {
        "formal": "FinEnum.Finset.mem_enum ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 xs : List \u03b1 \u22a2 s \u2208 enum xs \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 xs ** induction' xs with xs_hd generalizing s <;> simp [*, Finset.enum] ** case nil \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d s : Finset \u03b1 \u22a2 s = \u2205 \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 False ** simp [Finset.eq_empty_iff_forall_not_mem] ** case cons \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 \u22a2 (\u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d ** constructor ** case cons.mp \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 \u22a2 (\u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a)) \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d ** rintro \u27e8a, h, h'\u27e9 x hx ** case cons.mp.intro.intro \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s a : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d h' : s = a \u2228 s = {xs_hd} \u222a a x : \u03b1 hx : x \u2208 s \u22a2 x = xs_hd \u2228 x \u2208 tail\u271d ** cases' h' with _ h' a b ** case cons.mp.intro.intro.inl \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s a : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d x : \u03b1 hx : x \u2208 s h\u271d : s = a \u22a2 x = xs_hd \u2228 x \u2208 tail\u271d ** right ** case cons.mp.intro.intro.inl.h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s a : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d x : \u03b1 hx : x \u2208 s h\u271d : s = a \u22a2 x \u2208 tail\u271d ** apply h ** case cons.mp.intro.intro.inl.h.a \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s a : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d x : \u03b1 hx : x \u2208 s h\u271d : s = a \u22a2 x \u2208 a ** subst a ** case cons.mp.intro.intro.inl.h.a \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 x : \u03b1 hx : x \u2208 s h : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d \u22a2 x \u2208 s ** exact hx ** case cons.mp.intro.intro.inr \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s a : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d x : \u03b1 hx : x \u2208 s h' : s = {xs_hd} \u222a a \u22a2 x = xs_hd \u2228 x \u2208 tail\u271d ** simp only [h', mem_union, mem_singleton] at hx \u22a2 ** case cons.mp.intro.intro.inr \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s a : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d x : \u03b1 h' : s = {xs_hd} \u222a a hx : x = xs_hd \u2228 x \u2208 a \u22a2 x = xs_hd \u2228 x \u2208 tail\u271d ** cases' hx with hx hx' ** case cons.mp.intro.intro.inr.inl \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s a : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d x : \u03b1 h' : s = {xs_hd} \u222a a hx : x = xs_hd \u22a2 x = xs_hd \u2228 x \u2208 tail\u271d ** exact Or.inl hx ** case cons.mp.intro.intro.inr.inr \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s a : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d x : \u03b1 h' : s = {xs_hd} \u222a a hx' : x \u2208 a \u22a2 x = xs_hd \u2228 x \u2208 tail\u271d ** exact Or.inr (h _ hx') ** case cons.mpr \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 \u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d) \u2192 \u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a) ** intro h ** case cons.mpr \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d \u22a2 \u2203 a, (\u2200 (x : \u03b1), x \u2208 a \u2192 x \u2208 tail\u271d) \u2227 (s = a \u2228 s = {xs_hd} \u222a a) ** exists s \\ ({xs_hd} : Finset \u03b1) ** case cons.mpr \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d \u22a2 (\u2200 (x : \u03b1), x \u2208 s \\ {xs_hd} \u2192 x \u2208 tail\u271d) \u2227 (s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}) ** simp only [and_imp, mem_sdiff, mem_singleton] ** case cons.mpr \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 x = xs_hd \u2228 x \u2208 tail\u271d \u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d) \u2227 (s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}) ** simp only [or_iff_not_imp_left] at h ** case cons.mpr \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d \u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d) \u2227 (s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd}) ** exists h ** case cons.mpr \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d \u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd} ** by_cases h : xs_hd \u2208 s ** case pos \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d h : xs_hd \u2208 s \u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd} ** have : {xs_hd} \u2286 s := by\n  simp only [HasSubset.Subset, *, forall_eq, mem_singleton] ** case pos \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d h : xs_hd \u2208 s this : {xs_hd} \u2286 s \u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd} ** simp only [union_sdiff_of_subset this, or_true_iff, Finset.union_sdiff_of_subset,\n  eq_self_iff_true] ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d h : xs_hd \u2208 s \u22a2 {xs_hd} \u2286 s ** simp only [HasSubset.Subset, *, forall_eq, mem_singleton] ** case neg \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d h : \u00acxs_hd \u2208 s \u22a2 s = s \\ {xs_hd} \u2228 s = {xs_hd} \u222a s \\ {xs_hd} ** left ** case neg.h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d h : \u00acxs_hd \u2208 s \u22a2 s = s \\ {xs_hd} ** symm ** case neg.h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d h : \u00acxs_hd \u2208 s \u22a2 s \\ {xs_hd} = s ** simp only [sdiff_eq_self] ** case neg.h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d h : \u00acxs_hd \u2208 s \u22a2 s \u2229 {xs_hd} \u2286 \u2205 ** intro a ** case neg.h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d h : \u00acxs_hd \u2208 s a : \u03b1 \u22a2 a \u2208 s \u2229 {xs_hd} \u2192 a \u2208 \u2205 ** simp only [and_imp, mem_inter, mem_singleton] ** case neg.h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 xs_hd : \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = xs_hd \u2192 x \u2208 tail\u271d h : \u00acxs_hd \u2208 s a : \u03b1 \u22a2 a \u2208 s \u2192 a = xs_hd \u2192 a \u2208 \u2205 ** rintro h\u2080 rfl ** case neg.h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u271d : Finset \u03b1 tail\u271d : List \u03b1 tail_ih\u271d : \u2200 (s : Finset \u03b1), s \u2208 enum tail\u271d \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 tail\u271d s : Finset \u03b1 a : \u03b1 h\u2080 : a \u2208 s h\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 x \u2208 tail\u271d h : \u00aca \u2208 s \u22a2 a \u2208 \u2205 ** exact (h h\u2080).elim ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.mapAccumr_bisim ** \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 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2 ** induction xs using Vector.revInductionOn generalizing s\u2081 s\u2082 ** case nil \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 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop 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 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr f\u2081 nil s\u2081).1 (mapAccumr f\u2082 nil s\u2082).1 \u2227 (mapAccumr f\u2081 nil s\u2081).2 = (mapAccumr f\u2082 nil s\u2082).2  case snoc \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 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop 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 n\u271d : \u2115 xs\u271d : Vector \u03b1 n\u271d x\u271d : \u03b1 a\u271d :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs\u271d s\u2081).1 (mapAccumr f\u2082 xs\u271d s\u2082).1 \u2227 (mapAccumr f\u2081 xs\u271d s\u2081).2 = (mapAccumr f\u2082 xs\u271d s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).1 (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).1 \u2227     (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).2 = (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).2 ** next => exact \u27e8h\u2080, rfl\u27e9 ** case snoc \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 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop 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 n\u271d : \u2115 xs\u271d : Vector \u03b1 n\u271d x\u271d : \u03b1 a\u271d :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs\u271d s\u2081).1 (mapAccumr f\u2082 xs\u271d s\u2082).1 \u2227 (mapAccumr f\u2081 xs\u271d s\u2081).2 = (mapAccumr f\u2082 xs\u271d s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).1 (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).1 \u2227     (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).2 = (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).2 ** next xs x ih =>\n  rcases (hR x h\u2080) with \u27e8hR, _\u27e9\n  simp only [mapAccumr_snoc, ih hR, true_and]\n  congr 1 ** \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 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop 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 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr f\u2081 nil s\u2081).1 (mapAccumr f\u2082 nil s\u2082).1 \u2227 (mapAccumr f\u2081 nil s\u2081).2 = (mapAccumr f\u2082 nil s\u2082).2 ** exact \u27e8h\u2080, rfl\u27e9 ** \u03b1 : Type u_2 n : \u2115 xs\u271d : 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 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop 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 n\u271d : \u2115 xs : Vector \u03b1 n\u271d x : \u03b1 ih :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr f\u2081 (snoc xs x) s\u2081).1 (mapAccumr f\u2082 (snoc xs x) s\u2082).1 \u2227     (mapAccumr f\u2081 (snoc xs x) s\u2081).2 = (mapAccumr f\u2082 (snoc xs x) s\u2082).2 ** rcases (hR x h\u2080) with \u27e8hR, _\u27e9 ** case intro \u03b1 : Type u_2 n : \u2115 xs\u271d : 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 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop hR\u271d : \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 n\u271d : \u2115 xs : Vector \u03b1 n\u271d x : \u03b1 ih :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 hR : R (f\u2081 x s\u2081).1 (f\u2082 x s\u2082).1 right\u271d : (f\u2081 x s\u2081).2 = (f\u2082 x s\u2082).2 \u22a2 R (mapAccumr f\u2081 (snoc xs x) s\u2081).1 (mapAccumr f\u2082 (snoc xs x) s\u2082).1 \u2227     (mapAccumr f\u2081 (snoc xs x) s\u2081).2 = (mapAccumr f\u2082 (snoc xs x) s\u2082).2 ** simp only [mapAccumr_snoc, ih hR, true_and] ** case intro \u03b1 : Type u_2 n : \u2115 xs\u271d : 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 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop hR\u271d : \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 n\u271d : \u2115 xs : Vector \u03b1 n\u271d x : \u03b1 ih :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).1 (mapAccumr f\u2082 xs s\u2082).1 \u2227 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 hR : R (f\u2081 x s\u2081).1 (f\u2082 x s\u2082).1 right\u271d : (f\u2081 x s\u2081).2 = (f\u2082 x s\u2082).2 \u22a2 snoc (mapAccumr f\u2082 xs (f\u2082 x s\u2082).1).2 (f\u2081 x s\u2081).2 = snoc (mapAccumr f\u2082 xs (f\u2082 x s\u2082).1).2 (f\u2082 x s\u2082).2 ** congr 1 ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.pos_iff_ne_zero ** n : Nat a : Fin (n + 1) \u22a2 0 < a \u2194 a \u2260 0 ** rw [lt_def, val_zero, Nat.pos_iff_ne_zero, \u2190 val_ne_iff] ** n : Nat a : Fin (n + 1) \u22a2 \u2191a \u2260 0 \u2194 \u2191a \u2260 \u21910 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.iIndepSets.indepSets ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j \u22a2 IndepSets (s i) (s j) \u03ba ** intro t\u2081 t\u2082 ht\u2081 ht\u2082 ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082 ** have h1 : t\u2081 = ite (i = i) t\u2081 t\u2082 := by simp only [if_true, eq_self_iff_true] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082 ** have h2 : t\u2082 = ite (j = i) t\u2081 t\u2082 := by simp only [hij.symm, if_false] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 h2 : t\u2082 = if j = i then t\u2081 else t\u2082 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082 ** have h_inter : \u22c2 (t : \u03b9) (_ : t \u2208 ({i, j} : Finset \u03b9)), ite (t = i) t\u2081 t\u2082 =\n    ite (i = i) t\u2081 t\u2082 \u2229 ite (j = i) t\u2081 t\u2082 := by\n  simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 h2 : t\u2082 = if j = i then t\u2081 else t\u2082 h_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082 ** filter_upwards [h_indep {i, j} hf_m] with a h_indep' ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 h2 : t\u2082 = if j = i then t\u2081 else t\u2082 h_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082 a : \u03b1 h_indep' :   \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082) \u22a2 \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082 ** have h_prod : (\u220f t : \u03b9 in ({i, j} : Finset \u03b9), \u03ba a (ite (t = i) t\u2081 t\u2082))\n    = \u03ba a (ite (i = i) t\u2081 t\u2082) * \u03ba a (ite (j = i) t\u2081 t\u2082) := by\n  simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff,\n    Finset.mem_singleton] ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 h2 : t\u2082 = if j = i then t\u2081 else t\u2082 h_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082 a : \u03b1 h_indep' :   \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082) h_prod :   \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =     \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082) \u22a2 \u2191\u2191(\u2191\u03ba a) (t\u2081 \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) t\u2081 * \u2191\u2191(\u2191\u03ba a) t\u2082 ** rw [h1] ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 h2 : t\u2082 = if j = i then t\u2081 else t\u2082 h_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082 a : \u03b1 h_indep' :   \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082) h_prod :   \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =     \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082) \u22a2 \u2191\u2191(\u2191\u03ba a) ((if i = i then t\u2081 else t\u2082) \u2229 t\u2082) = \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) t\u2082 ** nth_rw 2 [h2] ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 h2 : t\u2082 = if j = i then t\u2081 else t\u2082 h_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082 a : \u03b1 h_indep' :   \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082) h_prod :   \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =     \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082) \u22a2 \u2191\u2191(\u2191\u03ba a) ((if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082) = \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) t\u2082 ** nth_rw 4 [h2] ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 h2 : t\u2082 = if j = i then t\u2081 else t\u2082 h_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082 a : \u03b1 h_indep' :   \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082) h_prod :   \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =     \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082) \u22a2 \u2191\u2191(\u2191\u03ba a) ((if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082) =     \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082) ** rw [\u2190 h_inter, \u2190 h_prod, h_indep'] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j \u22a2 \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x ** intro x hx ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j x : \u03b9 hx : x \u2208 {i, j} \u22a2 (if x = i then t\u2081 else t\u2082) \u2208 s x ** cases' Finset.mem_insert.mp hx with hx hx ** case inl \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j x : \u03b9 hx\u271d : x \u2208 {i, j} hx : x = i \u22a2 (if x = i then t\u2081 else t\u2082) \u2208 s x ** simp [hx, ht\u2081] ** case inr \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j x : \u03b9 hx\u271d : x \u2208 {i, j} hx : x \u2208 {j} \u22a2 (if x = i then t\u2081 else t\u2082) \u2208 s x ** simp [Finset.mem_singleton.mp hx, hij.symm, ht\u2082] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x \u22a2 t\u2081 = if i = i then t\u2081 else t\u2082 ** simp only [if_true, eq_self_iff_true] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 \u22a2 t\u2082 = if j = i then t\u2081 else t\u2082 ** simp only [hij.symm, if_false] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 h2 : t\u2082 = if j = i then t\u2081 else t\u2082 \u22a2 (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082 ** simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 s : \u03b9 \u2192 Set (Set \u03a9) _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 h_indep : iIndepSets s \u03ba i j : \u03b9 hij : i \u2260 j t\u2081 t\u2082 : Set \u03a9 ht\u2081 : t\u2081 \u2208 s i ht\u2082 : t\u2082 \u2208 s j hf_m : \u2200 (x : \u03b9), x \u2208 {i, j} \u2192 (if x = i then t\u2081 else t\u2082) \u2208 s x h1 : t\u2081 = if i = i then t\u2081 else t\u2082 h2 : t\u2082 = if j = i then t\u2081 else t\u2082 h_inter : (\u22c2 t \u2208 {i, j}, if t = i then t\u2081 else t\u2082) = (if i = i then t\u2081 else t\u2082) \u2229 if j = i then t\u2081 else t\u2082 a : \u03b1 h_indep' :   \u2191\u2191(\u2191\u03ba a) (\u22c2 i_1 \u2208 {i, j}, if i_1 = i then t\u2081 else t\u2082) = \u220f i_1 in {i, j}, \u2191\u2191(\u2191\u03ba a) (if i_1 = i then t\u2081 else t\u2082) \u22a2 \u220f t in {i, j}, \u2191\u2191(\u2191\u03ba a) (if t = i then t\u2081 else t\u2082) =     \u2191\u2191(\u2191\u03ba a) (if i = i then t\u2081 else t\u2082) * \u2191\u2191(\u2191\u03ba a) (if j = i then t\u2081 else t\u2082) ** simp only [hij, Finset.prod_singleton, Finset.prod_insert, not_false_iff,\n  Finset.mem_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.setToL1_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\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \u211d F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : CompleteSpace F T\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C' C'' : \u211d G' : Type u_7 G'' : Type u_8 inst\u271d\u2074 : NormedLatticeAddCommGroup G'' inst\u271d\u00b3 : NormedSpace \u211d G'' inst\u271d\u00b2 : CompleteSpace G'' inst\u271d\u00b9 : NormedLatticeAddCommGroup G' inst\u271d : NormedSpace \u211d G' T : Set \u03b1 \u2192 G' \u2192L[\u211d] G'' C : \u211d hT : DominatedFinMeasAdditive \u03bc T C 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 Lp G' 1 } hf : 0 \u2264 f \u22a2 0 \u2264 \u2191(setToL1 hT) f ** suffices : \u2200 f : { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g }, 0 \u2264 setToL1 hT 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\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \u211d F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : CompleteSpace F T\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C' C'' : \u211d G' : Type u_7 G'' : Type u_8 inst\u271d\u2074 : NormedLatticeAddCommGroup G'' inst\u271d\u00b3 : NormedSpace \u211d G'' inst\u271d\u00b2 : CompleteSpace G'' inst\u271d\u00b9 : NormedLatticeAddCommGroup G' inst\u271d : NormedSpace \u211d G' T : Set \u03b1 \u2192 G' \u2192L[\u211d] G'' C : \u211d hT : DominatedFinMeasAdditive \u03bc T C 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 Lp G' 1 } hf : 0 \u2264 f this : \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f \u22a2 0 \u2264 \u2191(setToL1 hT) f  case 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\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \u211d F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : CompleteSpace F T\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C' C'' : \u211d G' : Type u_7 G'' : Type u_8 inst\u271d\u2074 : NormedLatticeAddCommGroup G'' inst\u271d\u00b3 : NormedSpace \u211d G'' inst\u271d\u00b2 : CompleteSpace G'' inst\u271d\u00b9 : NormedLatticeAddCommGroup G' inst\u271d : NormedSpace \u211d G' T : Set \u03b1 \u2192 G' \u2192L[\u211d] G'' C : \u211d hT : DominatedFinMeasAdditive \u03bc T C 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 Lp G' 1 } hf : 0 \u2264 f \u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f ** exact this (\u27e8f, hf\u27e9 : { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g }) ** case 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\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \u211d F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : CompleteSpace F T\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C' C'' : \u211d G' : Type u_7 G'' : Type u_8 inst\u271d\u2074 : NormedLatticeAddCommGroup G'' inst\u271d\u00b3 : NormedSpace \u211d G'' inst\u271d\u00b2 : CompleteSpace G'' inst\u271d\u00b9 : NormedLatticeAddCommGroup G' inst\u271d : NormedSpace \u211d G' T : Set \u03b1 \u2192 G' \u2192L[\u211d] G'' C : \u211d hT : DominatedFinMeasAdditive \u03bc T C 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 Lp G' 1 } hf : 0 \u2264 f \u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f ** refine' fun g =>\n  @isClosed_property { g : \u03b1 \u2192\u2081\u209b[\u03bc] G' // 0 \u2264 g } { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g } _ _\n    (fun g => 0 \u2264 setToL1 hT g)\n    (denseRange_coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' one_ne_top) _ _ g ** case this.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\u00b9\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \u211d F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : CompleteSpace F T\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C' C'' : \u211d G' : Type u_7 G'' : Type u_8 inst\u271d\u2074 : NormedLatticeAddCommGroup G'' inst\u271d\u00b3 : NormedSpace \u211d G'' inst\u271d\u00b2 : CompleteSpace G'' inst\u271d\u00b9 : NormedLatticeAddCommGroup G' inst\u271d : NormedSpace \u211d G' T : Set \u03b1 \u2192 G' \u2192L[\u211d] G'' C : \u211d hT : DominatedFinMeasAdditive \u03bc T C 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 Lp G' 1 } hf : 0 \u2264 f g : { g // 0 \u2264 g } \u22a2 IsClosed {x | (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) x} ** exact isClosed_le continuous_zero ((setToL1 hT).continuous.comp continuous_induced_dom) ** case this.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\u00b9\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \u211d F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : CompleteSpace F T\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C' C'' : \u211d G' : Type u_7 G'' : Type u_8 inst\u271d\u2074 : NormedLatticeAddCommGroup G'' inst\u271d\u00b3 : NormedSpace \u211d G'' inst\u271d\u00b2 : CompleteSpace G'' inst\u271d\u00b9 : NormedLatticeAddCommGroup G' inst\u271d : NormedSpace \u211d G' T : Set \u03b1 \u2192 G' \u2192L[\u211d] G'' C : \u211d hT : DominatedFinMeasAdditive \u03bc T C 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 Lp G' 1 } hf : 0 \u2264 f g : { g // 0 \u2264 g } \u22a2 \u2200 (a : { g // 0 \u2264 g }), (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) (coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' a) ** intro g ** case this.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\u00b9\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \u211d F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : CompleteSpace F T\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C' C'' : \u211d G' : Type u_7 G'' : Type u_8 inst\u271d\u2074 : NormedLatticeAddCommGroup G'' inst\u271d\u00b3 : NormedSpace \u211d G'' inst\u271d\u00b2 : CompleteSpace G'' inst\u271d\u00b9 : NormedLatticeAddCommGroup G' inst\u271d : NormedSpace \u211d G' T : Set \u03b1 \u2192 G' \u2192L[\u211d] G'' C : \u211d hT : DominatedFinMeasAdditive \u03bc T C 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 Lp G' 1 } hf : 0 \u2264 f g\u271d : { g // 0 \u2264 g } g : { g // 0 \u2264 g } \u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) ** have : (coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g : \u03b1 \u2192\u2081[\u03bc] G') = (g : \u03b1 \u2192\u2081\u209b[\u03bc] G') := rfl ** case this.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\u00b9\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \u211d F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : CompleteSpace F T\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C' C'' : \u211d G' : Type u_7 G'' : Type u_8 inst\u271d\u2074 : NormedLatticeAddCommGroup G'' inst\u271d\u00b3 : NormedSpace \u211d G'' inst\u271d\u00b2 : CompleteSpace G'' inst\u271d\u00b9 : NormedLatticeAddCommGroup G' inst\u271d : NormedSpace \u211d G' T : Set \u03b1 \u2192 G' \u2192L[\u211d] G'' C : \u211d hT : DominatedFinMeasAdditive \u03bc T C 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 Lp G' 1 } hf : 0 \u2264 f g\u271d : { g // 0 \u2264 g } g : { g // 0 \u2264 g } this : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g \u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) ** rw [this, setToL1_eq_setToL1SCLM] ** case this.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\u00b9\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \u211d F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : CompleteSpace F T\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C' C'' : \u211d G' : Type u_7 G'' : Type u_8 inst\u271d\u2074 : NormedLatticeAddCommGroup G'' inst\u271d\u00b3 : NormedSpace \u211d G'' inst\u271d\u00b2 : CompleteSpace G'' inst\u271d\u00b9 : NormedLatticeAddCommGroup G' inst\u271d : NormedSpace \u211d G' T : Set \u03b1 \u2192 G' \u2192L[\u211d] G'' C : \u211d hT : DominatedFinMeasAdditive \u03bc T C 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 Lp G' 1 } hf : 0 \u2264 f g\u271d : { g // 0 \u2264 g } g : { g // 0 \u2264 g } this : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g \u22a2 0 \u2264 \u2191(setToL1SCLM \u03b1 G' \u03bc hT) \u2191g ** exact setToL1S_nonneg (fun s => hT.eq_zero_of_measure_zero) hT.1 hT_nonneg g.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "UInt8.toChar_aux ** n : \u2115 h : n < size \u22a2 Nat.isValidChar \u2191(UInt32.ofNat n).val ** rw [UInt32.val_eq_of_lt] ** n : \u2115 h : n < size \u22a2 Nat.isValidChar n  n : \u2115 h : n < size \u22a2 n < UInt32.size ** exact Or.inl $ Nat.lt_trans h $ by decide ** n : \u2115 h : n < size \u22a2 n < UInt32.size ** exact Nat.lt_trans h $ by decide ** n : \u2115 h : n < size \u22a2 size < 55296 ** decide ** n : \u2115 h : n < size \u22a2 size < UInt32.size ** decide ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.measure_le_le_exp_cgf ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : t \u2264 0 h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | X \u03c9 \u2264 \u03b5}) \u2264 rexp (-t * \u03b5 + cgf X \u03bc t) ** refine' (measure_le_le_exp_mul_mgf \u03b5 ht h_int).trans _ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : t \u2264 0 h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) \u22a2 rexp (-t * \u03b5) * mgf (fun \u03c9 => X \u03c9) \u03bc t \u2264 rexp (-t * \u03b5 + cgf X \u03bc t) ** rw [exp_add] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : t \u2264 0 h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) \u22a2 rexp (-t * \u03b5) * mgf (fun \u03c9 => X \u03c9) \u03bc t \u2264 rexp (-t * \u03b5) * rexp (cgf X \u03bc t) ** exact mul_le_mul le_rfl (le_exp_log _) mgf_nonneg (exp_pos _).le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.stoppedValue_stoppedValue_leastGE ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c0 : \u03a9 \u2192 \u2115 r : \u211d n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n \u22a2 stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 = stoppedValue (stoppedProcess f (leastGE f r n)) \u03c0 ** ext1 \u03c9 ** case h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c0 : \u03a9 \u2192 \u2115 r : \u211d n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n \u03c9 : \u03a9 \u22a2 stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 \u03c9 = stoppedValue (stoppedProcess f (leastGE f r n)) \u03c0 \u03c9 ** simp_rw [stoppedProcess, stoppedValue] ** case h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c0 : \u03a9 \u2192 \u2115 r : \u211d n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n \u03c9 : \u03a9 \u22a2 f (leastGE f r (\u03c0 \u03c9) \u03c9) \u03c9 = f (min (\u03c0 \u03c9) (leastGE f r n \u03c9)) \u03c9 ** rw [leastGE_eq_min _ _ _ h\u03c0n] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_eq_zero ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 encard s = 0 \u2194 s = \u2205 ** rw [encard, \u2190PartENat.withTopEquiv.symm.injective.eq_iff, Equiv.symm_apply_apply,\n  PartENat.withTopEquiv_symm_zero, PartENat.card_eq_zero_iff_empty, isEmpty_subtype,\n  eq_empty_iff_forall_not_mem] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.rec_computable ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf \u22a2 let PR := fun a cf cg hf hg => pr a (cf, cg, hf, hg);   let CO := fun a cf cg hf hg => co a (cf, cg, hf, hg);   let PC := fun a cf cg hf hg => pc a (cf, cg, hf, hg);   let RF := fun a cf hf => rf a (cf, hf);   let F := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a);   Computable fun a => F a (c a) ** intros _ _ _ _ F ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) \u22a2 Computable fun a => F a (c a) ** let G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p =>\n  let a := p.1.1\n  let IH := p.1.2\n  let n := p.2.1\n  let m := p.2.2\n  (IH.get? m).bind fun s =>\n    (IH.get? m.unpair.1).bind fun s\u2081 =>\n      (IH.get? m.unpair.2).map fun s\u2082 =>\n        cond n.bodd\n          (cond n.div2.bodd (rf a (ofNat Code m, s))\n            (pc a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082)))\n          (cond n.div2.bodd (co a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082))\n            (pr a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082))) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) \u22a2 Computable fun a => F a (c a) ** have : Computable G\u2081 := by\n  refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n  unfold Computable\u2082\n  refine'\n    option_bind\n      ((list_get?.comp (snd.comp fst)\n        (fst.comp <| Computable.unpair.comp (snd.comp snd))).comp fst) _\n  unfold Computable\u2082\n  refine'\n    option_map\n      ((list_get?.comp (snd.comp fst)\n        (snd.comp <| Computable.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _\n  have a : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n    fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n  have n : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n    fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n    snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m\u2081 := fst.comp (Computable.unpair.comp m)\n  have m\u2082 := snd.comp (Computable.unpair.comp m)\n  have s : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) :=\n    snd.comp (fst.comp fst)\n  have s\u2081 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) :=\n    snd.comp fst\n  have s\u2082 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) :=\n    snd\n  exact\n    (nat_bodd.comp n).cond\n      ((nat_bodd.comp <| nat_div2.comp n).cond\n        (hrf.comp a (((Computable.ofNat Code).comp m).pair s))\n        (hpc.comp a\n          (((Computable.ofNat Code).comp m\u2081).pair <|\n            ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n      (Computable.cond (nat_bodd.comp <| nat_div2.comp n)\n        (hco.comp a\n          (((Computable.ofNat Code).comp m\u2081).pair <|\n            ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n        (hpr.comp a\n          (((Computable.ofNat Code).comp m\u2081).pair <|\n            ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this : Computable G\u2081 \u22a2 Computable fun a => F a (c a) ** let G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH =>\n  IH.length.casesOn (some (z a)) fun n =>\n    n.casesOn (some (s a)) fun n =>\n      n.casesOn (some (l a)) fun n =>\n        n.casesOn (some (r a)) fun n => G\u2081 ((a, IH), n, n.div2.div2) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Computable fun a => F a (c a) ** have : Computable\u2082 G :=\n  Computable.nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) <|\n    Computable.nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) <|\n      Computable.nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) <|\n        Computable.nat_casesOn snd\n          (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst)))\n          (this.comp <|\n            ((Computable.fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n              snd.pair <| nat_div2.comp <| nat_div2.comp snd) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G \u22a2 Computable fun a => F a (c a) ** refine'\n  ((nat_strong_rec (fun a n => F a (ofNat Code n)) this.to\u2082 fun a n => _).comp Computable.id <|\n    encode_iff.2 hc).of_eq fun a => by simp ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 \u22a2 G (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).1       (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).2 =     some ((fun a n => F a (ofNat Code n)) a n) ** simp (config := { zeta := false }) ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 \u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) =     some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) ** simp only [] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 \u22a2 Nat.rec (some (z a))       (fun n_1 n_ih =>         Nat.rec (some (s a))           (fun n_2 n_ih =>             Nat.rec (some (l a))               (fun n_3 n_ih =>                 Nat.rec (some (r a))                   (fun n_4 n_ih =>                     Option.bind                       (List.get?                         (List.map                           (fun n =>                             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))                               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))                               (fun cf hf => rf a (cf, hf)) (ofNat Code n))                           (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))                         (div2 (div2 n_4)))                       fun s_1 =>                       Option.bind                         (List.get?                           (List.map                             (fun n =>                               rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))                                 (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))                                 (fun cf hf => rf a (cf, hf)) (ofNat Code n))                             (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))                           (unpair (div2 (div2 n_4))).1)                         fun s\u2081 =>                         Option.map                           (fun s\u2082 =>                             bif bodd n_4 then                               bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)                               else                                 pc a                                   (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,                                     s\u2082)                             else                               bif bodd (div2 n_4) then                                 co a                                   (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,                                     s\u2082)                               else                                 pr a                                   (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,                                     s\u2082))                           (List.get?                             (List.map                               (fun n =>                                 rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))                                   (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))                                   (fun cf hf => rf a (cf, hf)) (ofNat Code n))                               (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))                             (unpair (div2 (div2 n_4))).2))                   n_3)               n_2)           n_1)       (List.length         (List.map           (fun n =>             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code n))           (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))) =     some       (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))         (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))         (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) ** rw [List.length_map, List.length_range] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 \u22a2 Nat.rec (some (z a))       (fun n_1 n_ih =>         Nat.rec (some (s a))           (fun n_2 n_ih =>             Nat.rec (some (l a))               (fun n_3 n_ih =>                 Nat.rec (some (r a))                   (fun n_4 n_ih =>                     Option.bind                       (List.get?                         (List.map                           (fun n =>                             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))                               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))                               (fun cf hf => rf a (cf, hf)) (ofNat Code n))                           (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))                         (div2 (div2 n_4)))                       fun s_1 =>                       Option.bind                         (List.get?                           (List.map                             (fun n =>                               rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))                                 (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))                                 (fun cf hf => rf a (cf, hf)) (ofNat Code n))                             (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))                           (unpair (div2 (div2 n_4))).1)                         fun s\u2081 =>                         Option.map                           (fun s\u2082 =>                             bif bodd n_4 then                               bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)                               else                                 pc a                                   (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,                                     s\u2082)                             else                               bif bodd (div2 n_4) then                                 co a                                   (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,                                     s\u2082)                               else                                 pr a                                   (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,                                     s\u2082))                           (List.get?                             (List.map                               (fun n =>                                 rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))                                   (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))                                   (fun cf hf => rf a (cf, hf)) (ofNat Code n))                               (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))                             (unpair (div2 (div2 n_4))).2))                   n_3)               n_2)           n_1)       (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =     some       (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))         (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))         (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) ** let m := n.div2.div2 ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 Nat.rec (some (z a))       (fun n_1 n_ih =>         Nat.rec (some (s a))           (fun n_2 n_ih =>             Nat.rec (some (l a))               (fun n_3 n_ih =>                 Nat.rec (some (r a))                   (fun n_4 n_ih =>                     Option.bind                       (List.get?                         (List.map                           (fun n =>                             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))                               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))                               (fun cf hf => rf a (cf, hf)) (ofNat Code n))                           (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))                         (div2 (div2 n_4)))                       fun s_1 =>                       Option.bind                         (List.get?                           (List.map                             (fun n =>                               rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))                                 (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))                                 (fun cf hf => rf a (cf, hf)) (ofNat Code n))                             (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))                           (unpair (div2 (div2 n_4))).1)                         fun s\u2081 =>                         Option.map                           (fun s\u2082 =>                             bif bodd n_4 then                               bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)                               else                                 pc a                                   (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,                                     s\u2082)                             else                               bif bodd (div2 n_4) then                                 co a                                   (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,                                     s\u2082)                               else                                 pr a                                   (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081,                                     s\u2082))                           (List.get?                             (List.map                               (fun n =>                                 rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))                                   (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))                                   (fun cf hf => rf a (cf, hf)) (ofNat Code n))                               (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))                             (unpair (div2 (div2 n_4))).2))                   n_3)               n_2)           n_1)       (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =     some       (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))         (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))         (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) ** show\n  G\u2081 ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m) =\n    some (F a (ofNat Code (n + 4))) ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (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 _ _)) ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 \u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4))) ** have m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 \u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4))) ** have m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 m2 : (unpair m).2 < n + 4 \u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4))) ** simp [List.get?_map, List.get?_range, hm, m1, m2] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 m2 : (unpair m).2 < n + 4 \u22a2 (bif bodd n then       bif bodd (div2 n) then         rf a           (ofNat Code (div2 (div2 n)),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))       else         pc a           (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))     else       bif bodd (div2 n) then         co a           (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))       else         pr a           (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) =     rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))       (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (n + 4)) ** rw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 m2 : (unpair m).2 < n + 4 \u22a2 (bif bodd n then       bif bodd (div2 n) then         rf a           (ofNat Code (div2 (div2 n)),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))       else         pc a           (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))     else       bif bodd (div2 n) then         co a           (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))       else         pr a           (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) =     rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))       (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNatCode (n + 4)) ** simp [ofNatCode] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 m2 : (unpair m).2 < n + 4 \u22a2 (bif bodd n then       bif bodd (div2 n) then         rf a           (ofNat Code (div2 (div2 n)),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))       else         pc a           (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))     else       bif bodd (div2 n) then         co a           (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))       else         pr a           (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2,             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1),             rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))               (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))               (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) =     rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))       (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))       (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)))) ** cases n.bodd <;> cases n.div2.bodd <;> rfl ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) \u22a2 Computable G\u2081 ** refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) \u22a2 Computable\u2082 fun p s =>     Option.bind (List.get? p.1.2 (unpair p.2.2).1) fun s\u2081 =>       Option.map         (fun s\u2082 =>           bif bodd p.2.1 then             bif bodd (div2 p.2.1) then rf p.1.1 (ofNat Code p.2.2, s)             else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)           else             bif bodd (div2 p.2.1) then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)             else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082))         (List.get? p.1.2 (unpair p.2.2).2) ** unfold Computable\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) \u22a2 Computable fun p =>     (fun p s =>         Option.bind (List.get? p.1.2 (unpair p.2.2).1) fun s\u2081 =>           Option.map             (fun s\u2082 =>               bif bodd p.2.1 then                 bif bodd (div2 p.2.1) then rf p.1.1 (ofNat Code p.2.2, s)                 else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)               else                 bif bodd (div2 p.2.1) then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)                 else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082))             (List.get? p.1.2 (unpair p.2.2).2))       p.1 p.2 ** refine'\n  option_bind\n    ((list_get?.comp (snd.comp fst)\n      (fst.comp <| Computable.unpair.comp (snd.comp snd))).comp fst) _ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) \u22a2 Computable\u2082 fun p s\u2081 =>     Option.map       (fun s\u2082 =>         bif bodd p.1.2.1 then           bif bodd (div2 p.1.2.1) then rf p.1.1.1 (ofNat Code p.1.2.2, p.2)           else pc p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)         else           bif bodd (div2 p.1.2.1) then co p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)           else pr p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082))       (List.get? p.1.1.2 (unpair p.1.2.2).2) ** unfold Computable\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) \u22a2 Computable fun p =>     (fun p s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd p.1.2.1 then               bif bodd (div2 p.1.2.1) then rf p.1.1.1 (ofNat Code p.1.2.2, p.2)               else pc p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)             else               bif bodd (div2 p.1.2.1) then                 co p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)               else pr p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082))           (List.get? p.1.1.2 (unpair p.1.2.2).2))       p.1 p.2 ** refine'\n  option_map\n    ((list_get?.comp (snd.comp fst)\n      (snd.comp <| Computable.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) \u22a2 Computable\u2082 fun p s\u2082 =>     bif bodd p.1.1.2.1 then       bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)       else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)     else       bif bodd (div2 p.1.1.2.1) then         co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)       else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have a : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n  fst.comp (fst.comp <| fst.comp <| fst.comp fst) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) a : Computable fun p => p.1.1.1.1.1 \u22a2 Computable\u2082 fun p s\u2082 =>     bif bodd p.1.1.2.1 then       bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)       else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)     else       bif bodd (div2 p.1.1.2.1) then         co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)       else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have n : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n  fst.comp (snd.comp <| fst.comp <| fst.comp fst) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) a : Computable fun p => p.1.1.1.1.1 n : Computable fun p => p.1.1.1.2.1 \u22a2 Computable\u2082 fun p s\u2082 =>     bif bodd p.1.1.2.1 then       bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)       else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)     else       bif bodd (div2 p.1.1.2.1) then         co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)       else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have m : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n  snd.comp (snd.comp <| fst.comp <| fst.comp fst) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) a : Computable fun p => p.1.1.1.1.1 n : Computable fun p => p.1.1.1.2.1 m : Computable fun p => p.1.1.1.2.2 \u22a2 Computable\u2082 fun p s\u2082 =>     bif bodd p.1.1.2.1 then       bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)       else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)     else       bif bodd (div2 p.1.1.2.1) then         co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)       else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have m\u2081 := fst.comp (Computable.unpair.comp m) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) a : Computable fun p => p.1.1.1.1.1 n : Computable fun p => p.1.1.1.2.1 m : Computable fun p => p.1.1.1.2.2 m\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1 \u22a2 Computable\u2082 fun p s\u2082 =>     bif bodd p.1.1.2.1 then       bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)       else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)     else       bif bodd (div2 p.1.1.2.1) then         co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)       else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have m\u2082 := snd.comp (Computable.unpair.comp m) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) a : Computable fun p => p.1.1.1.1.1 n : Computable fun p => p.1.1.1.2.1 m : Computable fun p => p.1.1.1.2.2 m\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1 m\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2 \u22a2 Computable\u2082 fun p s\u2082 =>     bif bodd p.1.1.2.1 then       bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)       else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)     else       bif bodd (div2 p.1.1.2.1) then         co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)       else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have s : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) :=\n  snd.comp (fst.comp fst) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s\u271d : \u03b1 \u2192 \u03c3 hs : Computable s\u271d l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) a : Computable fun p => p.1.1.1.1.1 n : Computable fun p => p.1.1.1.2.1 m : Computable fun p => p.1.1.1.2.2 m\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1 m\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2 s : Computable fun p => p.1.1.2 \u22a2 Computable\u2082 fun p s\u2082 =>     bif bodd p.1.1.2.1 then       bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)       else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)     else       bif bodd (div2 p.1.1.2.1) then         co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)       else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have s\u2081 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) :=\n  snd.comp fst ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s\u271d : \u03b1 \u2192 \u03c3 hs : Computable s\u271d l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) a : Computable fun p => p.1.1.1.1.1 n : Computable fun p => p.1.1.1.2.1 m : Computable fun p => p.1.1.1.2.2 m\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1 m\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2 s : Computable fun p => p.1.1.2 s\u2081 : Computable fun p => p.1.2 \u22a2 Computable\u2082 fun p s\u2082 =>     bif bodd p.1.1.2.1 then       bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)       else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)     else       bif bodd (div2 p.1.1.2.1) then         co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)       else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have s\u2082 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) :=\n  snd ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s\u271d : \u03b1 \u2192 \u03c3 hs : Computable s\u271d l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) a : Computable fun p => p.1.1.1.1.1 n : Computable fun p => p.1.1.1.2.1 m : Computable fun p => p.1.1.1.2.2 m\u2081 : Computable fun a => (unpair a.1.1.1.2.2).1 m\u2082 : Computable fun a => (unpair a.1.1.1.2.2).2 s : Computable fun p => p.1.1.2 s\u2081 : Computable fun p => p.1.2 s\u2082 : Computable fun p => p.2 \u22a2 Computable\u2082 fun p s\u2082 =>     bif bodd p.1.1.2.1 then       bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2)       else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)     else       bif bodd (div2 p.1.1.2.1) then         co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)       else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** exact\n  (nat_bodd.comp n).cond\n    ((nat_bodd.comp <| nat_div2.comp n).cond\n      (hrf.comp a (((Computable.ofNat Code).comp m).pair s))\n      (hpc.comp a\n        (((Computable.ofNat Code).comp m\u2081).pair <|\n          ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n    (Computable.cond (nat_bodd.comp <| nat_div2.comp n)\n      (hco.comp a\n        (((Computable.ofNat Code).comp m\u2081).pair <|\n          ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n      (hpr.comp a\n        (((Computable.ofNat Code).comp m\u2081).pair <|\n          ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 \u22a2 F (id a) (ofNat Code (encode (c a))) = F a (c a) ** simp ** case succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 \u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ n))))) =     some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ n))))) ** cases' n with n ** case succ.succ.succ.zero \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 \u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) =     some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) ** simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode] ** case succ.succ.succ.zero \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 \u22a2 G a (List.map (fun n => F a (ofNatCode n)) (List.range (Nat.succ (Nat.succ (Nat.succ 0))))) = some (F a right) ** rfl ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 m < n + 4 ** simp only [div2_val] ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Computable c z : \u03b1 \u2192 \u03c3 hz : Computable z s : \u03b1 \u2192 \u03c3 hs : Computable s l : \u03b1 \u2192 \u03c3 hl : Computable l r : \u03b1 \u2192 \u03c3 hr : Computable r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Computable\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Computable\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Computable\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Computable\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=   fun p =>     let a := p.1.1;     let IH := p.1.2;     let n := p.2.1;     let m := p.2.2;     Option.bind (List.get? IH m) fun s =>       Option.bind (List.get? IH (unpair m).1) fun s\u2081 =>         Option.map           (fun s\u2082 =>             bif bodd n then               bif bodd (div2 n) then rf a (ofNat Code m, s)               else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)             else               bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)               else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082))           (List.get? IH (unpair m).2) this\u271d : Computable G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=   fun a IH =>     Nat.casesOn (List.length IH) (some (z a)) fun n =>       Nat.casesOn n (some (s a)) fun n =>         Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Computable\u2082 G a : \u03b1 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.FiniteMeasure.self_eq_mass_smul_normalize ** \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u22a2 \u03bc = mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc) ** apply eq_of_forall_apply_eq ** case h \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u22a2 \u2200 (s : Set \u03a9),     MeasurableSet s \u2192       (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s =         (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) s)) s ** intro s _s_mble ** case h \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 s : Set \u03a9 _s_mble : MeasurableSet s \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s =     (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) s)) s ** rw [\u03bc.self_eq_mass_mul_normalize s, coeFn_smul_apply, smul_eq_mul,\n  ProbabilityMeasure.coeFn_comp_toFiniteMeasure_eq_coeFn] ** Qed",
        "informal": ""
    },
    {
        "formal": "LipschitzWith.norm_compLp_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 g : E \u2192 F c : \u211d\u22650 hg : LipschitzWith c g g0 : g 0 = 0 f f' : { x // x \u2208 Lp E p } \u22a2 \u2016compLp hg g0 f - compLp hg g0 f'\u2016 \u2264 \u2191c * \u2016f - f'\u2016 ** apply Lp.norm_le_mul_norm_of_ae_le_mul ** 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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 hg : LipschitzWith c g g0 : g 0 = 0 f f' : { x // x \u2208 Lp E p } \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191(compLp hg g0 f - compLp hg g0 f') x\u2016 \u2264 \u2191c * \u2016\u2191\u2191(f - f') x\u2016 ** filter_upwards [hg.coeFn_compLp g0 f, hg.coeFn_compLp g0 f',\n  Lp.coeFn_sub (hg.compLp g0 f) (hg.compLp g0 f'), Lp.coeFn_sub f f'] with a ha1 ha2 ha3 ha4 ** 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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 hg : LipschitzWith c g g0 : g 0 = 0 f f' : { x // x \u2208 Lp E p } a : \u03b1 ha1 : \u2191\u2191(compLp hg g0 f) a = (g \u2218 \u2191\u2191f) a ha2 : \u2191\u2191(compLp hg g0 f') a = (g \u2218 \u2191\u2191f') a ha3 : \u2191\u2191(compLp hg g0 f - compLp hg g0 f') a = (\u2191\u2191(compLp hg g0 f) - \u2191\u2191(compLp hg g0 f')) a ha4 : \u2191\u2191(f - f') a = (\u2191\u2191f - \u2191\u2191f') a \u22a2 \u2016\u2191\u2191(compLp hg g0 f - compLp hg g0 f') a\u2016 \u2264 \u2191c * \u2016\u2191\u2191(f - f') a\u2016 ** simp only [ha1, ha2, ha3, ha4, \u2190 dist_eq_norm, Pi.sub_apply, Function.comp_apply] ** 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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 hg : LipschitzWith c g g0 : g 0 = 0 f f' : { x // x \u2208 Lp E p } a : \u03b1 ha1 : \u2191\u2191(compLp hg g0 f) a = (g \u2218 \u2191\u2191f) a ha2 : \u2191\u2191(compLp hg g0 f') a = (g \u2218 \u2191\u2191f') a ha3 : \u2191\u2191(compLp hg g0 f - compLp hg g0 f') a = (\u2191\u2191(compLp hg g0 f) - \u2191\u2191(compLp hg g0 f')) a ha4 : \u2191\u2191(f - f') a = (\u2191\u2191f - \u2191\u2191f') a \u22a2 dist (g (\u2191\u2191f a)) (g (\u2191\u2191f' a)) \u2264 \u2191c * dist (\u2191\u2191f a) (\u2191\u2191f' a) ** exact hg.dist_le_mul (f a) (f' a) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_mul_Ioi_of_neg ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a c : \u03b1 h : c < 0 \u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Ioi a = Iio (a / c) ** simpa only [mul_comm] using preimage_mul_const_Ioi_of_neg a h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_not_mem_null_le_lintegral ** \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\u22650\u221e inst\u271d : IsProbabilityMeasure \u03bc hf : AEMeasurable f hN : \u2191\u2191\u03bc N = 0 \u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc ** simpa only [laverage_eq_lintegral] using\n  exists_not_mem_null_le_laverage (IsProbabilityMeasure.ne_zero \u03bc) hf hN ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_fintype_prod_pi ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b3 : CommMonoid \u03b1 inst\u271d\u00b2 : CommMonoid \u03b2 inst\u271d\u00b9 : MonoidHomClass F \u03b1 \u03b2 inst\u271d : Fintype \u03b9 S : \u03b9 \u2192 Set \u03b1 \u22a2 (fun f => \u220f i : \u03b9, f i) '' pi univ S = \u220f i : \u03b9, S i ** simpa only [Finset.coe_univ] using image_finset_prod_pi Finset.univ S ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Icc_ssubset_Icc_left ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 hI : a\u2082 \u2264 b\u2082 ha : a\u2082 < a\u2081 hb : b\u2081 \u2264 b\u2082 \u22a2 Icc a\u2081 b\u2081 \u2282 Icc a\u2082 b\u2082 ** rw [\u2190 coe_ssubset, coe_Icc, coe_Icc] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 hI : a\u2082 \u2264 b\u2082 ha : a\u2082 < a\u2081 hb : b\u2081 \u2264 b\u2082 \u22a2 Set.Icc a\u2081 b\u2081 \u2282 Set.Icc a\u2082 b\u2082 ** exact Set.Icc_ssubset_Icc_left hI ha hb ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.tr_supports ** 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 S : Finset \u039b ss : TM2.Supports M S l' : \u039b' h : l' \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l') ** suffices \u2200 (q) (_ : TM2.SupportsStmt S q) (_ : \u2200 x \u2208 trStmts\u2081 q, x \u2208 trSupp M S),\n    TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n    \u2200 l' \u2208 trStmts\u2081 q, TM1.SupportsStmt (trSupp M S) (tr M l') by\n  rcases Finset.mem_biUnion.1 h with \u27e8l, lS, h\u27e9\n  have :=\n    this _ (ss.2 l lS) fun x hx \u21a6 Finset.mem_biUnion.2 \u27e8_, lS, Finset.mem_insert_of_mem hx\u27e9\n  rcases Finset.mem_insert.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ 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 M : \u039b \u2192 Stmt\u2082 S : Finset \u039b ss : TM2.Supports M S l' : \u039b' h : l' \u2208 trSupp M S \u22a2 \u2200 (q : Stmt\u2082),     TM2.SupportsStmt S q \u2192       (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192         TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227           \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** clear h 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 M : \u039b \u2192 Stmt\u2082 S : Finset \u039b ss : TM2.Supports M S \u22a2 \u2200 (q : Stmt\u2082),     TM2.SupportsStmt S q \u2192       (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192         TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227           \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** refine' stmtStRec _ _ _ _ _ ** 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 S : Finset \u039b ss : TM2.Supports M S l' : \u039b' h : l' \u2208 trSupp M S this :   \u2200 (q : Stmt\u2082),     TM2.SupportsStmt S q \u2192       (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192         TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227           \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l') ** rcases Finset.mem_biUnion.1 h with \u27e8l, lS, h\u27e9 ** 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 S : Finset \u039b ss : TM2.Supports M S l' : \u039b' h\u271d : l' \u2208 trSupp M S this :   \u2200 (q : Stmt\u2082),     TM2.SupportsStmt S q \u2192       (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192         TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227           \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') l : \u039b lS : l \u2208 S h : l' \u2208 insert (normal l) (trStmts\u2081 (M l)) \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l') ** have :=\n  this _ (ss.2 l lS) fun x hx \u21a6 Finset.mem_biUnion.2 \u27e8_, lS, Finset.mem_insert_of_mem hx\u27e9 ** 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 S : Finset \u039b ss : TM2.Supports M S l' : \u039b' h\u271d : l' \u2208 trSupp M S this\u271d :   \u2200 (q : Stmt\u2082),     TM2.SupportsStmt S q \u2192       (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192         TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227           \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') l : \u039b lS : l \u2208 S h : l' \u2208 insert (normal l) (trStmts\u2081 (M l)) this :   TM1.SupportsStmt (trSupp M S) (trNormal (M l)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (M l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l') ** rcases Finset.mem_insert.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ h] ** case refine'_1 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 S : Finset \u039b ss : TM2.Supports M S \u22a2 \u2200 (k : K) (s : StAct k) (q : Stmt\u2082),     (TM2.SupportsStmt S q \u2192         (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192           TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227             \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192       TM2.SupportsStmt S (stRun s q) \u2192         (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q) \u2192 x \u2208 trSupp M S) \u2192           TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q)) \u2227             \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** intro _ s _ IH ss' sub ** case refine'_1 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S (stRun s q\u271d) sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** rw [TM2to1.supports_run] at ss' ** case refine'_1 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** simp only [TM2to1.trStmts\u2081_run, Finset.mem_union, Finset.mem_insert, Finset.mem_singleton]\n  at sub ** case refine'_1 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** have hgo := sub _ (Or.inl <| Or.inl rfl) ** case refine'_1 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** have hret := sub _ (Or.inl <| Or.inr rfl) ** case refine'_1 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** cases' IH ss' fun x hx \u21a6 sub x <| Or.inr hx with IH\u2081 IH\u2082 ** case refine'_1.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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** refine' \u27e8by simp only [trNormal_run, TM1.SupportsStmt]; intros; exact hgo, fun l h \u21a6 _\u27e9 ** case refine'_1.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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') l : \u039b' h : l \u2208 trStmts\u2081 (stRun s q\u271d) \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l) ** rw [trStmts\u2081_run] at h ** case refine'_1.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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') l : \u039b' h : l \u2208 {go k\u271d s q\u271d, ret q\u271d} \u222a trStmts\u2081 q\u271d \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l) ** simp only [TM2to1.trStmts\u2081_run, Finset.mem_union, Finset.mem_insert, Finset.mem_singleton]\n  at h ** case refine'_1.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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') l : \u039b' h : (l = go k\u271d s q\u271d \u2228 l = ret q\u271d) \u2228 l \u2208 trStmts\u2081 q\u271d \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l) ** rcases h with (\u27e8rfl | rfl\u27e9 | 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 M : \u039b \u2192 Stmt\u2082 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) ** simp only [trNormal_run, TM1.SupportsStmt] ** 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 \u0393' \u2192 \u03c3 \u2192 go k\u271d s q\u271d \u2208 trSupp M S ** intros ** 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') a\u271d : \u0393' v\u271d : \u03c3 \u22a2 go k\u271d s q\u271d \u2208 trSupp M S ** exact hgo ** case refine'_1.intro.inl.inl 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d s q\u271d)) ** cases s ** case refine'_1.intro.inl.inl.push 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') a\u271d : \u03c3 \u2192 \u0393 k\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.push a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d (StAct.push a\u271d) q\u271d \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.push a\u271d) q\u271d)) ** exact \u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hgo\u27e9 ** case refine'_1.intro.inl.inl.peek 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') a\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3 sub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.peek a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d (StAct.peek a\u271d) q\u271d \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.peek a\u271d) q\u271d)) ** exact \u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hgo\u27e9 ** case refine'_1.intro.inl.inl.pop 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') a\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3 sub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.pop a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d (StAct.pop a\u271d) q\u271d \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.pop a\u271d) q\u271d)) ** exact \u27e8\u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hret\u27e9, fun _ _ \u21a6 hgo\u27e9 ** case refine'_1.intro.inl.inr 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 TM1.SupportsStmt (trSupp M S) (tr M (ret q\u271d)) ** unfold TM1.SupportsStmt TM2to1.tr ** case refine'_1.intro.inl.inr 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 match     match ret q\u271d with     | normal q => trNormal (M q)     | go k s q =>       branch (fun a x => Option.isNone (a.2 k)) (trStAct (goto fun x x => ret q) s)         (move Dir.right (goto fun x x => go k s q))     | ret q => branch (fun a x => a.1) (trNormal q) (move Dir.left (goto fun x x => ret q)) with   | move a q => TM1.SupportsStmt (trSupp M S) q   | write a q => TM1.SupportsStmt (trSupp M S) q   | load a q => TM1.SupportsStmt (trSupp M S) q   | branch a q\u2081 q\u2082 => TM1.SupportsStmt (trSupp M S) q\u2081 \u2227 TM1.SupportsStmt (trSupp M S) q\u2082   | goto l => \u2200 (a : \u0393') (v : \u03c3), l a v \u2208 trSupp M S   | halt => True ** exact \u27e8IH\u2081, fun _ _ \u21a6 hret\u27e9 ** case refine'_1.intro.inr 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 S : Finset \u039b ss : TM2.Supports M S k\u271d : K s : StAct k\u271d q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S q\u271d sub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S hgo : go k\u271d s q\u271d \u2208 trSupp M S hret : ret q\u271d \u2208 trSupp M S IH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) IH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') l : \u039b' h : l \u2208 trStmts\u2081 q\u271d \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l) ** exact IH\u2082 _ h ** case refine'_2 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 S : Finset \u039b ss : TM2.Supports M S \u22a2 \u2200 (a : \u03c3 \u2192 \u03c3) (q : Stmt\u2082),     (TM2.SupportsStmt S q \u2192         (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192           TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227             \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192       TM2.SupportsStmt S (TM2.Stmt.load a q) \u2192         (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.load a q) \u2192 x \u2208 trSupp M S) \u2192           TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a q)) \u2227             \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.load a q) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** intro _ _ IH ss' sub ** case refine'_2 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 S : Finset \u039b ss : TM2.Supports M S a\u271d : \u03c3 \u2192 \u03c3 q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S (TM2.Stmt.load a\u271d q\u271d) sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.load a\u271d q\u271d) \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a\u271d q\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.load a\u271d q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** unfold TM2to1.trStmts\u2081 at ss' sub \u22a2 ** case refine'_2 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 S : Finset \u039b ss : TM2.Supports M S a\u271d : \u03c3 \u2192 \u03c3 q\u271d : Stmt\u2082 IH :   TM2.SupportsStmt S q\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S (TM2.Stmt.load a\u271d q\u271d) sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a\u271d q\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** exact IH ss' sub ** case refine'_3 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 S : Finset \u039b ss : TM2.Supports M S \u22a2 \u2200 (p : \u03c3 \u2192 Bool) (q\u2081 q\u2082 : Stmt\u2082),     (TM2.SupportsStmt S q\u2081 \u2192         (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081 \u2192 x \u2208 trSupp M S) \u2192           TM1.SupportsStmt (trSupp M S) (trNormal q\u2081) \u2227             \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081 \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192       (TM2.SupportsStmt S q\u2082 \u2192           (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082 \u2192 x \u2208 trSupp M S) \u2192             TM1.SupportsStmt (trSupp M S) (trNormal q\u2082) \u2227               \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082 \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192         TM2.SupportsStmt S (TM2.Stmt.branch p q\u2081 q\u2082) \u2192           (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.branch p q\u2081 q\u2082) \u2192 x \u2208 trSupp M S) \u2192             TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p q\u2081 q\u2082)) \u2227               \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p q\u2081 q\u2082) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** intro _ _ _ IH\u2081 IH\u2082 ss' sub ** case refine'_3 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 S : Finset \u039b ss : TM2.Supports M S p\u271d : \u03c3 \u2192 Bool q\u2081\u271d q\u2082\u271d : Stmt\u2082 IH\u2081 :   TM2.SupportsStmt S q\u2081\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') IH\u2082 :   TM2.SupportsStmt S q\u2082\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** unfold TM2to1.trStmts\u2081 at sub ** case refine'_3 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 S : Finset \u039b ss : TM2.Supports M S p\u271d : \u03c3 \u2192 Bool q\u2081\u271d q\u2082\u271d : Stmt\u2082 IH\u2081 :   TM2.SupportsStmt S q\u2081\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') IH\u2082 :   TM2.SupportsStmt S q\u2082\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** cases' IH\u2081 ss'.1 fun x hx \u21a6 sub x <| Finset.mem_union_left _ hx with IH\u2081\u2081 IH\u2081\u2082 ** case refine'_3.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 S : Finset \u039b ss : TM2.Supports M S p\u271d : \u03c3 \u2192 Bool q\u2081\u271d q\u2082\u271d : Stmt\u2082 IH\u2081 :   TM2.SupportsStmt S q\u2081\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') IH\u2082 :   TM2.SupportsStmt S q\u2082\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S IH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) IH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** cases' IH\u2082 ss'.2 fun x hx \u21a6 sub x <| Finset.mem_union_right _ hx with IH\u2082\u2081 IH\u2082\u2082 ** case refine'_3.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 S : Finset \u039b ss : TM2.Supports M S p\u271d : \u03c3 \u2192 Bool q\u2081\u271d q\u2082\u271d : Stmt\u2082 IH\u2081 :   TM2.SupportsStmt S q\u2081\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') IH\u2082 :   TM2.SupportsStmt S q\u2082\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S IH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) IH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') IH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) IH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** refine' \u27e8\u27e8IH\u2081\u2081, IH\u2082\u2081\u27e9, fun l h \u21a6 _\u27e9 ** case refine'_3.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 S : Finset \u039b ss : TM2.Supports M S p\u271d : \u03c3 \u2192 Bool q\u2081\u271d q\u2082\u271d : Stmt\u2082 IH\u2081 :   TM2.SupportsStmt S q\u2081\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') IH\u2082 :   TM2.SupportsStmt S q\u2082\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S IH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) IH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') IH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) IH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') l : \u039b' h : l \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l) ** rw [trStmts\u2081] at h ** case refine'_3.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 S : Finset \u039b ss : TM2.Supports M S p\u271d : \u03c3 \u2192 Bool q\u2081\u271d q\u2082\u271d : Stmt\u2082 IH\u2081 :   TM2.SupportsStmt S q\u2081\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') IH\u2082 :   TM2.SupportsStmt S q\u2082\u271d \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) sub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S IH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) IH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') IH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) IH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') l : \u039b' h : l \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u22a2 TM1.SupportsStmt (trSupp M S) (tr M l) ** rcases Finset.mem_union.1 h with (h | h) <;> [exact IH\u2081\u2082 _ h; exact IH\u2082\u2082 _ h] ** case refine'_4 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 S : Finset \u039b ss : TM2.Supports M S \u22a2 \u2200 (l : \u03c3 \u2192 \u039b),     TM2.SupportsStmt S (TM2.Stmt.goto l) \u2192       (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l) \u2192 x \u2208 trSupp M S) \u2192         TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l)) \u2227           \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.goto l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** intro _ ss' _ ** case refine'_4 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 S : Finset \u039b ss : TM2.Supports M S l\u271d : \u03c3 \u2192 \u039b ss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d) x\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d)) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** simp only [trStmts\u2081, Finset.not_mem_empty] ** case refine'_4 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 S : Finset \u039b ss : TM2.Supports M S l\u271d : \u03c3 \u2192 \u039b ss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d) x\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d)) \u2227     \u2200 (l' : \u039b'), False \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** refine' \u27e8_, fun _ \u21a6 False.elim\u27e9 ** case refine'_4 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 S : Finset \u039b ss : TM2.Supports M S l\u271d : \u03c3 \u2192 \u039b ss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d) x\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d)) ** exact fun _ v \u21a6 Finset.mem_biUnion.2 \u27e8_, ss' v, Finset.mem_insert_self _ _\u27e9 ** case refine'_5 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 S : Finset \u039b ss : TM2.Supports M S \u22a2 TM2.SupportsStmt S TM2.Stmt.halt \u2192     (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S) \u2192       TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227         \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** intro _ _ ** case refine'_5 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 S : Finset \u039b ss : TM2.Supports M S x\u271d\u00b9 : TM2.SupportsStmt S TM2.Stmt.halt x\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227     \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** simp only [trStmts\u2081, Finset.not_mem_empty] ** case refine'_5 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 S : Finset \u039b ss : TM2.Supports M S x\u271d\u00b9 : TM2.SupportsStmt S TM2.Stmt.halt x\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S \u22a2 TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227 \u2200 (l' : \u039b'), False \u2192 TM1.SupportsStmt (trSupp M S) (tr M l') ** exact \u27e8trivial, fun _ \u21a6 False.elim\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion\u2080 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) \u22a2 Set.Countable {i | 0 < \u2191\u2191\u03bc (As i)} ** have obs : { i : \u03b9 | 0 < \u03bc (As i) } \u2286 \u22c3 n, { i : \u03b9 | 0 < \u03bc (As i \u2229 spanningSets \u03bc n) } := by\n  intro i i_in_nonzeroes\n  by_contra con\n  simp only [mem_iUnion, mem_setOf_eq, not_exists, not_lt, nonpos_iff_eq_zero] at *\n  simp [(forall_measure_inter_spanningSets_eq_zero _).mp con] at i_in_nonzeroes ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) obs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)} \u22a2 Set.Countable {i | 0 < \u2191\u2191\u03bc (As i)} ** apply Countable.mono obs ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) obs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)} \u22a2 Set.Countable (\u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}) ** refine' countable_iUnion fun n => countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top\u2080 \u03bc _ _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) \u22a2 {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)} ** intro i i_in_nonzeroes ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) i : \u03b9 i_in_nonzeroes : i \u2208 {i | 0 < \u2191\u2191\u03bc (As i)} \u22a2 i \u2208 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)} ** by_contra con ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) i : \u03b9 i_in_nonzeroes : i \u2208 {i | 0 < \u2191\u2191\u03bc (As i)} con : \u00aci \u2208 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)} \u22a2 False ** simp only [mem_iUnion, mem_setOf_eq, not_exists, not_lt, nonpos_iff_eq_zero] at * ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) i : \u03b9 i_in_nonzeroes : 0 < \u2191\u2191\u03bc (As i) con : \u2200 (x : \u2115), \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc x) = 0 \u22a2 False ** simp [(forall_measure_inter_spanningSets_eq_zero _).mp con] at i_in_nonzeroes ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) obs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)} n : \u2115 \u22a2 \u2200 (i : \u03b9), NullMeasurableSet (As i \u2229 spanningSets \u03bc n) ** exact fun i \u21a6 NullMeasurableSet.inter (As_mble i)\n  (measurable_spanningSets \u03bc n).nullMeasurableSet ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) obs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)} n : \u2115 \u22a2 Pairwise (AEDisjoint \u03bc on fun i => As i \u2229 spanningSets \u03bc n) ** exact fun i j i_ne_j \u21a6 (As_disj i_ne_j).mono\n  (inter_subset_left (As i) (spanningSets \u03bc n)) (inter_subset_left (As j) (spanningSets \u03bc n)) ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) obs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)} n : \u2115 \u22a2 \u2191\u2191\u03bc (\u22c3 i, As i \u2229 spanningSets \u03bc n) \u2260 \u22a4 ** refine' (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top \u03bc n)).ne ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03b9 : Type u_8 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) obs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 n, {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)} n : \u2115 \u22a2 \u22c3 i, As i \u2229 spanningSets \u03bc n \u2286 spanningSets \u03bc n ** exact iUnion_subset fun i => inter_subset_right _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.eq_insert_of_ncard_eq_succ ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 h : ncard s = n + 1 \u22a2 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n ** have hsf := finite_of_ncard_pos (n.zero_lt_succ.trans_eq h.symm) ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 h : ncard s = n + 1 hsf : Set.Finite s \u22a2 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n ** rw [ncard_eq_toFinset_card _ hsf, Finset.card_eq_succ] at h ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hsf : Set.Finite s h\u271d : Finset.card (Finite.toFinset hsf) = n + 1 h : \u2203 a t, \u00aca \u2208 t \u2227 insert a t = Finite.toFinset hsf \u2227 Finset.card t = n \u22a2 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n ** obtain \u27e8a, t, hat, hts, rfl\u27e9 := h ** case intro.intro.intro.intro \u03b1 : Type u_1 s t\u271d : Set \u03b1 hsf : Set.Finite s a : \u03b1 t : Finset \u03b1 hat : \u00aca \u2208 t hts : insert a t = Finite.toFinset hsf h : Finset.card (Finite.toFinset hsf) = Finset.card t + 1 \u22a2 \u2203 a t_1, \u00aca \u2208 t_1 \u2227 insert a t_1 = s \u2227 ncard t_1 = Finset.card t ** simp only [Finset.ext_iff, Finset.mem_insert, Finite.mem_toFinset] at hts ** case intro.intro.intro.intro \u03b1 : Type u_1 s t\u271d : Set \u03b1 hsf : Set.Finite s a : \u03b1 t : Finset \u03b1 hat : \u00aca \u2208 t h : Finset.card (Finite.toFinset hsf) = Finset.card t + 1 hts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s \u22a2 \u2203 a t_1, \u00aca \u2208 t_1 \u2227 insert a t_1 = s \u2227 ncard t_1 = Finset.card t ** refine' \u27e8a, t, hat, _, _\u27e9 ** case intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 s t\u271d : Set \u03b1 hsf : Set.Finite s a : \u03b1 t : Finset \u03b1 hat : \u00aca \u2208 t h : Finset.card (Finite.toFinset hsf) = Finset.card t + 1 hts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s \u22a2 ncard \u2191t = Finset.card t ** simp ** case intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 s t\u271d : Set \u03b1 hsf : Set.Finite s a : \u03b1 t : Finset \u03b1 hat : \u00aca \u2208 t h : Finset.card (Finite.toFinset hsf) = Finset.card t + 1 hts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s \u22a2 insert a \u2191t = s ** simp only [Finset.mem_coe, ext_iff, mem_insert_iff] ** case intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 s t\u271d : Set \u03b1 hsf : Set.Finite s a : \u03b1 t : Finset \u03b1 hat : \u00aca \u2208 t h : Finset.card (Finite.toFinset hsf) = Finset.card t + 1 hts : \u2200 (a_1 : \u03b1), a_1 = a \u2228 a_1 \u2208 t \u2194 a_1 \u2208 s \u22a2 \u2200 (x : \u03b1), x = a \u2228 x \u2208 t \u2194 x \u2208 s ** tauto ** Qed",
        "informal": ""
    },
    {
        "formal": "Setoid.sup_def ** \u03b1 : Type u_1 \u03b2 : Type u_2 r s : Setoid \u03b1 \u22a2 r \u2294 s = EqvGen.Setoid (Rel r \u2294 Rel s) ** rw [sup_eq_eqvGen] ** \u03b1 : Type u_1 \u03b2 : Type u_2 r s : Setoid \u03b1 \u22a2 (EqvGen.Setoid fun x y => Rel r x y \u2228 Rel s x y) = EqvGen.Setoid (Rel r \u2294 Rel s) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.nonempty_iInter_clPrehaar ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G \u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 V, clPrehaar (\u2191K\u2080) V) ** have : IsCompact (haarProduct (K\u2080 : Set G)) := by\n  apply isCompact_univ_pi; intro K; apply isCompact_Icc ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) \u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 V, clPrehaar (\u2191K\u2080) V) ** refine' this.inter_iInter_nonempty (clPrehaar K\u2080) (fun s => isClosed_closure) fun t => _ ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) \u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i) ** let V\u2080 := \u22c2 V \u2208 t, (V : OpenNhdsOf (1 : G)).carrier ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier \u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i) ** have h1V\u2080 : IsOpen V\u2080 := isOpen_biInter_finset $ by rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _; exact hV\u2081 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 \u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i) ** have h2V\u2080 : (1 : G) \u2208 V\u2080 := by simp only [mem_iInter]; rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _; exact hV\u2082 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 \u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i) ** refine' \u27e8prehaar K\u2080 V\u2080, _\u27e9 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 \u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 haarProduct \u2191K\u2080 \u2229 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i ** constructor ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G \u22a2 IsCompact (haarProduct \u2191K\u2080) ** apply isCompact_univ_pi ** case h G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G \u22a2 \u2200 (i : Compacts G), IsCompact (Icc 0 \u2191(index \u2191i \u2191K\u2080)) ** intro K ** case 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 \u22a2 IsCompact (Icc 0 \u2191(index \u2191K \u2191K\u2080)) ** apply isCompact_Icc ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier \u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 IsOpen i.carrier ** rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _ ** case mk.mk G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier V : Set G hV\u2081 : IsOpen V hV\u2082 : 1 \u2208 { carrier := V, is_open' := hV\u2081 }.carrier a\u271d : { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 } \u2208 t \u22a2 IsOpen { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 }.toOpens.carrier ** exact hV\u2081 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 \u22a2 1 \u2208 V\u2080 ** simp only [mem_iInter] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 \u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 1 \u2208 i.carrier ** rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _ ** case mk.mk G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 V : Set G hV\u2081 : IsOpen V hV\u2082 : 1 \u2208 { carrier := V, is_open' := hV\u2081 }.carrier i\u271d : { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 } \u2208 t \u22a2 1 \u2208 { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 }.toOpens.carrier ** exact hV\u2082 ** case left G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 \u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 haarProduct \u2191K\u2080 ** apply prehaar_mem_haarProduct K\u2080 ** case left G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 \u22a2 Set.Nonempty (interior V\u2080) ** use 1 ** case h G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 \u22a2 1 \u2208 interior V\u2080 ** rwa [h1V\u2080.interior_eq] ** case right G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 \u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 \u22c2 i \u2208 t, clPrehaar (\u2191K\u2080) i ** simp only [mem_iInter] ** case right G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 \u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 prehaar (\u2191K\u2080) (\u22c2 V \u2208 t, V.carrier) \u2208 clPrehaar (\u2191K\u2080) i ** rintro \u27e8V, hV\u27e9 h2V ** case right.mk G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 V : Opens G hV : 1 \u2208 V.carrier h2V : { toOpens := V, mem' := hV } \u2208 t \u22a2 prehaar (\u2191K\u2080) (\u22c2 V \u2208 t, V.carrier) \u2208 clPrehaar \u2191K\u2080 { toOpens := V, mem' := hV } ** apply subset_closure ** case right.mk.a G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 V : Opens G hV : 1 \u2208 V.carrier h2V : { toOpens := V, mem' := hV } \u2208 t \u22a2 prehaar (\u2191K\u2080) (\u22c2 V \u2208 t, V.carrier) \u2208 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} ** apply mem_image_of_mem ** case right.mk.a.h G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 V : Opens G hV : 1 \u2208 V.carrier h2V : { toOpens := V, mem' := hV } \u2208 t \u22a2 \u22c2 V \u2208 t, V.carrier \u2208 {U | U \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} ** rw [mem_setOf_eq] ** case right.mk.a.h G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G this : IsCompact (haarProduct \u2191K\u2080) t : Finset (OpenNhdsOf 1) V\u2080 : Set G := \u22c2 V \u2208 t, V.carrier h1V\u2080 : IsOpen V\u2080 h2V\u2080 : 1 \u2208 V\u2080 V : Opens G hV : 1 \u2208 V.carrier h2V : { toOpens := V, mem' := hV } \u2208 t \u22a2 \u22c2 V \u2208 t, V.carrier \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen (\u22c2 V \u2208 t, V.carrier) \u2227 1 \u2208 \u22c2 V \u2208 t, V.carrier ** exact \u27e8Subset.trans (iInter_subset _ \u27e8V, hV\u27e9) (iInter_subset _ h2V), h1V\u2080, h2V\u2080\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.biUnion_null_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 : Set \u03b2 hs : Set.Countable s t : \u03b2 \u2192 Set \u03b1 \u22a2 \u2191m (\u22c3 i \u2208 s, t i) = 0 \u2194 \u2200 (i : \u03b2), i \u2208 s \u2192 \u2191m (t i) = 0 ** haveI := hs.toEncodable ** \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 : Set \u03b2 hs : Set.Countable s t : \u03b2 \u2192 Set \u03b1 this : Encodable \u2191s \u22a2 \u2191m (\u22c3 i \u2208 s, t i) = 0 \u2194 \u2200 (i : \u03b2), i \u2208 s \u2192 \u2191m (t i) = 0 ** rw [biUnion_eq_iUnion, iUnion_null_iff, SetCoe.forall'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PartrecToTM2.move_ok ** p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e : splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } ** induction' L\u2081 with a L\u2081 IH generalizing S s ** case nil p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = ([], o, L\u2082) \u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux [] (S k\u2082)) } ** rw [(_ : [].reverseAux _ = _), Function.update_eq_self] ** case nil p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = ([], o, L\u2082) \u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }     { l := some q, var := o, stk := update S k\u2081 L\u2082 }  p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = ([], o, L\u2082) \u22a2 List.reverseAux [] (S k\u2082) = update S k\u2081 L\u2082 k\u2082 ** swap ** case nil p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = ([], o, L\u2082) \u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }     { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** refine' TransGen.head' rfl _ ** case nil p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = ([], o, L\u2082) \u22a2 ReflTransGen (fun a b => b \u2208 TM2.step tr a) (TM2.stepAux (tr (\u039b'.move p k\u2081 k\u2082 q)) s S)     { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim, ne_eq] ** case nil p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = ([], o, L\u2082) \u22a2 ReflTransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (S k\u2081), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (S k\u2081), stk := update S k\u2081 (List.tail (S k\u2081)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (S k\u2081),         stk :=           update (update S k\u2081 (List.tail (S k\u2081))) k\u2082             (Option.iget (List.head? (S k\u2081)) :: update S k\u2081 (List.tail (S k\u2081)) k\u2082) })     { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** revert e ** case nil p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' \u22a2 splitAtPred p (S k\u2081) = ([], o, L\u2082) \u2192     ReflTransGen       (fun a b =>         (match a with           | { l := none, var := var, stk := stk } => none           | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =           some b)       (bif           match List.head? (S k\u2081), true, p with           | some x, x_1, f => f x           | none, y, x => y then         { l := some q, var := List.head? (S k\u2081), stk := update S k\u2081 (List.tail (S k\u2081)) }       else         { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (S k\u2081),           stk :=             update (update S k\u2081 (List.tail (S k\u2081))) k\u2082               (Option.iget (List.head? (S k\u2081)) :: update S k\u2081 (List.tail (S k\u2081)) k\u2082) })       { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** cases' S k\u2081 with a Sk <;> intro e ** case nil.cons p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' a : \u0393' Sk : List \u0393' e : splitAtPred p (a :: Sk) = ([], o, L\u2082) \u22a2 ReflTransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (a :: Sk), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (a :: Sk), stk := update S k\u2081 (List.tail (a :: Sk)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a :: Sk),         stk :=           update (update S k\u2081 (List.tail (a :: Sk))) k\u2082             (Option.iget (List.head? (a :: Sk)) :: update S k\u2081 (List.tail (a :: Sk)) k\u2082) })     { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** simp only [splitAtPred, Option.elim, List.head?, List.tail_cons, Option.iget_some] at e \u22a2 ** case nil.cons p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' a : \u0393' Sk : List \u0393' e :   (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) =     ([], o, L\u2082) \u22a2 ReflTransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif p a then { l := some q, var := some a, stk := update S k\u2081 Sk }     else { l := some (\u039b'.move p k\u2081 k\u2082 q), var := some a, stk := update (update S k\u2081 Sk) k\u2082 (a :: update S k\u2081 Sk k\u2082) })     { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** revert e ** case nil.cons p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' a : \u0393' Sk : List \u0393' \u22a2 (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) =       ([], o, L\u2082) \u2192     ReflTransGen       (fun a b =>         (match a with           | { l := none, var := var, stk := stk } => none           | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =           some b)       (bif p a then { l := some q, var := some a, stk := update S k\u2081 Sk }       else { l := some (\u039b'.move p k\u2081 k\u2082 q), var := some a, stk := update (update S k\u2081 Sk) k\u2082 (a :: update S k\u2081 Sk k\u2082) })       { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** cases p a <;> intro e <;>\nsimp only [cond_false, cond_true, Prod.mk.injEq, true_and, false_and] at e \u22a2 ** p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = ([], o, L\u2082) \u22a2 List.reverseAux [] (S k\u2082) = update S k\u2081 L\u2082 k\u2082 ** rw [Function.update_noteq h\u2081.symm] ** p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = ([], o, L\u2082) \u22a2 List.reverseAux [] (S k\u2082) = S k\u2082 ** rfl ** case nil.nil p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p [] = ([], o, L\u2082) \u22a2 ReflTransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? [], true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? [], stk := update S k\u2081 (List.tail []) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? [],         stk := update (update S k\u2081 (List.tail [])) k\u2082 (Option.iget (List.head? []) :: update S k\u2081 (List.tail []) k\u2082) })     { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** cases e ** case nil.nil.refl p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S\u271d k\u2081) = (L\u2081, none, []) \u22a2 ReflTransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? [], true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? [], stk := update S k\u2081 (List.tail []) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? [],         stk := update (update S k\u2081 (List.tail [])) k\u2082 (Option.iget (List.head? []) :: update S k\u2081 (List.tail []) k\u2082) })     { l := some q, var := none, stk := update S k\u2081 [] } ** rfl ** case nil.cons.true p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' a : \u0393' Sk : List \u0393' e : some a = o \u2227 Sk = L\u2082 \u22a2 ReflTransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     { l := some q, var := some a, stk := update S k\u2081 Sk } { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** simp only [e] ** case nil.cons.true p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081, o, L\u2082) s : Option \u0393' S : K' \u2192 List \u0393' a : \u0393' Sk : List \u0393' e : some a = o \u2227 Sk = L\u2082 \u22a2 ReflTransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     { l := some q, var := o, stk := update S k\u2081 L\u2082 } { l := some q, var := o, stk := update S k\u2081 L\u2082 } ** rfl ** case cons p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = (a :: L\u2081, o, L\u2082) \u22a2 Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux (a :: L\u2081) (S k\u2082)) } ** refine' TransGen.head rfl _ ** case cons p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = (a :: L\u2081, o, L\u2082) \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a) (TM2.stepAux (tr (\u039b'.move p k\u2081 k\u2082 q)) s S)     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux (a :: L\u2081) (S k\u2082)) } ** simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim, ne_eq, List.reverseAux_cons] ** case cons p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' e : splitAtPred p (S k\u2081) = (a :: L\u2081, o, L\u2082) \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (S k\u2081), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (S k\u2081), stk := update S k\u2081 (List.tail (S k\u2081)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (S k\u2081),         stk :=           update (update S k\u2081 (List.tail (S k\u2081))) k\u2082             (Option.iget (List.head? (S k\u2081)) :: update S k\u2081 (List.tail (S k\u2081)) k\u2082) })     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) } ** cases' e\u2081 : S k\u2081 with a' Sk <;> rw [e\u2081, splitAtPred] at e ** case cons.cons p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' a' : \u0393' Sk : List \u0393' e :   (bif p a' then ([], some a', Sk)     else       match splitAtPred p Sk with       | (l\u2081, o, l\u2082) => (a' :: l\u2081, o, l\u2082)) =     (a :: L\u2081, o, L\u2082) e\u2081 : S k\u2081 = a' :: Sk \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (a' :: Sk), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),         stk :=           update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082             (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) } ** cases e\u2082 : p a' <;> simp only [e\u2082, cond] at e ** case cons.cons.false p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' a' : \u0393' Sk : List \u0393' e\u2081 : S k\u2081 = a' :: Sk e\u2082 : p a' = false e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L\u2081, o, L\u2082) \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (a' :: Sk), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),         stk :=           update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082             (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) }  case cons.cons.true p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' a' : \u0393' Sk : List \u0393' e\u2081 : S k\u2081 = a' :: Sk e\u2082 : p a' = true e : ([], some a', Sk) = (a :: L\u2081, o, L\u2082) \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (a' :: Sk), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),         stk :=           update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082             (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) } ** swap ** case cons.cons.false p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' a' : \u0393' Sk : List \u0393' e\u2081 : S k\u2081 = a' :: Sk e\u2082 : p a' = false e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L\u2081, o, L\u2082) \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (a' :: Sk), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),         stk :=           update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082             (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) } ** rcases e\u2083 : splitAtPred p Sk with \u27e8_, _, _\u27e9 ** case cons.cons.false.mk.mk p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' a' : \u0393' Sk : List \u0393' e\u2081 : S k\u2081 = a' :: Sk e\u2082 : p a' = false e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L\u2081, o, L\u2082) fst\u271d\u00b9 : List \u0393' fst\u271d : Option \u0393' snd\u271d : List \u0393' e\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d) \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (a' :: Sk), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),         stk :=           update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082             (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) } ** rw [e\u2083] at e ** case cons.cons.false.mk.mk p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' a' : \u0393' Sk : List \u0393' e\u2081 : S k\u2081 = a' :: Sk e\u2082 : p a' = false fst\u271d\u00b9 : List \u0393' fst\u271d : Option \u0393' snd\u271d : List \u0393' e : (a' :: (fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) = (a :: L\u2081, o, L\u2082) e\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d) \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (a' :: Sk), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),         stk :=           update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082             (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) } ** cases e ** case cons.cons.false.mk.mk.refl p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 a : \u0393' s : Option \u0393' S : K' \u2192 List \u0393' Sk fst\u271d\u00b9 : List \u0393' fst\u271d : Option \u0393' snd\u271d : List \u0393' e\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d) e\u2081 : S k\u2081 = a :: Sk e\u2082 : p a = false e : splitAtPred p (S\u271d k\u2081) = (L\u2081, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = ((fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,           stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (S k\u2082)) } \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (a :: Sk), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (a :: Sk), stk := update S k\u2081 (List.tail (a :: Sk)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a :: Sk),         stk :=           update (update S k\u2081 (List.tail (a :: Sk))) k\u2082             (Option.iget (List.head? (a :: Sk)) :: update S k\u2081 (List.tail (a :: Sk)) k\u2082) })     { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,       stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (a :: S k\u2082)) } ** simp only [List.head?_cons, e\u2082, List.tail_cons, ne_eq, cond_false] ** case cons.cons.false.mk.mk.refl p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 a : \u0393' s : Option \u0393' S : K' \u2192 List \u0393' Sk fst\u271d\u00b9 : List \u0393' fst\u271d : Option \u0393' snd\u271d : List \u0393' e\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d) e\u2081 : S k\u2081 = a :: Sk e\u2082 : p a = false e : splitAtPred p (S\u271d k\u2081) = (L\u2081, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = ((fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,           stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (S k\u2082)) } \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     { l := some (\u039b'.move p k\u2081 k\u2082 q), var := some a,       stk := update (update S k\u2081 Sk) k\u2082 (Option.iget (some a) :: update S k\u2081 Sk k\u2082) }     { l := some q, var := fst\u271d, stk := update (update S k\u2081 snd\u271d) k\u2082 (List.reverseAux fst\u271d\u00b9 (a :: S k\u2082)) } ** convert @IH _ (update (update S k\u2081 Sk) k\u2082 (a :: S k\u2082)) _ using 2 <;>\n  simp [Function.update_noteq, h\u2081, h\u2081.symm, e\u2083, List.reverseAux] ** case h.e'_2.h.e'_7 p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 a : \u0393' s : Option \u0393' S : K' \u2192 List \u0393' Sk fst\u271d\u00b9 : List \u0393' fst\u271d : Option \u0393' snd\u271d : List \u0393' e\u2083 : splitAtPred p Sk = (fst\u271d\u00b9, fst\u271d, snd\u271d) e\u2081 : S k\u2081 = a :: Sk e\u2082 : p a = false e : splitAtPred p (S\u271d k\u2081) = (L\u2081, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = ((fst\u271d\u00b9, fst\u271d, snd\u271d).1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1, (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := (fst\u271d\u00b9, fst\u271d, snd\u271d).2.1,           stk := update (update S k\u2081 (fst\u271d\u00b9, fst\u271d, snd\u271d).2.2) k\u2082 (List.reverseAux (fst\u271d\u00b9, fst\u271d, snd\u271d).1 (S k\u2082)) } \u22a2 update (update S k\u2081 snd\u271d) k\u2082 (List.reverseAux fst\u271d\u00b9 (a :: S k\u2082)) =     update (update (update (update S k\u2081 Sk) k\u2082 (a :: S k\u2082)) k\u2081 snd\u271d) k\u2082 (List.reverseAux fst\u271d\u00b9 (a :: S k\u2082)) ** simp [Function.update_comm h\u2081.symm] ** case cons.nil p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' e : ([], none, []) = (a :: L\u2081, o, L\u2082) e\u2081 : S k\u2081 = [] \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? [], true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? [], stk := update S k\u2081 (List.tail []) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? [],         stk := update (update S k\u2081 (List.tail [])) k\u2082 (Option.iget (List.head? []) :: update S k\u2081 (List.tail []) k\u2082) })     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) } ** cases e ** case cons.cons.true p : \u0393' \u2192 Bool k\u2081 k\u2082 : K' q : \u039b' s\u271d : Option \u0393' L\u2081\u271d : List \u0393' o : Option \u0393' L\u2082 : List \u0393' S\u271d : K' \u2192 List \u0393' h\u2081 : k\u2081 \u2260 k\u2082 e\u271d : splitAtPred p (S\u271d k\u2081) = (L\u2081\u271d, o, L\u2082) a : \u0393' L\u2081 : List \u0393' IH :   \u2200 {s : Option \u0393'} {S : K' \u2192 List \u0393'},     splitAtPred p (S k\u2081) = (L\u2081, o, L\u2082) \u2192       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.move p k\u2081 k\u2082 q), var := s, stk := S }         { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (S k\u2082)) } s : Option \u0393' S : K' \u2192 List \u0393' a' : \u0393' Sk : List \u0393' e\u2081 : S k\u2081 = a' :: Sk e\u2082 : p a' = true e : ([], some a', Sk) = (a :: L\u2081, o, L\u2082) \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     (bif         match List.head? (a' :: Sk), true, p with         | some x, x_1, f => f x         | none, y, x => y then       { l := some q, var := List.head? (a' :: Sk), stk := update S k\u2081 (List.tail (a' :: Sk)) }     else       { l := some (\u039b'.move p k\u2081 k\u2082 q), var := List.head? (a' :: Sk),         stk :=           update (update S k\u2081 (List.tail (a' :: Sk))) k\u2082             (Option.iget (List.head? (a' :: Sk)) :: update S k\u2081 (List.tail (a' :: Sk)) k\u2082) })     { l := some q, var := o, stk := update (update S k\u2081 L\u2082) k\u2082 (List.reverseAux L\u2081 (a :: S k\u2082)) } ** cases e ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_insert_le ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 \u22a2 ncard (insert a s) \u2264 ncard s + 1 ** obtain hs | hs := s.finite_or_infinite ** case inr \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 hs : Set.Infinite s \u22a2 ncard (insert a s) \u2264 ncard s + 1 ** rw [(hs.mono (subset_insert a s)).ncard] ** case inr \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 hs : Set.Infinite s \u22a2 0 \u2264 ncard s + 1 ** exact Nat.zero_le _ ** case inl \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 hs : Set.Finite s \u22a2 ncard (insert a s) \u2264 ncard s + 1 ** to_encard_tac ** case inl \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 hs : Set.Finite s \u22a2 \u2191(ncard (insert a s)) \u2264 \u2191(ncard s) + 1 ** rw [hs.cast_ncard_eq, (hs.insert _).cast_ncard_eq] ** case inl \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 hs : Set.Finite s \u22a2 encard (insert a s) \u2264 encard s + 1 ** apply encard_insert_le ** Qed",
        "informal": ""
    },
    {
        "formal": "hasSum_two_pi_I_cauchyPowerSeries_integral ** 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R \u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z)     (\u222e (z : \u2102) in C(c, R), (z - (c + w))\u207b\u00b9 \u2022 f z) ** have hR : 0 < R := (Complex.abs.nonneg w).trans_lt hw ** 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R \u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z)     (\u222e (z : \u2102) in C(c, R), (z - (c + w))\u207b\u00b9 \u2022 f z) ** have hwR : abs w / R \u2208 Ico (0 : \u211d) 1 :=\n  \u27e8div_nonneg (Complex.abs.nonneg w) hR.le, (div_lt_one hR).2 hw\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 : \u211d w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z)     (\u222e (z : \u2102) in C(c, R), (z - (c + w))\u207b\u00b9 \u2022 f z) ** refine' intervalIntegral.hasSum_integral_of_dominated_convergence\n    (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (abs w / R) ^ n) (fun n => _) (fun n => _) _ _ _ ** 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 : \u211d w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 n : \u2115 \u22a2 AEStronglyMeasurable     (fun \u03b8 => deriv (circleMap c R) \u03b8 \u2022 (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R \u03b8))     (Measure.restrict volume (\u0399 0 (2 * \u03c0))) ** simp only [deriv_circleMap] ** 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 : \u211d w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 n : \u2115 \u22a2 AEStronglyMeasurable     (fun \u03b8 => (circleMap 0 R \u03b8 * I) \u2022 (w / (circleMap c R \u03b8 - c)) ^ n \u2022 (circleMap c R \u03b8 - c)\u207b\u00b9 \u2022 f (circleMap c R \u03b8))     (Measure.restrict volume (\u0399 0 (2 * \u03c0))) ** apply_rules [AEStronglyMeasurable.smul, hf.def.1] <;> apply Measurable.aestronglyMeasurable ** case refine'_1.hf.hf 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 n : \u2115 \u22a2 Measurable fun x => circleMap 0 R x * I ** exact (measurable_circleMap 0 R).mul_const I ** case refine'_1.hg.hf.hf 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 n : \u2115 \u22a2 Measurable fun x => (w / (circleMap c R x - c)) ^ n ** exact (((measurable_circleMap c R).sub measurable_const).const_div w).pow measurable_const ** case refine'_1.hg.hg.hf.hf 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 n : \u2115 \u22a2 Measurable fun x => (circleMap c R x - c)\u207b\u00b9 ** exact ((measurable_circleMap c R).sub measurable_const).inv ** 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 : \u211d w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 n : \u2115 \u22a2 \u2200\u1d50 (t : \u211d),     t \u2208 \u0399 0 (2 * \u03c0) \u2192       \u2016deriv (circleMap c R) t \u2022 (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R t)\u2016 \u2264         (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (\u2191Complex.abs w / R) ^ n) n t ** simp [norm_smul, abs_of_pos hR, mul_left_comm R, inv_mul_cancel_left\u2080 hR.ne', mul_comm \u2016_\u2016] ** 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 : \u211d w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u22a2 \u2200\u1d50 (t : \u211d), t \u2208 \u0399 0 (2 * \u03c0) \u2192 Summable fun n => (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (\u2191Complex.abs w / R) ^ n) n t ** exact eventually_of_forall fun _ _ => (summable_geometric_of_lt_1 hwR.1 hwR.2).mul_left _ ** case refine'_4 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u22a2 IntervalIntegrable (fun t => \u2211' (n : \u2115), (fun n \u03b8 => \u2016f (circleMap c R \u03b8)\u2016 * (\u2191Complex.abs w / R) ^ n) n t) volume 0     (2 * \u03c0) ** simpa only [tsum_mul_left, tsum_geometric_of_lt_1 hwR.1 hwR.2] using\n  hf.norm.mul_continuousOn continuousOn_const ** case refine'_5 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u22a2 \u2200\u1d50 (t : \u211d),     t \u2208 \u0399 0 (2 * \u03c0) \u2192       HasSum (fun n => deriv (circleMap c R) t \u2022 (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R t))         (deriv (circleMap c R) t \u2022 (fun z => (z - (c + w))\u207b\u00b9 \u2022 f z) (circleMap c R t)) ** refine' eventually_of_forall fun \u03b8 _ => HasSum.const_smul _ _ ** case refine'_5 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u03b8 : \u211d x\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0) \u22a2 HasSum (fun n => (fun z => (w / (z - c)) ^ n \u2022 (z - c)\u207b\u00b9 \u2022 f z) (circleMap c R \u03b8))     ((fun z => (z - (c + w))\u207b\u00b9 \u2022 f z) (circleMap c R \u03b8)) ** simp only [smul_smul] ** case refine'_5 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u03b8 : \u211d x\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0) \u22a2 HasSum (fun n => ((w / (circleMap c R \u03b8 - c)) ^ n * (circleMap c R \u03b8 - c)\u207b\u00b9) \u2022 f (circleMap c R \u03b8))     ((circleMap c R \u03b8 - (c + w))\u207b\u00b9 \u2022 f (circleMap c R \u03b8)) ** refine' HasSum.smul_const _ _ ** case refine'_5 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u03b8 : \u211d x\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0) \u22a2 HasSum (fun n => (w / (circleMap c R \u03b8 - c)) ^ n * (circleMap c R \u03b8 - c)\u207b\u00b9) (circleMap c R \u03b8 - (c + w))\u207b\u00b9 ** have : \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1 := by simpa [abs_of_pos hR] using hwR.2 ** case refine'_5 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u03b8 : \u211d x\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0) this : \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1 \u22a2 HasSum (fun n => (w / (circleMap c R \u03b8 - c)) ^ n * (circleMap c R \u03b8 - c)\u207b\u00b9) (circleMap c R \u03b8 - (c + w))\u207b\u00b9 ** convert (hasSum_geometric_of_norm_lt_1 this).mul_right _ using 1 ** case h.e'_6 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u03b8 : \u211d x\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0) this : \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1 \u22a2 (circleMap c R \u03b8 - (c + w))\u207b\u00b9 = (1 - w / (circleMap c R \u03b8 - c))\u207b\u00b9 * (circleMap c R \u03b8 - c)\u207b\u00b9 ** simp [\u2190 sub_sub, \u2190 mul_inv, sub_mul, div_mul_cancel _ (circleMap_ne_center hR.ne')] ** 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 w : \u2102 hf : CircleIntegrable f c R hw : \u2191Complex.abs w < R hR : 0 < R hwR : \u2191Complex.abs w / R \u2208 Ico 0 1 \u03b8 : \u211d x\u271d : \u03b8 \u2208 \u0399 0 (2 * \u03c0) \u22a2 \u2016w / (circleMap c R \u03b8 - c)\u2016 < 1 ** simpa [abs_of_pos hR] using hwR.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toMeasure_map_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 p : PMF \u03b1 b : \u03b2 s : Set \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 hf : Measurable f hs : MeasurableSet s \u22a2 \u2191\u2191(toMeasure (map f p)) s = \u2191\u2191(toMeasure p) (f \u207b\u00b9' s) ** rw [toMeasure_apply_eq_toOuterMeasure_apply _ s hs,\n  toMeasure_apply_eq_toOuterMeasure_apply _ (f \u207b\u00b9' s) (measurableSet_preimage hf hs)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 p : PMF \u03b1 b : \u03b2 s : Set \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 hf : Measurable f hs : MeasurableSet s \u22a2 \u2191(toOuterMeasure (map f p)) s = \u2191(toOuterMeasure p) (f \u207b\u00b9' s) ** exact toOuterMeasure_map_apply f p s ** Qed",
        "informal": ""
    },
    {
        "formal": "BoundedContinuousFunction.toLp_denseRange ** \u03b1 : Type u_1 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b1 inst\u271d\u2078 : T4Space \u03b1 inst\u271d\u2077 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 \ud835\udd5c : Type u_3 inst\u271d\u2074 : NormedField \ud835\udd5c inst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : Measure.WeaklyRegular \u03bc inst\u271d : IsFiniteMeasure \u03bc \u22a2 DenseRange \u2191(toLp p \u03bc \ud835\udd5c) ** haveI : NormedSpace \u211d E := RestrictScalars.normedSpace \u211d \ud835\udd5c E ** \u03b1 : Type u_1 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b1 inst\u271d\u2078 : T4Space \u03b1 inst\u271d\u2077 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 \ud835\udd5c : Type u_3 inst\u271d\u2074 : NormedField \ud835\udd5c inst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : Measure.WeaklyRegular \u03bc inst\u271d : IsFiniteMeasure \u03bc this : NormedSpace \u211d E \u22a2 DenseRange \u2191(toLp p \u03bc \ud835\udd5c) ** rw [denseRange_iff_closure_range] ** \u03b1 : Type u_1 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b1 inst\u271d\u2078 : T4Space \u03b1 inst\u271d\u2077 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 \ud835\udd5c : Type u_3 inst\u271d\u2074 : NormedField \ud835\udd5c inst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : Measure.WeaklyRegular \u03bc inst\u271d : IsFiniteMeasure \u03bc this : NormedSpace \u211d E \u22a2 closure (range \u2191(toLp p \u03bc \ud835\udd5c)) = univ ** suffices (LinearMap.range (toLp p \u03bc \ud835\udd5c : _ \u2192L[\ud835\udd5c] Lp E p \u03bc)).toAddSubgroup.topologicalClosure = \u22a4\n  by exact congr_arg ((\u2191) : AddSubgroup (Lp E p \u03bc) \u2192 Set (Lp E p \u03bc)) this ** \u03b1 : Type u_1 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b1 inst\u271d\u2078 : T4Space \u03b1 inst\u271d\u2077 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 \ud835\udd5c : Type u_3 inst\u271d\u2074 : NormedField \ud835\udd5c inst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : Measure.WeaklyRegular \u03bc inst\u271d : IsFiniteMeasure \u03bc this : NormedSpace \u211d E \u22a2 AddSubgroup.topologicalClosure (Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u22a4 ** simpa [range_toLp p \u03bc] using MeasureTheory.Lp.boundedContinuousFunction_dense E hp ** \u03b1 : Type u_1 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b1 inst\u271d\u2078 : T4Space \u03b1 inst\u271d\u2077 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 \ud835\udd5c : Type u_3 inst\u271d\u2074 : NormedField \ud835\udd5c inst\u271d\u00b3 : NormedAlgebra \u211d \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : Measure.WeaklyRegular \u03bc inst\u271d : IsFiniteMeasure \u03bc this\u271d : NormedSpace \u211d E this : AddSubgroup.topologicalClosure (Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u22a4 \u22a2 closure (range \u2191(toLp p \u03bc \ud835\udd5c)) = univ ** exact congr_arg ((\u2191) : AddSubgroup (Lp E p \u03bc) \u2192 Set (Lp E p \u03bc)) this ** Qed",
        "informal": ""
    },
    {
        "formal": "Char.ofNat_toNat ** c : Char h : isValidCharNat (toNat c) \u22a2 ofNat (toNat c) = c ** rw [Char.ofNat, dif_pos h] ** c : Char h : isValidCharNat (toNat c) \u22a2 ofNatAux (toNat c) h = c ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec'.rfindOpt ** n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b : \u2115 \u22a2 (b \u2208 Part.bind (Nat.rfind fun n_1 => Part.some (decide (1 - f (n_1 ::\u1d65 v) = 0))) fun a => \u2191pred (f (a ::\u1d65 v))) \u2194     b \u2208 Nat.rfindOpt fun a => ofNat (Option \u2115) (f (a ::\u1d65 v)) ** simp only [Nat.rfindOpt, exists_prop, tsub_eq_zero_iff_le, PFun.coe_val, Part.mem_bind_iff,\n  Part.mem_some_iff, Option.mem_def, Part.mem_coe] ** n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b : \u2115 \u22a2 (\u2203 a, (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) \u2227 b = pred (f (a ::\u1d65 v))) \u2194     \u2203 a,       (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))) \u2227         ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b ** refine'\n  exists_congr fun a => (and_congr (iff_of_eq _) Iff.rfl).trans (and_congr_right fun h => _) ** case refine'_1 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 \u22a2 (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) =     (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))) ** congr ** case refine'_1.e_a.e_p n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 \u22a2 (fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) = fun n_1 =>     \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) ** funext n ** case refine'_1.e_a.e_p.h n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n\u271d b a n : \u2115 \u22a2 Part.some (decide (1 \u2264 f (n ::\u1d65 v))) = \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n ::\u1d65 v))))) ** cases f (n ::\u1d65 v) <;> simp [Nat.succ_le_succ] ** case refine'_1.e_a.e_p.h.succ n\u271d\u00b9 : \u2115 f : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n\u271d\u00b9 b a n n\u271d : \u2115 \u22a2 true = Option.isSome (ofNat (Option \u2115) (succ n\u271d)) ** rfl ** case refine'_2 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) \u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b ** have := Nat.rfind_spec h ** case refine'_2 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) this : true \u2208 \u2191(some (Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))))) \u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b ** simp only [Part.coe_some, Part.mem_some_iff] at this ** case refine'_2 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) this : true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))) \u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b ** revert this ** case refine'_2 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) \u22a2 true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))) \u2192     (b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b) ** cases' f (a ::\u1d65 v) with c <;> intro this ** case refine'_2.succ n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) c : \u2115 this : true = Option.isSome (ofNat (Option \u2115) (succ c)) \u22a2 b = pred (succ c) \u2194 ofNat (Option \u2115) (succ c) = some b ** rw [\u2190 Option.some_inj, eq_comm] ** case refine'_2.succ n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) c : \u2115 this : true = Option.isSome (ofNat (Option \u2115) (succ c)) \u22a2 some (pred (succ c)) = some b \u2194 ofNat (Option \u2115) (succ c) = some b ** rfl ** case refine'_2.zero n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 hf : Partrec' \u2191f v : Vector \u2115 n b a : \u2115 h : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))) this : true = Option.isSome (ofNat (Option \u2115) zero) \u22a2 b = pred zero \u2194 ofNat (Option \u2115) zero = some b ** cases this ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.snorm_exponent_top_lim_le_liminf_snorm_exponent_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\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedAddCommGroup G \u03b9 : Type u_5 inst\u271d\u00b2 : Nonempty \u03b9 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 F f_lim : \u03b1 \u2192 F h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) \u22a2 snorm f_lim \u22a4 \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop ** rw [snorm_exponent_top_lim_eq_essSup_liminf h_lim] ** \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 \u03b9 : Type u_5 inst\u271d\u00b2 : Nonempty \u03b9 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 F f_lim : \u03b1 \u2192 F h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) \u22a2 essSup (fun x => liminf (fun m => \u2191\u2016f m x\u2016\u208a) atTop) \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop ** simp_rw [snorm_exponent_top, snormEssSup] ** \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 \u03b9 : Type u_5 inst\u271d\u00b2 : Nonempty \u03b9 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 F f_lim : \u03b1 \u2192 F h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) \u22a2 essSup (fun x => liminf (fun m => \u2191\u2016f m x\u2016\u208a) atTop) \u03bc \u2264 liminf (fun n => essSup (fun x => \u2191\u2016f n x\u2016\u208a) \u03bc) atTop ** exact ENNReal.essSup_liminf_le fun n => fun x => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.singularPart_mutuallySingular ** \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 \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** by_cases hl : s.HaveLebesgueDecomposition \u03bc ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hl : HaveLebesgueDecomposition s \u03bc \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** haveI := hl ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hl this : HaveLebesgueDecomposition s \u03bc \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** obtain \u27e8i, hi, hpos, hneg\u27e9 := s.toJordanDecomposition.mutuallySingular ** case pos.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hl this : HaveLebesgueDecomposition s \u03bc i : Set \u03b1 hi : MeasurableSet i hpos : \u2191\u2191(toJordanDecomposition s).posPart i = 0 hneg : \u2191\u2191(toJordanDecomposition s).negPart i\u1d9c = 0 \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** rw [s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add \u03bc] at hpos ** case pos.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hl this : HaveLebesgueDecomposition s \u03bc i : Set \u03b1 hi : MeasurableSet i hpos :   \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +             withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))       i =     0 hneg : \u2191\u2191(toJordanDecomposition s).negPart i\u1d9c = 0 \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** rw [s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add \u03bc] at hneg ** case pos.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hl this : HaveLebesgueDecomposition s \u03bc i : Set \u03b1 hi : MeasurableSet i hpos :   \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +             withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))       i =     0 hneg :   \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc +             withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc))       i\u1d9c =     0 \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** rw [add_apply, add_eq_zero_iff] at hpos hneg ** case pos.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hl this : HaveLebesgueDecomposition s \u03bc i : Set \u03b1 hi : MeasurableSet i hpos :   \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc) i = 0 \u2227     \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc)) i = 0 hneg :   \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc) i\u1d9c = 0 \u2227     \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc)) i\u1d9c = 0 \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** exact \u27e8i, hi, hpos.1, hneg.1\u27e9 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hl : \u00acHaveLebesgueDecomposition s \u03bc \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** rw [not_haveLebesgueDecomposition_iff] at hl ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hl :   \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc \u2228     \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** cases' hl with hp hn ** case neg.inl \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** rw [Measure.singularPart, dif_neg hp] ** case neg.inl \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc \u22a2 0 \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** exact MutuallySingular.zero_left ** case neg.inr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** rw [Measure.singularPart, Measure.singularPart, dif_neg hn] ** case neg.inr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 hn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc \u22a2 (if h : Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc then       (Classical.choose           (_ : \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bc \u2227 (toJordanDecomposition s).posPart = p.1 + withDensity \u03bc p.2)).1     else 0) \u27c2\u2098     0 ** exact MutuallySingular.zero_right ** Qed",
        "informal": ""
    },
    {
        "formal": "AlternatingMap.measure_parallelepiped ** E : Type u_1 inst\u271d\u00b9\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : MeasurableSpace E inst\u271d\u00b9\u00b2 : BorelSpace E inst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \u211d F inst\u271d\u2077 : CompleteSpace F s : Set E \u03b9 : Type u_3 G : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : BorelSpace G inst\u271d : FiniteDimensional \u211d G n : \u2115 _i : Fact (finrank \u211d G = n) \u03c9 : AlternatingMap \u211d G \u211d (Fin n) v : Fin n \u2192 G \u22a2 \u2191\u2191(AlternatingMap.measure \u03c9) (parallelepiped v) = ENNReal.ofReal |\u2191\u03c9 v| ** conv_rhs => rw [\u03c9.eq_smul_basis_det (finBasisOfFinrankEq \u211d G _i.out)] ** E : Type u_1 inst\u271d\u00b9\u2075 : NormedAddCommGroup E inst\u271d\u00b9\u2074 : NormedSpace \u211d E inst\u271d\u00b9\u00b3 : MeasurableSpace E inst\u271d\u00b9\u00b2 : BorelSpace E inst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \u211d F inst\u271d\u2077 : CompleteSpace F s : Set E \u03b9 : Type u_3 G : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : DecidableEq \u03b9 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : BorelSpace G inst\u271d : FiniteDimensional \u211d G n : \u2115 _i : Fact (finrank \u211d G = n) \u03c9 : AlternatingMap \u211d G \u211d (Fin n) v : Fin n \u2192 G \u22a2 \u2191\u2191(AlternatingMap.measure \u03c9) (parallelepiped v) =     ENNReal.ofReal       |\u2191(\u2191\u03c9 \u2191(finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)) \u2022               Basis.det (finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)))           v| ** simp only [addHaar_parallelepiped, AlternatingMap.measure, coe_nnreal_smul_apply,\n  AlternatingMap.smul_apply, Algebra.id.smul_eq_mul, abs_mul, ENNReal.ofReal_mul (abs_nonneg _),\n  Real.ennnorm_eq_ofReal_abs] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.variance_zero ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc\u271d \u03bc : Measure \u03a9 \u22a2 variance 0 \u03bc = 0 ** simp only [variance, evariance_zero, ENNReal.zero_toReal] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hasFiniteIntegral_of_fintype ** \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 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016f a\u2016 \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\u2016\u208a) ** apply ae_of_all \u03bc ** \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 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 \u22a2 \u2200 (a : \u03b1), \u2016f a\u2016 \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\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\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : NormedAddCommGroup \u03b2 inst\u271d\u00b2 : NormedAddCommGroup \u03b3 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 x : \u03b1 \u22a2 \u2016f x\u2016 \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\u2016\u208a) ** rw [\u2190 coe_nnnorm (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 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : NormedAddCommGroup \u03b2 inst\u271d\u00b2 : NormedAddCommGroup \u03b3 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 x : \u03b1 \u22a2 \u2191\u2016f x\u2016\u208a \u2264 \u2191(Finset.sup Finset.univ fun a => \u2016f a\u2016\u208a) ** apply NNReal.toReal_le_toReal ** 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\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : NormedAddCommGroup \u03b2 inst\u271d\u00b2 : NormedAddCommGroup \u03b3 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 x : \u03b1 \u22a2 \u2016f x\u2016\u208a \u2264 Finset.sup Finset.univ fun a => \u2016f a\u2016\u208a ** apply Finset.le_sup (Finset.mem_univ x) ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.mapIdx_induction ** \u03b1 : Type u_1 \u03b2 : Type u_2 as : Array \u03b1 f : Fin (size as) \u2192 \u03b1 \u2192 \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 p i (f i as[i]) \u2227 motive (i.val + 1) \u22a2 motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdx as f)[i] ** have := SatisfiesM_mapIdxM (m := Id) (as := as) (f := f) motive h0 ** \u03b1 : Type u_1 \u03b2 : Type u_2 as : Array \u03b1 f : Fin (size as) \u2192 \u03b1 \u2192 \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 p i (f i as[i]) \u2227 motive (i.val + 1) this :   \u2200 (p : Fin (size as) \u2192 \u03b2 \u2192 Prop),     (\u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f i as[i])) \u2192       SatisfiesM (fun arr => motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } arr[i])         (mapIdxM as f) \u22a2 motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdx as f)[i] ** simp [SatisfiesM_Id_eq] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 as : Array \u03b1 f : Fin (size as) \u2192 \u03b1 \u2192 \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 p i (f i as[i]) \u2227 motive (i.val + 1) this :   \u2200 (p : Fin (size as) \u2192 \u03b2 \u2192 Prop),     (\u2200 (i : Fin (size as)), motive i.val \u2192 p i (f i as[i.val]) \u2227 motive (i.val + 1)) \u2192       motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdxM as f)[i] \u22a2 motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } (mapIdx as f)[i] ** exact this _ hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.measurePreserving_zpow ** G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 \u22a2 map (fun g => g ^ n) \u03bc = \u03bc ** let f := @zpowGroupHom G _ n ** G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n \u22a2 map (fun g => g ^ n) \u03bc = \u03bc ** have hf : Continuous f := continuous_zpow n ** G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f \u22a2 map (fun g => g ^ n) \u03bc = \u03bc ** haveI : (\u03bc.map f).IsHaarMeasure :=\n  isHaarMeasure_map \u03bc f hf (RootableBy.surjective_pow G \u2124 hn) (by simp) ** G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f this : IsHaarMeasure (map (\u2191f) \u03bc) \u22a2 map (fun g => g ^ n) \u03bc = \u03bc ** obtain \u27e8C, -, -, hC\u27e9 := isHaarMeasure_eq_smul_isHaarMeasure (\u03bc.map f) \u03bc ** case intro.intro.intro G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f this : IsHaarMeasure (map (\u2191f) \u03bc) C : \u211d\u22650\u221e hC : map (\u2191f) \u03bc = C \u2022 \u03bc \u22a2 map (fun g => g ^ n) \u03bc = \u03bc ** suffices C = 1 by rwa [this, one_smul] at hC ** case intro.intro.intro G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f this : IsHaarMeasure (map (\u2191f) \u03bc) C : \u211d\u22650\u221e hC : map (\u2191f) \u03bc = C \u2022 \u03bc \u22a2 C = 1 ** have h_univ : (\u03bc.map f) univ = \u03bc univ := by\n  rw [map_apply_of_aemeasurable hf.measurable.aemeasurable MeasurableSet.univ,\n    preimage_univ] ** case intro.intro.intro G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f this : IsHaarMeasure (map (\u2191f) \u03bc) C : \u211d\u22650\u221e hC : map (\u2191f) \u03bc = C \u2022 \u03bc h_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ \u22a2 C = 1 ** have h\u03bc\u2080 : \u03bc univ \u2260 0 := IsOpenPosMeasure.open_pos univ isOpen_univ univ_nonempty ** case intro.intro.intro G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f this : IsHaarMeasure (map (\u2191f) \u03bc) C : \u211d\u22650\u221e hC : map (\u2191f) \u03bc = C \u2022 \u03bc h_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ h\u03bc\u2080 : \u2191\u2191\u03bc univ \u2260 0 \u22a2 C = 1 ** have h\u03bc\u2081 : \u03bc univ \u2260 \u221e := CompactSpace.isFiniteMeasure.measure_univ_lt_top.ne ** case intro.intro.intro G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f this : IsHaarMeasure (map (\u2191f) \u03bc) C : \u211d\u22650\u221e hC : map (\u2191f) \u03bc = C \u2022 \u03bc h_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ h\u03bc\u2080 : \u2191\u2191\u03bc univ \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc univ \u2260 \u22a4 \u22a2 C = 1 ** rwa [hC, smul_apply, Algebra.id.smul_eq_mul, mul_comm, \u2190 ENNReal.eq_div_iff h\u03bc\u2080 h\u03bc\u2081,\n  ENNReal.div_self h\u03bc\u2080 h\u03bc\u2081] at h_univ ** G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f \u22a2 Filter.Tendsto (\u2191f) (Filter.cocompact G) (Filter.cocompact G) ** simp ** G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f this\u271d : IsHaarMeasure (map (\u2191f) \u03bc) C : \u211d\u22650\u221e hC : map (\u2191f) \u03bc = C \u2022 \u03bc this : C = 1 \u22a2 map (fun g => g ^ n) \u03bc = \u03bc ** rwa [this, one_smul] at hC ** G : Type u_1 inst\u271d\u2079 : CommGroup G inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : TopologicalGroup G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : BorelSpace G inst\u271d\u00b3 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b2 : IsHaarMeasure \u03bc inst\u271d\u00b9 : CompactSpace G inst\u271d : RootableBy G \u2124 n : \u2124 hn : n \u2260 0 f : G \u2192* G := zpowGroupHom n hf : Continuous \u2191f this : IsHaarMeasure (map (\u2191f) \u03bc) C : \u211d\u22650\u221e hC : map (\u2191f) \u03bc = C \u2022 \u03bc \u22a2 \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ ** rw [map_apply_of_aemeasurable hf.measurable.aemeasurable MeasurableSet.univ,\n  preimage_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.Range.forIn'_eq_forIn_range' ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : (a : Nat) \u2192 a \u2208 r \u2192 \u03b2 \u2192 m (ForInStep \u03b2) \u22a2 forIn' r init f =     forIn       (List.pmap Subtype.mk (List.range' r.start (numElems r) r.step)         (_ : \u2200 (x : Nat), x \u2208 List.range' r.start (numElems r) r.step \u2192 x \u2208 r))       init fun x =>       match (motive := { x // x \u2208 r } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with       | { val := a, property := h } => f a h ** let \u27e8start, stop, step\u27e9 := r ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) \u22a2 forIn' { start := start, stop := stop, step := step } init f =     forIn       (List.pmap Subtype.mk         (List.range' { start := start, stop := stop, step := step }.start           (numElems { start := start, stop := stop, step := step }) { start := start, stop := stop, step := step }.step)         (_ :           \u2200 (x : Nat),             x \u2208                 List.range' { start := start, stop := stop, step := step }.start                   (numElems { start := start, stop := stop, step := step })                   { start := start, stop := stop, step := step }.step \u2192               x \u2208 { start := start, stop := stop, step := step }))       init fun x =>       match (motive := { x // x \u2208 { start := start, stop := stop, step := step } } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with       | { val := a, property := h } => f a h ** let L := List.range' start (numElems \u27e8start, stop, step\u27e9) step ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step \u22a2 forIn' { start := start, stop := stop, step := step } init f =     forIn       (List.pmap Subtype.mk         (List.range' { start := start, stop := stop, step := step }.start           (numElems { start := start, stop := stop, step := step }) { start := start, stop := stop, step := step }.step)         (_ :           \u2200 (x : Nat),             x \u2208                 List.range' { start := start, stop := stop, step := step }.start                   (numElems { start := start, stop := stop, step := step })                   { start := start, stop := stop, step := step }.step \u2192               x \u2208 { start := start, stop := stop, step := step }))       init fun x =>       match (motive := { x // x \u2208 { start := start, stop := stop, step := step } } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with       | { val := a, property := h } => f a h ** let f' : { a // start \u2264 a \u2227 a < stop } \u2192 _ := fun \u27e8a, h\u27e9 => f a h ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h \u22a2 forIn' { start := start, stop := stop, step := step } init f =     forIn       (List.pmap Subtype.mk         (List.range' { start := start, stop := stop, step := step }.start           (numElems { start := start, stop := stop, step := step }) { start := start, stop := stop, step := step }.step)         (_ :           \u2200 (x : Nat),             x \u2208                 List.range' { start := start, stop := stop, step := step }.start                   (numElems { start := start, stop := stop, step := step })                   { start := start, stop := stop, step := step }.step \u2192               x \u2208 { start := start, stop := stop, step := step }))       init fun x =>       match (motive := { x // x \u2208 { start := start, stop := stop, step := step } } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with       | { val := a, property := h } => f a h ** suffices \u2200 H, forIn' [start:stop:step] init f = forIn (L.pmap Subtype.mk H) init f' from this _ ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h \u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop),     forIn' { start := start, stop := stop, step := step } init f = forIn (List.pmap Subtype.mk L H) init f' ** intro H ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop \u22a2 forIn' { start := start, stop := stop, step := step } init f = forIn (List.pmap Subtype.mk L H) init f' ** dsimp only [forIn', Range.forIn'] ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop \u22a2 forIn'.loop start stop step f stop start       (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)       init =     forIn (List.pmap Subtype.mk (List.range' start (numElems { start := start, stop := stop, step := step }) step) H)       init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** simp [numElems, Nat.not_le.2 h] ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop \u22a2 forIn'.loop start stop step f stop start       (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)       init =     forIn       (List.pmap Subtype.mk (List.range' start (if step = 0 then stop else (stop - start + step - 1) / step) step)         (_ :           \u2200 (a : Nat),             a \u2208 List.range' start (if step = 0 then stop else (stop - start + step - 1) / step) step \u2192               (fun a => start \u2264 a \u2227 a < stop) a))       init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** split ** case inl m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop h\u271d : step = 0 \u22a2 forIn'.loop start stop step f stop start       (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)       init =     forIn       (List.pmap Subtype.mk (List.range' start stop step)         (_ : \u2200 (a : Nat), a \u2208 List.range' start stop step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))       init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** subst step ** case inl m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop : Nat h : start < stop f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0 f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop \u22a2 forIn'.loop start stop 0 f stop start       (_ : { start := start, stop := stop, step := 0 }.start \u2264 { start := start, stop := stop, step := 0 }.start) init =     forIn       (List.pmap Subtype.mk (List.range' start stop 0)         (_ : \u2200 (a : Nat), a \u2208 List.range' start stop 0 \u2192 (fun a => start \u2264 a \u2227 a < stop) a))       init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** suffices \u2200 n H init,\n    forIn'.loop start stop 0 f n start (Nat.le_refl _) init =\n    forIn ((List.range' start n 0).pmap Subtype.mk H) init f' from this _ .. ** case inl m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop : Nat h : start < stop f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0 f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop \u22a2 \u2200 (n : Nat) (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),     forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =       forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f' ** intro n ** case inl m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop : Nat h : start < stop f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0 f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop n : Nat \u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),     forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =       forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f' ** induction n with (intro H init; unfold forIn'.loop; simp [*])\n| succ n ih => simp [ih (List.forall_mem_cons.1 H).2]; rfl ** case inl.zero m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop : Nat h : start < stop f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0 f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop \u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start Nat.zero 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),     forIn'.loop start stop 0 f Nat.zero start (_ : start \u2264 start) init =       forIn (List.pmap Subtype.mk (List.range' start Nat.zero 0) H) init f' ** intro H init ** case inl.zero m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop : Nat h : start < stop f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0 f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop H : \u2200 (a : Nat), a \u2208 List.range' start Nat.zero 0 \u2192 start \u2264 a \u2227 a < stop init : \u03b2 \u22a2 forIn'.loop start stop 0 f Nat.zero start (_ : start \u2264 start) init =     forIn (List.pmap Subtype.mk (List.range' start Nat.zero 0) H) init f' ** unfold forIn'.loop ** case inl.zero m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop : Nat h : start < stop f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0 f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop H : \u2200 (a : Nat), a \u2208 List.range' start Nat.zero 0 \u2192 start \u2264 a \u2227 a < stop init : \u03b2 \u22a2 (if hu : start < stop then       match Nat.zero with       | 0 => pure init       | Nat.succ fuel => do         let __do_lift \u2190 f start (_ : start \u2264 start \u2227 start < stop) init         match __do_lift with           | ForInStep.done b => pure b           | ForInStep.yield b => forIn'.loop start stop 0 f fuel (start + 0) (_ : start \u2264 start + 0) b     else pure init) =     forIn (List.pmap Subtype.mk (List.range' start Nat.zero 0) H) init f' ** simp [*] ** case inl.succ m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop : Nat h : start < stop f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0 f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop n : Nat ih :   \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),     forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =       forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f' H : \u2200 (a : Nat), a \u2208 List.range' start (Nat.succ n) 0 \u2192 start \u2264 a \u2227 a < stop init : \u03b2 \u22a2 (do       let __do_lift \u2190 f start (_ : start \u2264 start \u2227 start < stop) init       match __do_lift with         | ForInStep.done b => pure b         | ForInStep.yield b => forIn'.loop start stop 0 f n start (_ : start \u2264 start) b) =     do     let x \u2190       f start           (_ :             start \u2264 { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val \u2227               { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val < stop)           init     match x with       | ForInStep.done b => pure b       | ForInStep.yield b =>         forIn           (List.pmap Subtype.mk (List.range' start n 0)             (_ : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 (fun a => (fun a => start \u2264 a \u2227 a < stop) a) a))           b fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** simp [ih (List.forall_mem_cons.1 H).2] ** case inl.succ m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop : Nat h : start < stop f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := 0 } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := 0 }) 0 f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop n : Nat ih :   \u2200 (H : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 start \u2264 a \u2227 a < stop) (init : \u03b2),     forIn'.loop start stop 0 f n start (_ : start \u2264 start) init =       forIn (List.pmap Subtype.mk (List.range' start n 0) H) init f' H : \u2200 (a : Nat), a \u2208 List.range' start (Nat.succ n) 0 \u2192 start \u2264 a \u2227 a < stop init : \u03b2 \u22a2 (do       let __do_lift \u2190 f start (_ : start \u2264 start \u2227 start < stop) init       match __do_lift with         | ForInStep.done b => pure b         | ForInStep.yield b =>           forIn             (List.pmap Subtype.mk (List.range' start n 0)               (_ : \u2200 (x : Nat), x \u2208 List.range' (start + 0) n 0 \u2192 start \u2264 x \u2227 x < stop))             b fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop)) =     do     let x \u2190       f start           (_ :             start \u2264 { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val \u2227               { val := start, property := (_ : start \u2264 start \u2227 start < stop) }.val < stop)           init     match x with       | ForInStep.done b => pure b       | ForInStep.yield b =>         forIn           (List.pmap Subtype.mk (List.range' start n 0)             (_ : \u2200 (a : Nat), a \u2208 List.range' start n 0 \u2192 (fun a => (fun a => start \u2264 a \u2227 a < stop) a) a))           b fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** rfl ** case inr m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop h\u271d : \u00acstep = 0 \u22a2 forIn'.loop start stop step f stop start       (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)       init =     forIn       (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)         (_ :           \u2200 (a : Nat),             a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))       init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** next step0 =>\nhave hstep := Nat.pos_of_ne_zero step0\nsuffices \u2200 fuel l i hle H, l \u2264 fuel \u2192\n    (\u2200 j, stop \u2264 i + step * j \u2194 l \u2264 j) \u2192 \u2200 init,\n    forIn'.loop start stop step f fuel i hle init =\n    List.forIn ((List.range' i l step).pmap Subtype.mk H) init f' by\n  refine this _ _ _ _ _\n    ((numElems_le_iff hstep).2 (Nat.le_trans ?_ (Nat.le_add_left ..)))\n    (fun _ => (numElems_le_iff hstep).symm) _\n  conv => lhs; rw [\u2190 Nat.one_mul stop]\n  exact Nat.mul_le_mul_right stop hstep\nintro fuel; induction fuel with intro l i hle H h1 h2 init\n| zero => simp [forIn'.loop, Nat.le_zero.1 h1]; split <;> simp\n| succ fuel ih =>\n  cases l with\n  | zero => rw [forIn'.loop]; simp [Nat.not_lt.2 <| by simpa using (h2 0).2 (Nat.le_refl _)]\n  | succ l =>\n    have ih := ih _ _ (Nat.le_trans hle (Nat.le_add_right ..))\n      (List.forall_mem_cons.1 H).2 (Nat.le_of_succ_le_succ h1) fun i => by\n        rw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff]\n    have := h2 0; simp at this\n    rw [forIn'.loop]; simp [List.forIn, this, ih]; rfl ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 \u22a2 forIn'.loop start stop step f stop start       (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)       init =     forIn       (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)         (_ :           \u2200 (a : Nat),             a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))       init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** have hstep := Nat.pos_of_ne_zero step0 ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step \u22a2 forIn'.loop start stop step f stop start       (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)       init =     forIn       (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)         (_ :           \u2200 (a : Nat),             a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))       init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** suffices \u2200 fuel l i hle H, l \u2264 fuel \u2192\n    (\u2200 j, stop \u2264 i + step * j \u2194 l \u2264 j) \u2192 \u2200 init,\n    forIn'.loop start stop step f fuel i hle init =\n    List.forIn ((List.range' i l step).pmap Subtype.mk H) init f' by\n  refine this _ _ _ _ _\n    ((numElems_le_iff hstep).2 (Nat.le_trans ?_ (Nat.le_add_left ..)))\n    (fun _ => (numElems_le_iff hstep).symm) _\n  conv => lhs; rw [\u2190 Nat.one_mul stop]\n  exact Nat.mul_le_mul_right stop hstep ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step \u22a2 \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' ** intro fuel ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat \u22a2 \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' ** induction fuel with intro l i hle H h1 h2 init\n| zero => simp [forIn'.loop, Nat.le_zero.1 h1]; split <;> simp\n| succ fuel ih =>\ncases l with\n| zero => rw [forIn'.loop]; simp [Nat.not_lt.2 <| by simpa using (h2 0).2 (Nat.le_refl _)]\n| succ l =>\nhave ih := ih _ _ (Nat.le_trans hle (Nat.le_add_right ..))\n(List.forall_mem_cons.1 H).2 (Nat.le_of_succ_le_succ h1) fun i => by\nrw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff]\nhave := h2 0; simp at this\nrw [forIn'.loop]; simp [List.forIn, this, ih]; rfl ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step this :   \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' \u22a2 forIn'.loop start stop step f stop start       (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)       init =     forIn       (List.pmap Subtype.mk (List.range' start ((stop - start + step - 1) / step) step)         (_ :           \u2200 (a : Nat),             a \u2208 List.range' start ((stop - start + step - 1) / step) step \u2192 (fun a => start \u2264 a \u2227 a < stop) a))       init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** refine this _ _ _ _ _\n  ((numElems_le_iff hstep).2 (Nat.le_trans ?_ (Nat.le_add_left ..)))\n  (fun _ => (numElems_le_iff hstep).symm) _ ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step this :   \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' \u22a2 stop \u2264 step * stop ** conv => lhs; rw [\u2190 Nat.one_mul stop] ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step this :   \u2200 (fuel l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' \u22a2 1 * stop \u2264 step * stop ** exact Nat.mul_le_mul_right stop hstep ** case zero m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step l i : Nat hle : start \u2264 i H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop h1 : l \u2264 Nat.zero h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j init : \u03b2 \u22a2 forIn'.loop start stop step f Nat.zero i hle init = List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' ** simp [forIn'.loop, Nat.le_zero.1 h1] ** case zero m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step l i : Nat hle : start \u2264 i H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop h1 : l \u2264 Nat.zero h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j init : \u03b2 \u22a2 (if hu : i < stop then pure init else pure init) = pure init ** split <;> simp ** case succ m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' l i : Nat hle : start \u2264 i H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop h1 : l \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j init : \u03b2 \u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =     List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' ** cases l with\n| zero => rw [forIn'.loop]; simp [Nat.not_lt.2 <| by simpa using (h2 0).2 (Nat.le_refl _)]\n| succ l =>\n  have ih := ih _ _ (Nat.le_trans hle (Nat.le_add_right ..))\n    (List.forall_mem_cons.1 H).2 (Nat.le_of_succ_le_succ h1) fun i => by\n      rw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff]\n  have := h2 0; simp at this\n  rw [forIn'.loop]; simp [List.forIn, this, ih]; rfl ** case succ.zero m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i : Nat hle : start \u2264 i init : \u03b2 H : \u2200 (a : Nat), a \u2208 List.range' i Nat.zero step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.zero \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.zero \u2264 j \u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =     List.forIn (List.pmap Subtype.mk (List.range' i Nat.zero step) H) init f' ** rw [forIn'.loop] ** case succ.zero m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i : Nat hle : start \u2264 i init : \u03b2 H : \u2200 (a : Nat), a \u2208 List.range' i Nat.zero step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.zero \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.zero \u2264 j \u22a2 (if hu : i < stop then       match Nat.succ fuel with       | 0 => pure init       | Nat.succ fuel => do         let __do_lift \u2190 f i (_ : start \u2264 i \u2227 i < stop) init         match __do_lift with           | ForInStep.done b => pure b           | ForInStep.yield b => forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) b     else pure init) =     List.forIn (List.pmap Subtype.mk (List.range' i Nat.zero step) H) init f' ** simp [Nat.not_lt.2 <| by simpa using (h2 0).2 (Nat.le_refl _)] ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i : Nat hle : start \u2264 i init : \u03b2 H : \u2200 (a : Nat), a \u2208 List.range' i Nat.zero step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.zero \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.zero \u2264 j \u22a2 ?m.22038 \u2264 ?m.22037 ** simpa using (h2 0).2 (Nat.le_refl _) ** case succ.succ m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i : Nat hle : start \u2264 i init : \u03b2 l : Nat H : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.succ l \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j \u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =     List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f' ** have ih := ih _ _ (Nat.le_trans hle (Nat.le_add_right ..))\n  (List.forall_mem_cons.1 H).2 (Nat.le_of_succ_le_succ h1) fun i => by\n    rw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff] ** case succ.succ m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih\u271d :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i : Nat hle : start \u2264 i init : \u03b2 l : Nat H : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.succ l \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j ih :   \u2200 (init : \u03b2),     forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =       List.forIn         (List.pmap Subtype.mk (List.range' (i + step) l step)           (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))         init f' \u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =     List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f' ** have := h2 0 ** case succ.succ m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih\u271d :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i : Nat hle : start \u2264 i init : \u03b2 l : Nat H : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.succ l \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j ih :   \u2200 (init : \u03b2),     forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =       List.forIn         (List.pmap Subtype.mk (List.range' (i + step) l step)           (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))         init f' this : stop \u2264 i + step * 0 \u2194 Nat.succ l \u2264 0 \u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =     List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f' ** simp at this ** case succ.succ m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih\u271d :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i : Nat hle : start \u2264 i init : \u03b2 l : Nat H : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.succ l \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j ih :   \u2200 (init : \u03b2),     forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =       List.forIn         (List.pmap Subtype.mk (List.range' (i + step) l step)           (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))         init f' this : i < stop \u22a2 forIn'.loop start stop step f (Nat.succ fuel) i hle init =     List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f' ** rw [forIn'.loop] ** case succ.succ m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih\u271d :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i : Nat hle : start \u2264 i init : \u03b2 l : Nat H : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.succ l \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j ih :   \u2200 (init : \u03b2),     forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =       List.forIn         (List.pmap Subtype.mk (List.range' (i + step) l step)           (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))         init f' this : i < stop \u22a2 (if hu : i < stop then       match Nat.succ fuel with       | 0 => pure init       | Nat.succ fuel => do         let __do_lift \u2190 f i (_ : start \u2264 i \u2227 i < stop) init         match __do_lift with           | ForInStep.done b => pure b           | ForInStep.yield b => forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) b     else pure init) =     List.forIn (List.pmap Subtype.mk (List.range' i (Nat.succ l) step) H) init f' ** simp [List.forIn, this, ih] ** case succ.succ m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih\u271d :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i : Nat hle : start \u2264 i init : \u03b2 l : Nat H : \u2200 (a : Nat), a \u2208 List.range' i (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.succ l \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i + step * j \u2194 Nat.succ l \u2264 j ih :   \u2200 (init : \u03b2),     forIn'.loop start stop step f fuel (i + step) (_ : start \u2264 i + step) init =       List.forIn         (List.pmap Subtype.mk (List.range' (i + step) l step)           (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))         init f' this : i < stop \u22a2 (do       let __do_lift \u2190 f i (_ : start \u2264 i \u2227 i < stop) init       match __do_lift with         | ForInStep.done b => pure b         | ForInStep.yield b =>           List.forIn.loop (fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop))             (List.pmap Subtype.mk (List.range' (i + step) l step)               (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))             b) =     List.forIn.loop (fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop))       ({ val := i, property := (_ : start \u2264 i \u2227 i < stop) } ::         List.pmap Subtype.mk (List.range' (i + step) l step)           (_ : \u2200 (x : Nat), x \u2208 List.range' (i + step) l step \u2192 start \u2264 x \u2227 x < stop))       init ** rfl ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init\u271d : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H\u271d : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : start < stop step0 : \u00acstep = 0 hstep : 0 < step fuel : Nat ih :   \u2200 (l i : Nat) (hle : start \u2264 i) (H : \u2200 (a : Nat), a \u2208 List.range' i l step \u2192 start \u2264 a \u2227 a < stop),     l \u2264 fuel \u2192       (\u2200 (j : Nat), stop \u2264 i + step * j \u2194 l \u2264 j) \u2192         \u2200 (init : \u03b2),           forIn'.loop start stop step f fuel i hle init =             List.forIn (List.pmap Subtype.mk (List.range' i l step) H) init f' i\u271d : Nat hle : start \u2264 i\u271d init : \u03b2 l : Nat H : \u2200 (a : Nat), a \u2208 List.range' i\u271d (Nat.succ l) step \u2192 start \u2264 a \u2227 a < stop h1 : Nat.succ l \u2264 Nat.succ fuel h2 : \u2200 (j : Nat), stop \u2264 i\u271d + step * j \u2194 Nat.succ l \u2264 j i : Nat \u22a2 stop \u2264 i\u271d + step + step * i \u2194 l \u2264 i ** rw [Nat.add_right_comm, Nat.add_assoc, \u2190 Nat.mul_succ, h2, Nat.succ_le_succ_iff] ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : \u00acstart < stop \u22a2 forIn'.loop start stop step f stop start       (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)       init =     forIn (List.pmap Subtype.mk (List.range' start (numElems { start := start, stop := stop, step := step }) step) H)       init fun x => f x.val (_ : start \u2264 x.val \u2227 x.val < stop) ** simp [List.range', h, numElems_stop_le_start \u27e8start, stop, step\u27e9 (Nat.not_lt.1 h)] ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 start stop step : Nat f : (a : Nat) \u2192 a \u2208 { start := start, stop := stop, step := step } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat := List.range' start (numElems { start := start, stop := stop, step := step }) step f' : { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2) :=   fun x =>     match (motive := { a // start \u2264 a \u2227 a < stop } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with     | { val := a, property := h } => f a h H : \u2200 (a : Nat), a \u2208 L \u2192 start \u2264 a \u2227 a < stop h : \u00acstart < stop \u22a2 forIn'.loop start stop step f stop start       (_ : { start := start, stop := stop, step := step }.start \u2264 { start := start, stop := stop, step := step }.start)       init =     pure init ** cases stop <;> unfold forIn'.loop <;> simp [List.forIn', h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.tendsto_Icc_vitaliFamily_right ** x : \u211d \u22a2 Tendsto (fun y => Icc x y) (\ud835\udcdd[Ioi x] x) (VitaliFamily.filterAt (vitaliFamily volume 1) x) ** refine' (VitaliFamily.tendsto_filterAt_iff _).2 \u27e8_, _\u27e9 ** case refine'_1 x : \u211d \u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2208 VitaliFamily.setsAt (vitaliFamily volume 1) x ** filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy ** case refine'_2 x : \u211d \u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2286 Metric.closedBall x \u03b5 ** intro \u03b5 \u03b5pos ** case refine'_2 x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2286 Metric.closedBall x \u03b5 ** have : x \u2208 Ico x (x + \u03b5) := \u27e8le_refl _, by linarith\u27e9 ** case refine'_2 x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 this : x \u2208 Ico x (x + \u03b5) \u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Ioi x] x, Icc x i \u2286 Metric.closedBall x \u03b5 ** filter_upwards [Icc_mem_nhdsWithin_Ioi this] with y hy ** case h x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 this : x \u2208 Ico x (x + \u03b5) y : \u211d hy : y \u2208 Icc x (x + \u03b5) \u22a2 Icc x y \u2286 Metric.closedBall x \u03b5 ** rw [closedBall_eq_Icc] ** case h x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 this : x \u2208 Ico x (x + \u03b5) y : \u211d hy : y \u2208 Icc x (x + \u03b5) \u22a2 Icc x y \u2286 Icc (x - \u03b5) (x + \u03b5) ** exact Icc_subset_Icc (by linarith) hy.2 ** x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 x < x + \u03b5 ** linarith ** x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 this : x \u2208 Ico x (x + \u03b5) y : \u211d hy : y \u2208 Icc x (x + \u03b5) \u22a2 x - \u03b5 \u2264 x ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.map_traverse ** \u03b1 \u03b2 \u03b3 : Type u F G : Type u \u2192 Type u inst\u271d\u00b3 : Applicative F inst\u271d\u00b2 : Applicative G inst\u271d\u00b9 : CommApplicative F inst\u271d : CommApplicative G g : \u03b1 \u2192 G \u03b2 h : \u03b2 \u2192 \u03b3 s : Finset \u03b1 \u22a2 Functor.map h <$> traverse g s = traverse (Functor.map h \u2218 g) s ** unfold traverse ** \u03b1 \u03b2 \u03b3 : Type u F G : Type u \u2192 Type u inst\u271d\u00b3 : Applicative F inst\u271d\u00b2 : Applicative G inst\u271d\u00b9 : CommApplicative F inst\u271d : CommApplicative G g : \u03b1 \u2192 G \u03b2 h : \u03b2 \u2192 \u03b3 s : Finset \u03b1 \u22a2 Functor.map h <$> Multiset.toFinset <$> Multiset.traverse g s.val =     Multiset.toFinset <$> Multiset.traverse (Functor.map h \u2218 g) s.val ** simp only [map_comp_coe, functor_norm] ** \u03b1 \u03b2 \u03b3 : Type u F G : Type u \u2192 Type u inst\u271d\u00b3 : Applicative F inst\u271d\u00b2 : Applicative G inst\u271d\u00b9 : CommApplicative F inst\u271d : CommApplicative G g : \u03b1 \u2192 G \u03b2 h : \u03b2 \u2192 \u03b3 s : Finset \u03b1 \u22a2 (Multiset.toFinset \u2218 Functor.map h) <$> Multiset.traverse g s.val =     Multiset.toFinset <$> Multiset.traverse (Functor.map h \u2218 g) s.val ** rw [LawfulFunctor.comp_map, Multiset.map_traverse] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.data_join ** ss : List String \u22a2 (join ss).data = List.join (List.map data ss) ** rw [join_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.zero_testAgainstNN_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 : TopologicalSpace \u03a9 f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 testAgainstNN 0 f = 0 ** simp only [testAgainstNN, toMeasure_zero, lintegral_zero_measure, ENNReal.zero_toNNReal] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.maps_range_to ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 s\u271d s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 p : Set \u03b3 f\u271d f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2 g\u271d g\u2081 g\u2082 : \u03b2 \u2192 \u03b3 f' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1 g' : \u03b3 \u2192 \u03b2 a : \u03b1 b : \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b3 \u2192 \u03b1 s : Set \u03b2 \u22a2 MapsTo f (range g) s \u2194 MapsTo (f \u2218 g) univ s ** rw [\u2190 image_univ, maps_image_to] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vitali.exists_disjoint_covering_ae ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x this : \u2200 (x : \u03b1), \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4 \u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0 ** choose R hR0 hR1 hR\u03bc using this ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 \u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0 ** let t' := { a \u2208 t | r a \u2264 R (c a) } ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} \u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0 ** obtain \u27e8u, ut', u_disj, hu\u27e9 : \u2203 (u : _) (_ : u \u2286 t'),\n    u.PairwiseDisjoint B \u2227 \u2200 a \u2208 t', \u2203 b \u2208 u, (B a \u2229 B b).Nonempty \u2227 r a \u2264 2 * r b := by\n  have A : \u2200 a \u2208 t', r a \u2264 1 := by\n    intro a ha\n    apply ha.2.trans (hR1 (c a))\n  have A' : \u2200 a \u2208 t', (B a).Nonempty :=\n    fun a hat' => Set.Nonempty.mono interior_subset (ht a hat'.1)\n  refine' exists_disjoint_subfamily_covering_enlargment B t' r 2 one_lt_two (fun a ha => _) 1 A A'\n  exact nonempty_closedBall.1 ((A' a ha).mono (hB a ha.1)) ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b \u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0 ** have ut : u \u2286 t := fun a hau => (ut' hau).1 ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t \u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0 ** have u_count : u.Countable := u_disj.countable_of_nonempty_interior fun a ha => ht a (ut ha) ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u \u22a2 \u2203 u x, Set.Countable u \u2227 PairwiseDisjoint u B \u2227 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0 ** refine' \u27e8u, fun a hat' => (ut' hat').1, u_count, u_disj, _\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 a \u2208 u, B a) = 0 ** refine' null_of_locally_null _ fun x _ => _ ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a \u22a2 \u2203 u_1, u_1 \u2208 \ud835\udcdd[s \\ \u22c3 a \u2208 u, B a] x \u2227 \u2191\u2191\u03bc u_1 = 0 ** let v := { a \u2208 u | (B a \u2229 ball x (R x)).Nonempty } ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} \u22a2 \u2203 u_1, u_1 \u2208 \ud835\udcdd[s \\ \u22c3 a \u2208 u, B a] x \u2227 \u2191\u2191\u03bc u_1 = 0 ** have vu : v \u2286 u := fun a ha => ha.1 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u22a2 \u2203 u_1, u_1 \u2208 \ud835\udcdd[s \\ \u22c3 a \u2208 u, B a] x \u2227 \u2191\u2191\u03bc u_1 = 0 ** refine' \u27e8_ \u2229 ball x (R x), inter_mem_nhdsWithin _ (ball_mem_nhds _ (hR0 _)),\n  nonpos_iff_eq_zero.mp (le_of_forall_le_of_dense fun \u03b5 \u03b5pos => _)\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 \u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5 ** have I : (\u2211' a : v, \u03bc (B a)) < \u221e := by\n  calc\n    (\u2211' a : v, \u03bc (B a)) = \u03bc (\u22c3 a \u2208 v, B a) := by\n      rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]\n      exact u_disj.subset vu\n    _ \u2264 \u03bc (closedBall x K) := (measure_mono (iUnion\u2082_subset fun a ha => hK a (vu ha) ha.2))\n    _ < \u221e := \u03bcK ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 \u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5 ** obtain \u27e8w, hw\u27e9 : \u2203 w : Finset v, (\u2211' a : { a // a \u2209 w }, \u03bc (B a)) < \u03b5 / C :=\n  haveI : 0 < \u03b5 / C := by\n    simp only [ENNReal.div_pos_iff, \u03b5pos.ne', ENNReal.coe_ne_top, Ne.def, not_false_iff,\n      and_self_iff]\n  ((tendsto_order.1 (ENNReal.tendsto_tsum_compl_atTop_zero I.ne)).2 _ this).exists ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C \u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5 ** have M : (s \\ \u22c3 a \u2208 u, B a) \u2229\n    ball x (R x) \u2286 \u22c3 a : { a // a \u2209 w }, closedBall (c a) (3 * r a) := by\n  intro z hz\n  set k := \u22c3 (a : v) (_ : a \u2208 w), B a\n  have k_closed : IsClosed k := isClosed_biUnion_finset fun i _ => h't _ (ut (vu i.2))\n  have z_notmem_k : z \u2209 k := by\n    simp only [not_exists, exists_prop, mem_iUnion, mem_sep_iff, forall_exists_index,\n      SetCoe.exists, not_and, exists_and_right, Subtype.coe_mk]\n    intro b hbv _ h'z\n    have : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a :=\n      mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z)\n    simpa only [diff_inter_self]\n  have : ball x (R x) \\ k \u2208 \ud835\udcdd z := by\n    apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _\n    exact (mem_diff _).2 \u27e8mem_of_mem_inter_right hz, z_notmem_k\u27e9\n  obtain \u27e8d, dpos, hd\u27e9 : \u2203 d, 0 < d \u2227 closedBall z d \u2286 ball x (R x) \\ k :=\n    nhds_basis_closedBall.mem_iff.1 this\n  obtain \u27e8a, hat, ad, rfl\u27e9 : \u2203 a \u2208 t, r a \u2264 min d (R z) \u2227 c a = z\n  exact hf z ((mem_diff _).1 (mem_of_mem_inter_left hz)).1 (min d (R z)) (lt_min dpos (hR0 z))\n  have ax : B a \u2286 ball x (R x) := by\n    refine' (hB a hat).trans _\n    refine' Subset.trans _ (hd.trans (diff_subset (ball x (R x)) k))\n    exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))\n  obtain \u27e8b, bu, ab, bdiam\u27e9 : \u2203 b \u2208 u, (B a \u2229 B b).Nonempty \u2227 r a \u2264 2 * r b\n  exact hu a \u27e8hat, ad.trans (min_le_right _ _)\u27e9\n  have bv : b \u2208 v := by\n    refine' \u27e8bu, ab.mono _\u27e9\n    rw [inter_comm]\n    exact inter_subset_inter_right _ ax\n  let b' : v := \u27e8b, bv\u27e9\n  have b'_notmem_w : b' \u2209 w := by\n    intro b'w\n    have b'k : B b' \u2286 k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w\n    have : (ball x (R x) \\ k \u2229 k).Nonempty := by\n      apply ab.mono (inter_subset_inter _ b'k)\n      refine' ((hB _ hat).trans _).trans hd\n      exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))\n    simpa only [diff_inter_self, Set.not_nonempty_empty]\n  let b'' : { a // a \u2209 w } := \u27e8b', b'_notmem_w\u27e9\n  have zb : c a \u2208 closedBall (c b) (3 * r b) := by\n    rcases ab with \u27e8e, \u27e8ea, eb\u27e9\u27e9\n    have A : dist (c a) e \u2264 r a := mem_closedBall'.1 (hB a hat ea)\n    have B : dist e (c b) \u2264 r b := mem_closedBall.1 (hB b (ut bu) eb)\n    simp only [mem_closedBall]\n    linarith only [dist_triangle (c a) e (c b), A, B, bdiam]\n  suffices H : closedBall (c b'') (3 * r b'') \u2286 \u22c3 a : { a // a \u2209 w }, closedBall (c a) (3 * r a)\n  exact H zb\n  exact subset_iUnion (fun a : { a // a \u2209 w } => closedBall (c a) (3 * r a)) b'' ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C M : (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) \u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5 ** haveI : Encodable v := (u_count.mono vu).toEncodable ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C M : (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) this : Encodable \u2191v \u22a2 \u2191\u2191\u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03b5 ** calc\n  \u03bc ((s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x)) \u2264 \u03bc (\u22c3 a : { a // a \u2209 w }, closedBall (c a) (3 * r a)) :=\n    measure_mono M\n  _ \u2264 \u2211' a : { a // a \u2209 w }, \u03bc (closedBall (c a) (3 * r a)) := (measure_iUnion_le _)\n  _ \u2264 \u2211' a : { a // a \u2209 w }, C * \u03bc (B a) := (ENNReal.tsum_le_tsum fun a => \u03bcB a (ut (vu a.1.2)))\n  _ = C * \u2211' a : { a // a \u2209 w }, \u03bc (B a) := ENNReal.tsum_mul_left\n  _ \u2264 C * (\u03b5 / C) := (mul_le_mul_left' hw.le _)\n  _ \u2264 \u03b5 := ENNReal.mul_div_le ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x \u22a2 \u2200 (x : \u03b1), \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4 ** intro x ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x x : \u03b1 \u22a2 \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4 ** obtain \u27e8R, Rpos, \u03bcR\u27e9 : \u2203 R, 0 < R \u2227 \u03bc (closedBall x R) < \u221e :=\n  (\u03bc.finiteAt_nhds x).exists_mem_basis nhds_basis_closedBall ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x x : \u03b1 R : \u211d Rpos : 0 < R \u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4 \u22a2 \u2203 R, 0 < R \u2227 R \u2264 1 \u2227 \u2191\u2191\u03bc (closedBall x (20 * R)) < \u22a4 ** refine' \u27e8min 1 (R / 20), _, min_le_left _ _, _\u27e9 ** case intro.intro.refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x x : \u03b1 R : \u211d Rpos : 0 < R \u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4 \u22a2 0 < min 1 (R / 20) ** simp only [true_and_iff, lt_min_iff, zero_lt_one] ** case intro.intro.refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x x : \u03b1 R : \u211d Rpos : 0 < R \u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4 \u22a2 0 < R / 20 ** linarith ** case intro.intro.refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x x : \u03b1 R : \u211d Rpos : 0 < R \u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4 \u22a2 \u2191\u2191\u03bc (closedBall x (20 * min 1 (R / 20))) < \u22a4 ** apply lt_of_le_of_lt (measure_mono _) \u03bcR ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x x : \u03b1 R : \u211d Rpos : 0 < R \u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4 \u22a2 closedBall x (20 * min 1 (R / 20)) \u2286 closedBall x R ** apply closedBall_subset_closedBall ** case h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x x : \u03b1 R : \u211d Rpos : 0 < R \u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4 \u22a2 20 * min 1 (R / 20) \u2264 R ** calc\n  20 * min 1 (R / 20) \u2264 20 * (R / 20) :=\n    mul_le_mul_of_nonneg_left (min_le_right _ _) (by norm_num)\n  _ = R := by ring ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x x : \u03b1 R : \u211d Rpos : 0 < R \u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4 \u22a2 0 \u2264 20 ** norm_num ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x x : \u03b1 R : \u211d Rpos : 0 < R \u03bcR : \u2191\u2191\u03bc (closedBall x R) < \u22a4 \u22a2 20 * (R / 20) = R ** ring ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c 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 r a \u2264 2 * r b ** have A : \u2200 a \u2208 t', r a \u2264 1 := by\n  intro a ha\n  apply ha.2.trans (hR1 (c a)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} A : \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1 \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 r a \u2264 2 * r b ** have A' : \u2200 a \u2208 t', (B a).Nonempty :=\n  fun a hat' => Set.Nonempty.mono interior_subset (ht a hat'.1) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} A : \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1 A' : \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 r a \u2264 2 * r b ** refine' exists_disjoint_subfamily_covering_enlargment B t' r 2 one_lt_two (fun a ha => _) 1 A A' ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} A : \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1 A' : \u2200 (a : \u03b9), a \u2208 t' \u2192 Set.Nonempty (B a) a : \u03b9 ha : a \u2208 t' \u22a2 0 \u2264 r a ** exact nonempty_closedBall.1 ((A' a ha).mono (hB a ha.1)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} \u22a2 \u2200 (a : \u03b9), a \u2208 t' \u2192 r a \u2264 1 ** intro a ha ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} a : \u03b9 ha : a \u2208 t' \u22a2 r a \u2264 1 ** apply ha.2.trans (hR1 (c a)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u \u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K ** have Idist_v : \u2200 a \u2208 v, dist (c a) x \u2264 r a + R x := by\n  intro a hav\n  apply dist_le_add_of_nonempty_closedBall_inter_closedBall\n  refine' hav.2.mono _\n  apply inter_subset_inter _ ball_subset_closedBall\n  exact hB a (ut (vu hav)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x \u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K ** set R0 := sSup (r '' v) with R0_def ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) \u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K ** have R0_bdd : BddAbove (r '' v) := by\n  refine' \u27e81, fun r' hr' => _\u27e9\n  rcases (mem_image _ _ _).1 hr' with \u27e8b, hb, rfl\u27e9\n  exact le_trans (ut' (vu hb)).2 (hR1 (c b)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) \u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K ** rcases le_total R0 (R x) with (H | H) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u \u22a2 \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x ** intro a hav ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u a : \u03b9 hav : a \u2208 v \u22a2 dist (c a) x \u2264 r a + R x ** apply dist_le_add_of_nonempty_closedBall_inter_closedBall ** case h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u a : \u03b9 hav : a \u2208 v \u22a2 Set.Nonempty (closedBall (c a) (r a) \u2229 closedBall x (R x)) ** refine' hav.2.mono _ ** case h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u a : \u03b9 hav : a \u2208 v \u22a2 B a \u2229 ball x (R x) \u2286 closedBall (c a) (r a) \u2229 closedBall x (R x) ** apply inter_subset_inter _ ball_subset_closedBall ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u a : \u03b9 hav : a \u2208 v \u22a2 B a \u2286 closedBall (c a) (r a) ** exact hB a (ut (vu hav)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) \u22a2 BddAbove (r '' v) ** refine' \u27e81, fun r' hr' => _\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) r' : \u211d hr' : r' \u2208 r '' v \u22a2 r' \u2264 1 ** rcases (mem_image _ _ _).1 hr' with \u27e8b, hb, rfl\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) b : \u03b9 hb : b \u2208 v hr' : r b \u2208 r '' v \u22a2 r b \u2264 1 ** exact le_trans (ut' (vu hb)).2 (hR1 (c b)) ** case inl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R0 \u2264 R x \u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K ** refine' \u27e820 * R x, hR\u03bc x, fun a au hax => _\u27e9 ** case inl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R0 \u2264 R x a : \u03b9 au : a \u2208 u hax : Set.Nonempty (B a \u2229 ball x (R x)) \u22a2 B a \u2286 closedBall x (20 * R x) ** refine' (hB a (ut au)).trans _ ** case inl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R0 \u2264 R x a : \u03b9 au : a \u2208 u hax : Set.Nonempty (B a \u2229 ball x (R x)) \u22a2 closedBall (c a) (r a) \u2286 closedBall x (20 * R x) ** apply closedBall_subset_closedBall' ** case inl.h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R0 \u2264 R x a : \u03b9 au : a \u2208 u hax : Set.Nonempty (B a \u2229 ball x (R x)) \u22a2 r a + dist (c a) x \u2264 20 * R x ** have : r a \u2264 R0 := le_csSup R0_bdd (mem_image_of_mem _ \u27e8au, hax\u27e9) ** case inl.h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R0 \u2264 R x a : \u03b9 au : a \u2208 u hax : Set.Nonempty (B a \u2229 ball x (R x)) this : r a \u2264 R0 \u22a2 r a + dist (c a) x \u2264 20 * R x ** linarith [Idist_v a \u27e8au, hax\u27e9, hR0 x] ** case inr \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 \u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K ** have R0pos : 0 < R0 := (hR0 x).trans_le H ** case inr \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 \u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K ** have vnonempty : v.Nonempty := by\n  by_contra h\n  rw [nonempty_iff_ne_empty, Classical.not_not] at h\n  rw [h, image_empty, Real.sSup_empty] at R0_def\n  exact lt_irrefl _ (R0pos.trans_le (le_of_eq R0_def)) ** case inr \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v \u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K ** obtain \u27e8a, hav, R0a\u27e9 : \u2203 a \u2208 v, R0 / 2 < r a := by\n  obtain \u27e8r', r'mem, hr'\u27e9 : \u2203 r' \u2208 r '' v, R0 / 2 < r' :=\n    exists_lt_of_lt_csSup (nonempty_image_iff.2 vnonempty) (half_lt_self R0pos)\n  rcases (mem_image _ _ _).1 r'mem with \u27e8a, hav, rfl\u27e9\n  exact \u27e8a, hav, hr'\u27e9 ** case inr.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a \u22a2 \u2203 K, \u2191\u2191\u03bc (closedBall x K) < \u22a4 \u2227 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K ** refine' \u27e88 * R0, _, _\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 \u22a2 Set.Nonempty v ** by_contra h ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 h : \u00acSet.Nonempty v \u22a2 False ** rw [nonempty_iff_ne_empty, Classical.not_not] at h ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 h : v = \u2205 \u22a2 False ** rw [h, image_empty, Real.sSup_empty] at R0_def ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = 0 R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 h : v = \u2205 \u22a2 False ** exact lt_irrefl _ (R0pos.trans_le (le_of_eq R0_def)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v \u22a2 \u2203 a, a \u2208 v \u2227 R0 / 2 < r a ** obtain \u27e8r', r'mem, hr'\u27e9 : \u2203 r' \u2208 r '' v, R0 / 2 < r' :=\n  exists_lt_of_lt_csSup (nonempty_image_iff.2 vnonempty) (half_lt_self R0pos) ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v r' : \u211d r'mem : r' \u2208 r '' v hr' : R0 / 2 < r' \u22a2 \u2203 a, a \u2208 v \u2227 R0 / 2 < r a ** rcases (mem_image _ _ _).1 r'mem with \u27e8a, hav, rfl\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v r'mem : r a \u2208 r '' v hr' : R0 / 2 < r a \u22a2 \u2203 a, a \u2208 v \u2227 R0 / 2 < r a ** exact \u27e8a, hav, hr'\u27e9 ** case inr.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a \u22a2 \u2191\u2191\u03bc (closedBall x (8 * R0)) < \u22a4 ** apply lt_of_le_of_lt (measure_mono _) (hR\u03bc (c a)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a \u22a2 closedBall x (8 * R0) \u2286 closedBall (c a) (20 * R (c a)) ** apply closedBall_subset_closedBall' ** case h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a \u22a2 8 * R0 + dist x (c a) \u2264 20 * R (c a) ** rw [dist_comm] ** case h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a \u22a2 8 * R0 + dist (c a) x \u2264 20 * R (c a) ** linarith [Idist_v a hav, (ut' (vu hav)).2] ** case inr.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a \u22a2 \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x (8 * R0) ** intro b bu hbx ** case inr.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a b : \u03b9 bu : b \u2208 u hbx : Set.Nonempty (B b \u2229 ball x (R x)) \u22a2 B b \u2286 closedBall x (8 * R0) ** refine' (hB b (ut bu)).trans _ ** case inr.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a b : \u03b9 bu : b \u2208 u hbx : Set.Nonempty (B b \u2229 ball x (R x)) \u22a2 closedBall (c b) (r b) \u2286 closedBall x (8 * R0) ** apply closedBall_subset_closedBall' ** case inr.intro.intro.refine'_2.h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a b : \u03b9 bu : b \u2208 u hbx : Set.Nonempty (B b \u2229 ball x (R x)) \u22a2 r b + dist (c b) x \u2264 8 * R0 ** have : r b \u2264 R0 := le_csSup R0_bdd (mem_image_of_mem _ \u27e8bu, hbx\u27e9) ** case inr.intro.intro.refine'_2.h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u Idist_v : \u2200 (a : \u03b9), a \u2208 v \u2192 dist (c a) x \u2264 r a + R x R0 : \u211d := sSup (r '' v) R0_def : R0 = sSup (r '' v) R0_bdd : BddAbove (r '' v) H : R x \u2264 R0 R0pos : 0 < R0 vnonempty : Set.Nonempty v a : \u03b9 hav : a \u2208 v R0a : R0 / 2 < r a b : \u03b9 bu : b \u2208 u hbx : Set.Nonempty (B b \u2229 ball x (R x)) this : r b \u2264 R0 \u22a2 r b + dist (c b) x \u2264 8 * R0 ** linarith [Idist_v b \u27e8bu, hbx\u27e9] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 \u22a2 \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 ** calc\n  (\u2211' a : v, \u03bc (B a)) = \u03bc (\u22c3 a \u2208 v, B a) := by\n    rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]\n    exact u_disj.subset vu\n  _ \u2264 \u03bc (closedBall x K) := (measure_mono (iUnion\u2082_subset fun a ha => hK a (vu ha) ha.2))\n  _ < \u221e := \u03bcK ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 \u22a2 \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) = \u2191\u2191\u03bc (\u22c3 a \u2208 v, B a) ** rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 \u22a2 PairwiseDisjoint v fun a => B a ** exact u_disj.subset vu ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 \u22a2 0 < \u03b5 / \u2191C ** simp only [ENNReal.div_pos_iff, \u03b5pos.ne', ENNReal.coe_ne_top, Ne.def, not_false_iff,\n  and_self_iff] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C \u22a2 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** intro z hz ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) \u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** set k := \u22c3 (a : v) (_ : a \u2208 w), B a ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a \u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** have k_closed : IsClosed k := isClosed_biUnion_finset fun i _ => h't _ (ut (vu i.2)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k \u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** have z_notmem_k : z \u2209 k := by\n  simp only [not_exists, exists_prop, mem_iUnion, mem_sep_iff, forall_exists_index,\n    SetCoe.exists, not_and, exists_and_right, Subtype.coe_mk]\n  intro b hbv _ h'z\n  have : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a :=\n    mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z)\n  simpa only [diff_inter_self] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k z_notmem_k : \u00acz \u2208 k \u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** have : ball x (R x) \\ k \u2208 \ud835\udcdd z := by\n  apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _\n  exact (mem_diff _).2 \u27e8mem_of_mem_inter_right hz, z_notmem_k\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k z_notmem_k : \u00acz \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd z \u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** obtain \u27e8d, dpos, hd\u27e9 : \u2203 d, 0 < d \u2227 closedBall z d \u2286 ball x (R x) \\ k :=\n  nhds_basis_closedBall.mem_iff.1 this ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k z_notmem_k : \u00acz \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd z d : \u211d dpos : 0 < d hd : closedBall z d \u2286 ball x (R x) \\ k \u22a2 z \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** obtain \u27e8a, hat, ad, rfl\u27e9 : \u2203 a \u2208 t, r a \u2264 min d (R z) \u2227 c a = z ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k z_notmem_k : \u00acz \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd z d : \u211d dpos : 0 < d hd : closedBall z d \u2286 ball x (R x) \\ k \u22a2 \u2203 a, a \u2208 t \u2227 r a \u2264 min d (R z) \u2227 c a = z  case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** exact hf z ((mem_diff _).1 (mem_of_mem_inter_left hz)).1 (min d (R z)) (lt_min dpos (hR0 z)) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** have ax : B a \u2286 ball x (R x) := by\n  refine' (hB a hat).trans _\n  refine' Subset.trans _ (hd.trans (diff_subset (ball x (R x)) k))\n  exact closedBall_subset_closedBall (ad.trans (min_le_left _ _)) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** obtain \u27e8b, bu, ab, bdiam\u27e9 : \u2203 b \u2208 u, (B a \u2229 B b).Nonempty \u2227 r a \u2264 2 * r b ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) \u22a2 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b  case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** exact hu a \u27e8hat, ad.trans (min_le_right _ _)\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** have bv : b \u2208 v := by\n  refine' \u27e8bu, ab.mono _\u27e9\n  rw [inter_comm]\n  exact inter_subset_inter_right _ ax ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** let b' : v := \u27e8b, bv\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** have b'_notmem_w : b' \u2209 w := by\n  intro b'w\n  have b'k : B b' \u2286 k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w\n  have : (ball x (R x) \\ k \u2229 k).Nonempty := by\n    apply ab.mono (inter_subset_inter _ b'k)\n    refine' ((hB _ hat).trans _).trans hd\n    exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))\n  simpa only [diff_inter_self, Set.not_nonempty_empty] ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** let b'' : { a // a \u2209 w } := \u27e8b', b'_notmem_w\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** have zb : c a \u2208 closedBall (c b) (3 * r b) := by\n  rcases ab with \u27e8e, \u27e8ea, eb\u27e9\u27e9\n  have A : dist (c a) e \u2264 r a := mem_closedBall'.1 (hB a hat ea)\n  have B : dist e (c b) \u2264 r b := mem_closedBall.1 (hB b (ut bu) eb)\n  simp only [mem_closedBall]\n  linarith only [dist_triangle (c a) e (c b), A, B, bdiam] ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } zb : c a \u2208 closedBall (c b) (3 * r b) \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** suffices H : closedBall (c b'') (3 * r b'') \u2286 \u22c3 a : { a // a \u2209 w }, closedBall (c a) (3 * r a) ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } zb : c a \u2208 closedBall (c b) (3 * r b) H : closedBall (c \u2191\u2191b'') (3 * r \u2191\u2191b'') \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) \u22a2 c a \u2208 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a)  case H \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } zb : c a \u2208 closedBall (c b) (3 * r b) \u22a2 closedBall (c \u2191\u2191b'') (3 * r \u2191\u2191b'') \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** exact H zb ** case H \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } zb : c a \u2208 closedBall (c b) (3 * r b) \u22a2 closedBall (c \u2191\u2191b'') (3 * r \u2191\u2191b'') \u2286 \u22c3 a, closedBall (c \u2191\u2191a) (3 * r \u2191\u2191a) ** exact subset_iUnion (fun a : { a // a \u2209 w } => closedBall (c a) (3 * r a)) b'' ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k \u22a2 \u00acz \u2208 k ** simp only [not_exists, exists_prop, mem_iUnion, mem_sep_iff, forall_exists_index,\n  SetCoe.exists, not_and, exists_and_right, Subtype.coe_mk] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k \u22a2 \u2200 (x_1 : \u03b9) (x_2 : x_1 \u2208 u \u2227 Set.Nonempty (B x_1 \u2229 ball x (R x))),     { val := x_1, property := (_ : x_1 \u2208 {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}) } \u2208 w \u2192 \u00acz \u2208 B x_1 ** intro b hbv _ h'z ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k b : \u03b9 hbv : b \u2208 u \u2227 Set.Nonempty (B b \u2229 ball x (R x)) h\u271d : { val := b, property := (_ : b \u2208 {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}) } \u2208 w h'z : z \u2208 B b \u22a2 False ** have : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a :=\n  mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k b : \u03b9 hbv : b \u2208 u \u2227 Set.Nonempty (B b \u2229 ball x (R x)) h\u271d : { val := b, property := (_ : b \u2208 {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))}) } \u2208 w h'z : z \u2208 B b this : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 \u22c3 a \u2208 u, B a \u22a2 False ** simpa only [diff_inter_self] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k z_notmem_k : \u00acz \u2208 k \u22a2 ball x (R x) \\ k \u2208 \ud835\udcdd z ** apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C z : \u03b1 hz : z \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k z_notmem_k : \u00acz \u2208 k \u22a2 z \u2208 ball x (R x) \\ k ** exact (mem_diff _).2 \u27e8mem_of_mem_inter_right hz, z_notmem_k\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) \u22a2 B a \u2286 ball x (R x) ** refine' (hB a hat).trans _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) \u22a2 closedBall (c a) (r a) \u2286 ball x (R x) ** refine' Subset.trans _ (hd.trans (diff_subset (ball x (R x)) k)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) \u22a2 closedBall (c a) (r a) \u2286 closedBall (c a) d ** exact closedBall_subset_closedBall (ad.trans (min_le_left _ _)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b \u22a2 b \u2208 v ** refine' \u27e8bu, ab.mono _\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b \u22a2 B a \u2229 B b \u2286 B b \u2229 ball x (R x) ** rw [inter_comm] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b \u22a2 B b \u2229 B a \u2286 B b \u2229 ball x (R x) ** exact inter_subset_inter_right _ ax ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } \u22a2 \u00acb' \u2208 w ** intro b'w ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'w : b' \u2208 w \u22a2 False ** have b'k : B b' \u2286 k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'w : b' \u2208 w b'k : B \u2191b' \u2286 k \u22a2 False ** have : (ball x (R x) \\ k \u2229 k).Nonempty := by\n  apply ab.mono (inter_subset_inter _ b'k)\n  refine' ((hB _ hat).trans _).trans hd\n  exact closedBall_subset_closedBall (ad.trans (min_le_left _ _)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this\u271d : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'w : b' \u2208 w b'k : B \u2191b' \u2286 k this : Set.Nonempty (ball x (R x) \\ k \u2229 k) \u22a2 False ** simpa only [diff_inter_self, Set.not_nonempty_empty] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'w : b' \u2208 w b'k : B \u2191b' \u2286 k \u22a2 Set.Nonempty (ball x (R x) \\ k \u2229 k) ** apply ab.mono (inter_subset_inter _ b'k) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'w : b' \u2208 w b'k : B \u2191b' \u2286 k \u22a2 B a \u2286 ball x (R x) \\ k ** refine' ((hB _ hat).trans _).trans hd ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'w : b' \u2208 w b'k : B \u2191b' \u2286 k \u22a2 closedBall (c a) (r a) \u2286 closedBall (c a) d ** exact closedBall_subset_closedBall (ad.trans (min_le_left _ _)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u ab : Set.Nonempty (B a \u2229 B b) bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } \u22a2 c a \u2208 closedBall (c b) (3 * r b) ** rcases ab with \u27e8e, \u27e8ea, eb\u27e9\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } e : \u03b1 ea : e \u2208 B a eb : e \u2208 B b \u22a2 c a \u2208 closedBall (c b) (3 * r b) ** have A : dist (c a) e \u2264 r a := mem_closedBall'.1 (hB a hat ea) ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B a \u2229 ball x (R x)) \u2192 B a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } e : \u03b1 ea : e \u2208 B a eb : e \u2208 B b A : dist (c a) e \u2264 r a \u22a2 c a \u2208 closedBall (c b) (3 * r b) ** have B : dist e (c b) \u2264 r b := mem_closedBall.1 (hB b (ut bu) eb) ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B\u271d : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B\u271d a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B\u271d a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B\u271d a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B\u271d a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B\u271d hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 B\u271d b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B\u271d a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B\u271d a \u2229 ball x (R x)) \u2192 B\u271d a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B\u271d \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B\u271d \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B\u271d \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B\u271d a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B\u271d a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } e : \u03b1 ea : e \u2208 B\u271d a eb : e \u2208 B\u271d b A : dist (c a) e \u2264 r a B : dist e (c b) \u2264 r b \u22a2 c a \u2208 closedBall (c b) (3 * r b) ** simp only [mem_closedBall] ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : MetricSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc s : Set \u03b1 t : Set \u03b9 C : \u211d\u22650 r : \u03b9 \u2192 \u211d c : \u03b9 \u2192 \u03b1 B\u271d : \u03b9 \u2192 Set \u03b1 hB : \u2200 (a : \u03b9), a \u2208 t \u2192 B\u271d a \u2286 closedBall (c a) (r a) \u03bcB : \u2200 (a : \u03b9), a \u2208 t \u2192 \u2191\u2191\u03bc (closedBall (c a) (3 * r a)) \u2264 \u2191C * \u2191\u2191\u03bc (B\u271d a) ht : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (interior (B\u271d a)) h't : \u2200 (a : \u03b9), a \u2208 t \u2192 IsClosed (B\u271d a) hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 t \u2227 r a \u2264 \u03b5 \u2227 c a = x R : \u03b1 \u2192 \u211d hR0 : \u2200 (x : \u03b1), 0 < R x hR1 : \u2200 (x : \u03b1), R x \u2264 1 hR\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (20 * R x)) < \u22a4 t' : Set \u03b9 := {a | a \u2208 t \u2227 r a \u2264 R (c a)} u : Set \u03b9 ut' : u \u2286 t' u_disj : PairwiseDisjoint u B\u271d hu : \u2200 (a : \u03b9), a \u2208 t' \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 B\u271d b) \u2227 r a \u2264 2 * r b ut : u \u2286 t u_count : Set.Countable u x : \u03b1 x\u271d : x \u2208 s \\ \u22c3 a \u2208 u, B\u271d a v : Set \u03b9 := {a | a \u2208 u \u2227 Set.Nonempty (B\u271d a \u2229 ball x (R x))} vu : v \u2286 u K : \u211d \u03bcK : \u2191\u2191\u03bc (closedBall x K) < \u22a4 hK : \u2200 (a : \u03b9), a \u2208 u \u2192 Set.Nonempty (B\u271d a \u2229 ball x (R x)) \u2192 B\u271d a \u2286 closedBall x K \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 I : \u2211' (a : \u2191v), \u2191\u2191\u03bc (B\u271d \u2191a) < \u22a4 w : Finset \u2191v hw : \u2211' (a : { a // \u00aca \u2208 w }), \u2191\u2191\u03bc (B\u271d \u2191\u2191a) < \u03b5 / \u2191C k : Set \u03b1 := \u22c3 a \u2208 w, B\u271d \u2191a k_closed : IsClosed k d : \u211d dpos : 0 < d a : \u03b9 hat : a \u2208 t hz : c a \u2208 (s \\ \u22c3 a \u2208 u, B\u271d a) \u2229 ball x (R x) z_notmem_k : \u00acc a \u2208 k this : ball x (R x) \\ k \u2208 \ud835\udcdd (c a) hd : closedBall (c a) d \u2286 ball x (R x) \\ k ad : r a \u2264 min d (R (c a)) ax : B\u271d a \u2286 ball x (R x) b : \u03b9 bu : b \u2208 u bdiam : r a \u2264 2 * r b bv : b \u2208 v b' : \u2191v := { val := b, property := bv } b'_notmem_w : \u00acb' \u2208 w b'' : { a // \u00aca \u2208 w } := { val := b', property := b'_notmem_w } e : \u03b1 ea : e \u2208 B\u271d a eb : e \u2208 B\u271d b A : dist (c a) e \u2264 r a B : dist e (c b) \u2264 r b \u22a2 dist (c a) (c b) \u2264 3 * r b ** linarith only [dist_triangle (c a) e (c b), A, B, bdiam] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.lt_to_int ** \u03b1 : Type u_1 m n : ZNum h : Ordering.casesOn Ordering.lt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt ** simp at h ** \u03b1 : Type u_1 m n : ZNum h : \u2191m < \u2191n \u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt ** simp [h] ** \u03b1 : Type u_1 m n : ZNum h : Ordering.casesOn Ordering.eq (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt ** simp at h ** \u03b1 : Type u_1 m n : ZNum h : m = n \u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt ** simp [h, lt_irrefl] ** \u03b1 : Type u_1 m n : ZNum h : Ordering.casesOn Ordering.gt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u22a2 \u2191m < \u2191n \u2194 Ordering.gt = Ordering.lt ** simp [not_lt_of_gt h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.product_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s s' : Finset \u03b1 t t' : Finset \u03b2 a : \u03b1 b\u271d b : \u03b2 \u22a2 s \u00d7\u02e2 {b} = map { toFun := fun i => (i, b), inj' := (_ : Function.Injective fun a => (a, b)) } s ** ext \u27e8x, y\u27e9 ** case a.mk \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s s' : Finset \u03b1 t t' : Finset \u03b2 a : \u03b1 b\u271d b : \u03b2 x : \u03b1 y : \u03b2 \u22a2 (x, y) \u2208 s \u00d7\u02e2 {b} \u2194 (x, y) \u2208 map { toFun := fun i => (i, b), inj' := (_ : Function.Injective fun a => (a, b)) } s ** simp [and_left_comm, eq_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setLaverage_congr ** \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 : s =\u1da0[ae \u03bc] t \u22a2 \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc ** simp only [setLaverage_eq, set_lintegral_congr h, measure_congr h] ** 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 ::: (Cofix F \u03b1 \u2295 \u03b2)) x : \u03b2 \u22a2 dest (corec' g x) = (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x ** rw [Cofix.corec', Cofix.dest_corec] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x : \u03b2 \u22a2 (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g)) <$$>       Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g (Sum.inr x) =     (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x ** dsimp ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x : \u03b2 \u22a2 (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g)) <$$> g x =     (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x ** congr! ** case h.e'_6.h.e'_7 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x : \u03b2 \u22a2 corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) = Sum.elim _root_.id (corec' g) ** ext (i | i) <;> erw [corec_roll] <;> dsimp [Cofix.corec'] ** case h.e'_6.h.e'_7.h.inl n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x : \u03b2 i : Cofix F \u03b1 \u22a2 corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inl i) =     i ** mv_bisim i with R a b x Ha Hb ** case h.e'_6.h.e'_7.h.inl.intro.intro 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 ::: (Cofix F \u03b1 \u2295 \u03b2)) x\u271d : \u03b2 i : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=   fun a b =>     \u2203 x,       a =           corec             (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)             (Sum.inl x) \u2227         b = x a b x : Cofix F \u03b1 Ha :   a =     corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inl x) Hb : b = x \u22a2 LiftR' (RelLast' \u03b1 R) (dest a) (dest b) ** rw [Ha, Hb, Cofix.dest_corec] ** case h.e'_6.h.e'_7.h.inl.intro.intro 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 ::: (Cofix F \u03b1 \u2295 \u03b2)) x\u271d : \u03b2 i : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=   fun a b =>     \u2203 x,       a =           corec             (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)             (Sum.inl x) \u2227         b = x a b x : Cofix F \u03b1 Ha :   a =     corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inl x) Hb : b = x \u22a2 LiftR' (RelLast' \u03b1 R)     ((TypeVec.id :::         corec           (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218             Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)) <$$>       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218           Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)         (Sum.inl x))     (dest x) ** dsimp [Function.comp] ** case h.e'_6.h.e'_7.h.inl.intro.intro n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x\u271d : \u03b2 i : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=   fun a b =>     \u2203 x,       a =           corec             (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)             (Sum.inl x) \u2227         b = x a b x : Cofix F \u03b1 Ha :   a =     corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inl x) Hb : b = x \u22a2 LiftR'     (RelLast' \u03b1 fun a b =>       \u2203 x,         a =             corec               (fun x =>                 (TypeVec.id ::: fun x => x) <$$>                   Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)               (Sum.inl x) \u2227           b = x)     ((TypeVec.id :::         corec fun x =>           (TypeVec.id ::: fun x => x) <$$>             Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) <$$>       (TypeVec.id ::: fun x => x) <$$> (TypeVec.id ::: Sum.inl) <$$> dest x)     (dest x) ** repeat rw [MvFunctor.map_map, \u2190 appendFun_comp_id] ** case h.e'_6.h.e'_7.h.inl.intro.intro 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 ::: (Cofix F \u03b1 \u2295 \u03b2)) x\u271d : \u03b2 i : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=   fun a b =>     \u2203 x,       a =           corec             (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)             (Sum.inl x) \u2227         b = x a b x : Cofix F \u03b1 Ha :   a =     corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inl x) Hb : b = x \u22a2 LiftR'     (RelLast' \u03b1 fun a b =>       \u2203 x,         a =             corec               (fun x =>                 (TypeVec.id ::: fun x => x) <$$>                   Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)               (Sum.inl x) \u2227           b = x)     ((TypeVec.id :::         ((corec fun x =>               (TypeVec.id ::: fun x => x) <$$>                 Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218             fun x => x) \u2218           Sum.inl) <$$>       dest x)     (dest x) ** apply liftR_map_last' ** case h.e'_6.h.e'_7.h.inl.intro.intro.hh n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x\u271d : \u03b2 i : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=   fun a b =>     \u2203 x,       a =           corec             (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)             (Sum.inl x) \u2227         b = x a b x : Cofix F \u03b1 Ha :   a =     corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inl x) Hb : b = x \u22a2 \u2200 (x : Cofix F \u03b1),     \u2203 x_1,       (((corec fun x =>                   (TypeVec.id ::: fun x => x) <$$>                     Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218                 fun x => x) \u2218               Sum.inl)             x =           corec             (fun x =>               (TypeVec.id ::: fun x => x) <$$>                 Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)             (Sum.inl x_1) \u2227         x = x_1 ** dsimp [Function.comp] ** case h.e'_6.h.e'_7.h.inl.intro.intro.hh n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x\u271d : \u03b2 i : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=   fun a b =>     \u2203 x,       a =           corec             (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)             (Sum.inl x) \u2227         b = x a b x : Cofix F \u03b1 Ha :   a =     corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inl x) Hb : b = x \u22a2 \u2200 (x : Cofix F \u03b1),     \u2203 x_1,       corec             (fun x =>               (TypeVec.id ::: fun x => x) <$$>                 Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)             (Sum.inl x) =           corec             (fun x =>               (TypeVec.id ::: fun x => x) <$$>                 Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)             (Sum.inl x_1) \u2227         x = x_1 ** intros ** case h.e'_6.h.e'_7.h.inl.intro.intro.hh n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x\u271d\u00b9 : \u03b2 i : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=   fun a b =>     \u2203 x,       a =           corec             (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)             (Sum.inl x) \u2227         b = x a b x : Cofix F \u03b1 Ha :   a =     corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inl x) Hb : b = x x\u271d : Cofix F \u03b1 \u22a2 \u2203 x,     corec           (fun x =>             (TypeVec.id ::: fun x => x) <$$>               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)           (Sum.inl x\u271d) =         corec           (fun x =>             (TypeVec.id ::: fun x => x) <$$>               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)           (Sum.inl x) \u2227       x\u271d = x ** exact \u27e8_, rfl, rfl\u27e9 ** case h.e'_6.h.e'_7.h.inl.intro.intro n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x\u271d : \u03b2 i : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=   fun a b =>     \u2203 x,       a =           corec             (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)             (Sum.inl x) \u2227         b = x a b x : Cofix F \u03b1 Ha :   a =     corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inl x) Hb : b = x \u22a2 LiftR'     (RelLast' \u03b1 fun a b =>       \u2203 x,         a =             corec               (fun x =>                 (TypeVec.id ::: fun x => x) <$$>                   Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)               (Sum.inl x) \u2227           b = x)     ((TypeVec.id :::         (corec fun x =>             (TypeVec.id ::: fun x => x) <$$>               Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218           fun x => x) <$$>       (TypeVec.id ::: Sum.inl) <$$> dest x)     (dest x) ** rw [MvFunctor.map_map, \u2190 appendFun_comp_id] ** case h.e'_6.h.e'_7.h.inr n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x i : \u03b2 \u22a2 corec       (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       (Sum.inr i) =     corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) (Sum.inr i) ** congr with y ** case h.e'_6.h.e'_7.h.inr.e_g.h 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 ::: (Cofix F \u03b1 \u2295 \u03b2)) x i : \u03b2 y : Cofix F \u03b1 \u2295 \u03b2 \u22a2 (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218         Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)       y =     Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g y ** erw [appendFun_id_id] ** case h.e'_6.h.e'_7.h.inr.e_g.h n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2)) x i : \u03b2 y : Cofix F \u03b1 \u2295 \u03b2 \u22a2 (MvFunctor.map TypeVec.id \u2218 Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val) y =     Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g y ** simp [MvFunctor.id_map, Sum.elim] ** Qed",
        "informal": ""
    },
    {
        "formal": "VitaliFamily.withDensity_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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s ** have t_ne_zero' : t \u2260 0 := (zero_lt_one.trans ht).ne' ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s ** have t_ne_zero : (t : \u211d\u22650\u221e) \u2260 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero' ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s ** let \u03bd := \u03bc.withDensity (v.limRatioMeas h\u03c1) ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s ** let f := v.limRatioMeas h\u03c1 ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s ** have f_meas : Measurable f := v.limRatioMeas_measurable h\u03c1 ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s ** have A : \u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) := by\n  apply le_trans _ (zero_le _)\n  have M : MeasurableSet (s \u2229 f \u207b\u00b9' {0}) := hs.inter (f_meas (measurableSet_singleton _))\n  simp only [nonpos_iff_eq_zero, M, withDensity_apply, lintegral_eq_zero_iff f_meas]\n  apply (ae_restrict_iff' M).2\n  exact eventually_of_forall fun x hx => hx.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s ** have B : \u03bd (s \u2229 f \u207b\u00b9' {\u221e}) \u2264 ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u221e}) := by\n  apply le_trans (le_of_eq _) (zero_le _)\n  apply withDensity_absolutelyContinuous \u03bc _\n  rw [\u2190 nonpos_iff_eq_zero]\n  exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top h\u03c1).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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s ** have C :\n  \u2200 n : \u2124,\n    \u03bd (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) \u2264\n      ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) := by\n  intro n\n  let I := Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))\n  have M : MeasurableSet (s \u2229 f \u207b\u00b9' I) := hs.inter (f_meas measurableSet_Ico)\n  simp only [M, withDensity_apply, coe_nnreal_smul_apply]\n  calc\n    (\u222b\u207b x in s \u2229 f \u207b\u00b9' I, f x \u2202\u03bc) \u2264 \u222b\u207b x in s \u2229 f \u207b\u00b9' I, (t : \u211d\u22650\u221e) ^ (n + 1) \u2202\u03bc :=\n      lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => hx.2.2.le))\n    _ = (t : \u211d\u22650\u221e) ^ (n + 1) * \u03bc (s \u2229 f \u207b\u00b9' I) := by\n      simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n    _ = (t : \u211d\u22650\u221e) ^ (2 : \u2124) * ((t : \u211d\u22650\u221e) ^ (n - 1) * \u03bc (s \u2229 f \u207b\u00b9' I)) := by\n      rw [\u2190 mul_assoc, \u2190 ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]\n      congr 2\n      abel\n    _ \u2264 (t : \u211d\u22650\u221e) ^ 2 * \u03c1 (s \u2229 f \u207b\u00b9' I) := by\n      refine' mul_le_mul_left' _ _\n      rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne']\n      apply v.mul_measure_le_of_subset_lt_limRatioMeas h\u03c1\n      intro x hx\n      apply lt_of_lt_of_le _ hx.2.1\n      rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,\n        zpow_add\u2080 t_ne_zero']\n      conv_rhs => rw [\u2190 mul_one (t ^ n)]\n      refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)\n      rw [zpow_neg_one]\n      exact inv_lt_one ht ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) C : \u2200 (n : \u2124), \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 s ** calc\n  \u03bd s =\n    \u03bd (s \u2229 f \u207b\u00b9' {0}) + \u03bd (s \u2229 f \u207b\u00b9' {\u221e}) +\n      \u2211' n : \u2124, \u03bd (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) :=\n    measure_eq_measure_preimage_add_measure_tsum_Ico_zpow \u03bd f_meas hs ht\n  _ \u2264\n      ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) + ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u221e}) +\n        \u2211' n : \u2124, ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) :=\n    (add_le_add (add_le_add A B) (ENNReal.tsum_le_tsum C))\n  _ = ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) s :=\n    (measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ((t : \u211d\u22650\u221e) ^ 2 \u2022 \u03c1) f_meas hs ht).symm ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 \u22a2 \u2191t \u2260 0 ** simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero' ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f \u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) ** apply le_trans _ (zero_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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f \u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 0 ** have M : MeasurableSet (s \u2229 f \u207b\u00b9' {0}) := hs.inter (f_meas (measurableSet_singleton _)) ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f M : MeasurableSet (s \u2229 f \u207b\u00b9' {0}) \u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 0 ** simp only [nonpos_iff_eq_zero, M, withDensity_apply, lintegral_eq_zero_iff f_meas] ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f M : MeasurableSet (s \u2229 f \u207b\u00b9' {0}) \u22a2 limRatioMeas v h\u03c1 =\u1da0[ae (Measure.restrict \u03bc (s \u2229 limRatioMeas v h\u03c1 \u207b\u00b9' {0}))] 0 ** apply (ae_restrict_iff' M).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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f M : MeasurableSet (s \u2229 f \u207b\u00b9' {0}) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2229 f \u207b\u00b9' {0} \u2192 limRatioMeas v h\u03c1 x = OfNat.ofNat 0 x ** exact eventually_of_forall fun x hx => hx.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) \u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) ** apply le_trans (le_of_eq _) (zero_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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) \u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) = 0 ** apply withDensity_absolutelyContinuous \u03bc _ ** case 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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) \u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) = 0 ** rw [\u2190 nonpos_iff_eq_zero] ** case 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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) \u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 0 ** exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top h\u03c1).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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) \u22a2 \u2200 (n : \u2124), \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) ** intro n ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 \u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) ** let I := Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) \u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) ** have M : MeasurableSet (s \u2229 f \u207b\u00b9' I) := hs.inter (f_meas measurableSet_Ico) ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) ** simp only [M, withDensity_apply, coe_nnreal_smul_apply] ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 \u222b\u207b (a : \u03b1) in s \u2229 limRatioMeas v h\u03c1 \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1)), limRatioMeas v h\u03c1 a \u2202\u03bc \u2264     \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 limRatioMeas v h\u03c1 \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) ** calc\n  (\u222b\u207b x in s \u2229 f \u207b\u00b9' I, f x \u2202\u03bc) \u2264 \u222b\u207b x in s \u2229 f \u207b\u00b9' I, (t : \u211d\u22650\u221e) ^ (n + 1) \u2202\u03bc :=\n    lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => hx.2.2.le))\n  _ = (t : \u211d\u22650\u221e) ^ (n + 1) * \u03bc (s \u2229 f \u207b\u00b9' I) := by\n    simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n  _ = (t : \u211d\u22650\u221e) ^ (2 : \u2124) * ((t : \u211d\u22650\u221e) ^ (n - 1) * \u03bc (s \u2229 f \u207b\u00b9' I)) := by\n    rw [\u2190 mul_assoc, \u2190 ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]\n    congr 2\n    abel\n  _ \u2264 (t : \u211d\u22650\u221e) ^ 2 * \u03c1 (s \u2229 f \u207b\u00b9' I) := by\n    refine' mul_le_mul_left' _ _\n    rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne']\n    apply v.mul_measure_le_of_subset_lt_limRatioMeas h\u03c1\n    intro x hx\n    apply lt_of_lt_of_le _ hx.2.1\n    rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,\n      zpow_add\u2080 t_ne_zero']\n    conv_rhs => rw [\u2190 mul_one (t ^ n)]\n    refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)\n    rw [zpow_neg_one]\n    exact inv_lt_one ht ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 \u222b\u207b (x : \u03b1) in s \u2229 f \u207b\u00b9' I, \u2191t ^ (n + 1) \u2202\u03bc = \u2191t ^ (n + 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) ** simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter] ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 \u2191t ^ (n + 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) = \u2191t ^ 2 * (\u2191t ^ (n - 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I)) ** rw [\u2190 mul_assoc, \u2190 ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top] ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 \u2191t ^ (n + 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) = \u2191t ^ (2 + (n - 1)) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) ** congr 2 ** case e_a.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 n + 1 = 2 + (n - 1) ** abel ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 \u2191t ^ 2 * (\u2191t ^ (n - 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I)) \u2264 \u2191t ^ 2 * \u2191\u2191\u03c1 (s \u2229 f \u207b\u00b9' I) ** refine' mul_le_mul_left' _ _ ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 \u2191t ^ (n - 1) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) \u2264 \u2191\u2191\u03c1 (s \u2229 f \u207b\u00b9' I) ** rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne'] ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 \u2191(t ^ (n - 1)) * \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' I) \u2264 \u2191\u2191\u03c1 (s \u2229 f \u207b\u00b9' I) ** apply v.mul_measure_le_of_subset_lt_limRatioMeas h\u03c1 ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) \u22a2 s \u2229 f \u207b\u00b9' I \u2286 {x | \u2191(t ^ (n - 1)) < limRatioMeas v h\u03c1 x} ** intro x hx ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' I \u22a2 x \u2208 {x | \u2191(t ^ (n - 1)) < limRatioMeas v h\u03c1 x} ** apply lt_of_lt_of_le _ hx.2.1 ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' I \u22a2 \u2191(t ^ (n - 1)) < \u2191t ^ n ** rw [\u2190 ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,\n  zpow_add\u2080 t_ne_zero'] ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' I \u22a2 t ^ n * t ^ (-1) < t ^ n ** conv_rhs => rw [\u2190 mul_one (t ^ n)] ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' I \u22a2 t ^ n * t ^ (-1) < t ^ n * 1 ** refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _) ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' I \u22a2 t ^ (-1) < 1 ** rw [zpow_neg_one] ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t t_ne_zero' : t \u2260 0 t_ne_zero : \u2191t \u2260 0 \u03bd : Measure \u03b1 := withDensity \u03bc (limRatioMeas v h\u03c1) f : \u03b1 \u2192 \u211d\u22650\u221e := limRatioMeas v h\u03c1 f_meas : Measurable f A : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {0}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {0}) B : \u2191\u2191\u03bd (s \u2229 f \u207b\u00b9' {\u22a4}) \u2264 \u2191\u2191(\u2191t ^ 2 \u2022 \u03c1) (s \u2229 f \u207b\u00b9' {\u22a4}) n : \u2124 I : Set \u211d\u22650\u221e := Ico (\u2191t ^ n) (\u2191t ^ (n + 1)) M : MeasurableSet (s \u2229 f \u207b\u00b9' I) x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' I \u22a2 t\u207b\u00b9 < 1 ** exact inv_lt_one ht ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.ppred_to_nat ** \u03b1 : Type u_1 p : PosNum \u22a2 castNum <$> ppred (pos p) = Nat.ppred \u2191(pos p) ** rw [ppred, Option.map_some, Nat.ppred_eq_some.2] ** \u03b1 : Type u_1 p : PosNum \u22a2 Nat.succ \u2191(pred' p) = \u2191(pos p) ** rw [PosNum.pred'_to_nat, Nat.succ_pred_eq_of_pos (PosNum.to_nat_pos _)] ** \u03b1 : Type u_1 p : PosNum \u22a2 \u2191p = \u2191(pos p) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "List.foldr_sup_eq_sup_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 : SemilatticeSup \u03b1 inst\u271d\u00b9 : OrderBot \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 \u2294 x_1) \u22a5 l = sup (List.toFinset l) id ** rw [\u2190 coe_fold_r, \u2190 Multiset.fold_dedup_idem, sup_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 : SemilatticeSup \u03b1 inst\u271d\u00b9 : OrderBot \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 \u2294 x_1) \u22a5 (dedup \u2191l) = Multiset.sup (dedup \u2191l) ** rfl ** 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\u00b2 : SemilatticeSup \u03b1 s\u271d : Finset \u03b2 H : Finset.Nonempty s\u271d f\u271d : \u03b2 \u2192 \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b2 inst\u271d : SupHomClass F \u03b1 \u03b2 f : F s : Finset \u03b9 hs : Finset.Nonempty s g : \u03b9 \u2192 \u03b1 \u22a2 \u2191f (sup' s hs g) = sup' s hs (\u2191f \u2218 g) ** refine' hs.cons_induction _ _ <;> intros <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.totalDegree_finset_prod ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R \u03b9 : Type u_3 s : Finset \u03b9 f : \u03b9 \u2192 MvPolynomial \u03c3 R \u22a2 totalDegree (Finset.prod s f) \u2264 \u2211 i in s, totalDegree (f i) ** refine' le_trans (totalDegree_multiset_prod _) _ ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R \u03b9 : Type u_3 s : Finset \u03b9 f : \u03b9 \u2192 MvPolynomial \u03c3 R \u22a2 Multiset.sum (Multiset.map totalDegree (Multiset.map f s.val)) \u2264 \u2211 i in s, totalDegree (f i) ** rw [Multiset.map_map] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R \u03b9 : Type u_3 s : Finset \u03b9 f : \u03b9 \u2192 MvPolynomial \u03c3 R \u22a2 Multiset.sum (Multiset.map (totalDegree \u2218 f) s.val) \u2264 \u2211 i in s, totalDegree (f i) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.neg_le_sub_left_of_le_add ** a b c : Int h : c \u2264 a + b \u22a2 -a \u2264 b - c ** have h := Int.le_neg_add_of_add_le (Int.sub_left_le_of_le_add h) ** a b c : Int h\u271d : c \u2264 a + b h : -a \u2264 -c + b \u22a2 -a \u2264 b - c ** rwa [Int.add_comm] at h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_zero_testAgainstNN_of_tendsto_zero_mass ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd 0) ** apply tendsto_iff_dist_tendsto_zero.mpr ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0) ** have obs := fun i => (\u03bcs i).testAgainstNN_lipschitz_estimate f 0 ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 obs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 testAgainstNN (\u03bcs i) 0 + nndist f 0 * mass (\u03bcs i) \u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0) ** simp_rw [testAgainstNN_zero, zero_add] at obs ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 obs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i) \u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0) ** simp_rw [show \u2200 i, dist ((\u03bcs i).testAgainstNN f) 0 = (\u03bcs i).testAgainstNN f by\n    simp only [dist_nndist, NNReal.nndist_zero_eq_val', eq_self_iff_true, imp_true_iff]] ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 obs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i) \u22a2 Tendsto (fun b => \u2191(testAgainstNN (\u03bcs b) f)) F (\ud835\udcdd 0) ** refine' squeeze_zero (fun i => NNReal.coe_nonneg _) obs _ ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 obs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i) \u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0) ** have lim_pair : Tendsto (fun i => (\u27e8nndist f 0, (\u03bcs i).mass\u27e9 : \u211d \u00d7 \u211d)) F (\ud835\udcdd \u27e8nndist f 0, 0\u27e9) := by\n  refine' (Prod.tendsto_iff _ _).mpr \u27e8tendsto_const_nhds, _\u27e9\n  exact (NNReal.continuous_coe.tendsto 0).comp mass_lim ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 obs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i) lim_pair : Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0)) \u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0) ** have key := tendsto_mul.comp lim_pair ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 obs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i) lim_pair : Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0)) key : Tendsto ((fun p => p.1 * p.2) \u2218 fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0) * 0)) \u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0) ** rwa [mul_zero] at key ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 obs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i) \u22a2 \u2200 (i : \u03b3), dist (testAgainstNN (\u03bcs i) f) 0 = \u2191(testAgainstNN (\u03bcs i) f) ** simp only [dist_nndist, NNReal.nndist_zero_eq_val', eq_self_iff_true, imp_true_iff] ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 obs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i) \u22a2 Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0)) ** refine' (Prod.tendsto_iff _ _).mpr \u27e8tendsto_const_nhds, _\u27e9 ** \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 mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) f : \u03a9 \u2192\u1d47 \u211d\u22650 obs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i) \u22a2 Tendsto (fun n => (\u2191(nndist f 0), \u2191(mass (\u03bcs n))).2) F (\ud835\udcdd (\u2191(nndist f 0), 0).2) ** exact (NNReal.continuous_coe.tendsto 0).comp mass_lim ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Infinite.exists_superset_ncard_eq ** \u03b1 : Type u_1 s\u271d t\u271d s t : Set \u03b1 ht : Set.Infinite t hst : s \u2286 t hs : Set.Finite s k : \u2115 hsk : Set.ncard s \u2264 k \u22a2 \u2203 s', s \u2286 s' \u2227 s' \u2286 t \u2227 Set.ncard s' = k ** obtain \u27e8s\u2081, hs\u2081, hs\u2081fin, hs\u2081card\u27e9 := (ht.diff hs).exists_subset_ncard_eq (k - s.ncard) ** case intro.intro.intro \u03b1 : Type u_1 s\u271d t\u271d s t : Set \u03b1 ht : Set.Infinite t hst : s \u2286 t hs : Set.Finite s k : \u2115 hsk : Set.ncard s \u2264 k s\u2081 : Set \u03b1 hs\u2081 : s\u2081 \u2286 t \\ s hs\u2081fin : Set.Finite s\u2081 hs\u2081card : Set.ncard s\u2081 = k - Set.ncard s \u22a2 \u2203 s', s \u2286 s' \u2227 s' \u2286 t \u2227 Set.ncard s' = k ** refine' \u27e8s \u222a s\u2081, subset_union_left _ _, union_subset hst (hs\u2081.trans (diff_subset _ _)), _\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 s\u271d t\u271d s t : Set \u03b1 ht : Set.Infinite t hst : s \u2286 t hs : Set.Finite s k : \u2115 hsk : Set.ncard s \u2264 k s\u2081 : Set \u03b1 hs\u2081 : s\u2081 \u2286 t \\ s hs\u2081fin : Set.Finite s\u2081 hs\u2081card : Set.ncard s\u2081 = k - Set.ncard s \u22a2 Set.ncard (s \u222a s\u2081) = k ** rwa [ncard_union_eq (disjoint_of_subset_right hs\u2081 disjoint_sdiff_right) hs hs\u2081fin, hs\u2081card,\n  add_tsub_cancel_of_le] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.zero_dvd ** n : Int x\u271d : 0 \u2223 n k : Int e : n = 0 * k \u22a2 n = 0 ** rw [e, Int.zero_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.powerset_insert ** \u03b1 : Type u_1 s\u271d t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 \u22a2 powerset (insert a s) = powerset s \u222a image (insert a) (powerset s) ** ext t ** case a \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 \u22a2 t \u2208 powerset (insert a s) \u2194 t \u2208 powerset s \u222a image (insert a) (powerset s) ** simp only [exists_prop, mem_powerset, mem_image, mem_union, subset_insert_iff] ** case a \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 \u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t ** by_cases h : a \u2208 t ** case pos \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : a \u2208 t \u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t ** constructor ** case pos.mp \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : a \u2208 t \u22a2 erase t a \u2286 s \u2192 t \u2286 s \u2228 \u2203 a_2, a_2 \u2286 s \u2227 insert a a_2 = t ** exact fun H => Or.inr \u27e8_, H, insert_erase h\u27e9 ** case pos.mpr \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : a \u2208 t \u22a2 (t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t) \u2192 erase t a \u2286 s ** intro H ** case pos.mpr \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : a \u2208 t H : t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t \u22a2 erase t a \u2286 s ** cases' H with H H ** case pos.mpr.inl \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : a \u2208 t H : t \u2286 s \u22a2 erase t a \u2286 s ** exact Subset.trans (erase_subset a t) H ** case pos.mpr.inr \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : a \u2208 t H : \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t \u22a2 erase t a \u2286 s ** rcases H with \u27e8u, hu\u27e9 ** case pos.mpr.inr.intro \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : a \u2208 t u : Finset \u03b1 hu : u \u2286 s \u2227 insert a u = t \u22a2 erase t a \u2286 s ** rw [\u2190 hu.2] ** case pos.mpr.inr.intro \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : a \u2208 t u : Finset \u03b1 hu : u \u2286 s \u2227 insert a u = t \u22a2 erase (insert a u) a \u2286 s ** exact Subset.trans (erase_insert_subset a u) hu.1 ** case neg \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : \u00aca \u2208 t \u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t ** have : \u00ac\u2203 u : Finset \u03b1, u \u2286 s \u2227 insert a u = t := by simp [Ne.symm (ne_insert_of_not_mem _ _ h)] ** case neg \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : \u00aca \u2208 t this : \u00ac\u2203 u, u \u2286 s \u2227 insert a u = t \u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t ** simp [Finset.erase_eq_of_not_mem h, this] ** \u03b1 : Type u_1 s\u271d t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 t : Finset \u03b1 h : \u00aca \u2208 t \u22a2 \u00ac\u2203 u, u \u2286 s \u2227 insert a u = t ** simp [Ne.symm (ne_insert_of_not_mem _ _ h)] ** Qed",
        "informal": ""
    },
    {
        "formal": "WType.leftInverse_nat ** f : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2 \u22a2 ofNat (toNat (mk Nat\u03b1.zero f)) = mk Nat\u03b1.zero f ** rw [toNat, ofNat] ** f : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2 \u22a2 mk Nat\u03b1.zero Empty.elim = mk Nat\u03b1.zero f ** congr ** case e_f f : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2 \u22a2 Empty.elim = f ** ext x ** case e_f.h f : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2 x : Empty \u22a2 Empty.elim x = f x ** cases x ** f : Nat\u03b2 Nat\u03b1.succ \u2192 WType Nat\u03b2 \u22a2 ofNat (toNat (mk Nat\u03b1.succ f)) = mk Nat\u03b1.succ f ** simp only [toNat, ofNat, leftInverse_nat (f ()), mk.injEq, heq_eq_eq, true_and] ** f : Nat\u03b2 Nat\u03b1.succ \u2192 WType Nat\u03b2 \u22a2 (fun x => f ()) = f ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.comap_rename ** \u03c3 : Type u_1 \u03c4 : Type u_2 \u03c5 : Type u_3 R : Type u_4 inst\u271d : CommSemiring R f : \u03c3 \u2192 \u03c4 x : \u03c4 \u2192 R \u22a2 comap (rename f) x = x \u2218 f ** funext ** case h \u03c3 : Type u_1 \u03c4 : Type u_2 \u03c5 : Type u_3 R : Type u_4 inst\u271d : CommSemiring R f : \u03c3 \u2192 \u03c4 x : \u03c4 \u2192 R x\u271d : \u03c3 \u22a2 comap (rename f) x x\u271d = (x \u2218 f) x\u271d ** simp [rename_X, comap_apply, aeval_X] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint\u2080 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 \u03b9 : Type u_8 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) \u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (As i) \u2264 \u2191\u2191\u03bc (\u22c3 i, As i) ** rcases show Summable fun i => \u03bc (As i) from ENNReal.summable with \u27e8S, hS\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 \u03b9 : Type u_8 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) S : \u211d\u22650\u221e hS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S \u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (As i) \u2264 \u2191\u2191\u03bc (\u22c3 i, As i) ** rw [hS.tsum_eq] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 \u03b9 : Type u_8 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) S : \u211d\u22650\u221e hS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S \u22a2 S \u2264 \u2191\u2191\u03bc (\u22c3 i, As i) ** refine' tendsto_le_of_eventuallyLE hS tendsto_const_nhds (eventually_of_forall _) ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 \u03b9 : Type u_8 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) S : \u211d\u22650\u221e hS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S \u22a2 \u2200 (x : Finset \u03b9), (fun s => \u2211 b in s, (fun i => \u2191\u2191\u03bc (As i)) b) x \u2264 (fun x => \u2191\u2191\u03bc (\u22c3 i, As i)) x ** intro s ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 \u03b9 : Type u_8 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) S : \u211d\u22650\u221e hS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S s : Finset \u03b9 \u22a2 (fun s => \u2211 b in s, (fun i => \u2191\u2191\u03bc (As i)) b) s \u2264 (fun x => \u2191\u2191\u03bc (\u22c3 i, As i)) s ** simp only [\u2190 measure_biUnion_finset\u2080 (fun _i _hi _j _hj hij => As_disj hij) fun i _ => As_mble i] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9\u271d : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 \u03b9 : Type u_8 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 As : \u03b9 \u2192 Set \u03b1 As_mble : \u2200 (i : \u03b9), NullMeasurableSet (As i) As_disj : Pairwise (AEDisjoint \u03bc on As) S : \u211d\u22650\u221e hS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S s : Finset \u03b9 \u22a2 \u2191\u2191\u03bc (\u22c3 b \u2208 s, As b) \u2264 \u2191\u2191\u03bc (\u22c3 i, As i) ** exact measure_mono (iUnion\u2082_subset_iUnion (fun i : \u03b9 => i \u2208 s) fun i : \u03b9 => As i) ** Qed",
        "informal": ""
    },
    {
        "formal": "Function.LeftInverse.Prod_map ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b4 f\u2081 : \u03b1 \u2192 \u03b2 g\u2081 : \u03b3 \u2192 \u03b4 f\u2082 : \u03b2 \u2192 \u03b1 g\u2082 : \u03b4 \u2192 \u03b3 hf : LeftInverse f\u2081 f\u2082 hg : LeftInverse g\u2081 g\u2082 a : \u03b2 \u00d7 \u03b4 \u22a2 map f\u2081 g\u2081 (map f\u2082 g\u2082 a) = a ** rw [Prod.map_map, hf.comp_eq_id, hg.comp_eq_id, map_id, id] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.extractP_eq_find?_eraseP ** \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l : List \u03b1 \u22a2 extractP p l = (find? p l, eraseP p l) ** exact go #[] _ rfl ** \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l : List \u03b1 acc : Array \u03b1 h : l = acc.data ++ [] \u22a2 extractP.go p l [] acc = (find? p [], acc.data ++ eraseP p []) ** simp [extractP.go, find?, eraseP, h] ** \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l : List \u03b1 acc : Array \u03b1 x : \u03b1 xs : List \u03b1 \u22a2 l = acc.data ++ x :: xs \u2192 extractP.go p l (x :: xs) acc = (find? p (x :: xs), acc.data ++ eraseP p (x :: xs)) ** simp [extractP.go, find?, eraseP] ** \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l : List \u03b1 acc : Array \u03b1 x : \u03b1 xs : List \u03b1 \u22a2 l = acc.data ++ x :: xs \u2192     (bif p x then (some x, acc.data ++ xs) else extractP.go p l xs (Array.push acc x)) =       (match p x with         | true => some x         | false => find? p xs,         acc.data ++ bif p x then xs else x :: eraseP p xs) ** cases p x <;> simp ** case false \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l : List \u03b1 acc : Array \u03b1 x : \u03b1 xs : List \u03b1 \u22a2 l = acc.data ++ x :: xs \u2192 extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs) ** intro h ** case false \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l : List \u03b1 acc : Array \u03b1 x : \u03b1 xs : List \u03b1 h : l = acc.data ++ x :: xs \u22a2 extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs) ** rw [go _ xs] ** case false \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l : List \u03b1 acc : Array \u03b1 x : \u03b1 xs : List \u03b1 h : l = acc.data ++ x :: xs \u22a2 (find? p xs, (Array.push acc x).data ++ eraseP p xs) = (find? p xs, acc.data ++ x :: eraseP p xs)  case false \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l : List \u03b1 acc : Array \u03b1 x : \u03b1 xs : List \u03b1 h : l = acc.data ++ x :: xs \u22a2 l = (Array.push acc x).data ++ xs ** {simp} ** case false \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l : List \u03b1 acc : Array \u03b1 x : \u03b1 xs : List \u03b1 h : l = acc.data ++ x :: xs \u22a2 l = (Array.push acc x).data ++ xs ** simp [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "circleIntegral.integral_sub_zpow_of_ne ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0 ** rcases em (w \u2208 sphere c |R| \u2227 n < -1) with (\u27e8hw, hn\u27e9 | H) ** case inr E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : \u00ac(w \u2208 sphere c |R| \u2227 n < -1) \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0 ** push_neg at H ** case inr E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : w \u2208 sphere c |R| \u2192 -1 \u2264 n hd : \u2200 (z : \u2102), z \u2260 w \u2228 -1 \u2264 n \u2192 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0 ** refine' integral_eq_zero_of_hasDerivWithinAt' fun z hz => (hd z _).hasDerivWithinAt ** case inr E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : w \u2208 sphere c |R| \u2192 -1 \u2264 n hd : \u2200 (z : \u2102), z \u2260 w \u2228 -1 \u2264 n \u2192 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z z : \u2102 hz : z \u2208 sphere c |R| \u22a2 z \u2260 w \u2228 -1 \u2264 n ** exact (ne_or_eq z w).imp_right fun (h : z = w) => H <| h \u25b8 hz ** case inl.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn\u271d : n \u2260 -1 c w : \u2102 R : \u211d hw : w \u2208 sphere c |R| hn : n < -1 \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w) ^ n) = 0 ** exact integral_sub_zpow_of_undef (hn.trans (by decide)) hw ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn\u271d : n \u2260 -1 c w : \u2102 R : \u211d hw : w \u2208 sphere c |R| hn : n < -1 \u22a2 -1 < 0 ** decide ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : w \u2208 sphere c |R| \u2192 -1 \u2264 n \u22a2 \u2200 (z : \u2102), z \u2260 w \u2228 -1 \u2264 n \u2192 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z ** intro z hne ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : w \u2208 sphere c |R| \u2192 -1 \u2264 n z : \u2102 hne : z \u2260 w \u2228 -1 \u2264 n \u22a2 HasDerivAt (fun z => (z - w) ^ (n + 1) / (\u2191n + 1)) ((z - w) ^ n) z ** convert ((hasDerivAt_zpow (n + 1) _ (hne.imp _ _)).comp z\n  ((hasDerivAt_id z).sub_const w)).div_const _ using 1 ** case convert_1 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : w \u2208 sphere c |R| \u2192 -1 \u2264 n z : \u2102 hne : z \u2260 w \u2228 -1 \u2264 n \u22a2 z \u2260 w \u2192 id z - w \u2260 0  case convert_2 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : w \u2208 sphere c |R| \u2192 -1 \u2264 n z : \u2102 hne : z \u2260 w \u2228 -1 \u2264 n \u22a2 -1 \u2264 n \u2192 0 \u2264 n + 1 ** exacts [sub_ne_zero.2, neg_le_iff_add_nonneg.1] ** case h.e'_7 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : w \u2208 sphere c |R| \u2192 -1 \u2264 n z : \u2102 hne : z \u2260 w \u2228 -1 \u2264 n \u22a2 (z - w) ^ n = \u2191(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (\u2191n + 1) ** have hn' : (n + 1 : \u2102) \u2260 0 := by\n  rwa [Ne, \u2190 eq_neg_iff_add_eq_zero, \u2190 Int.cast_one, \u2190 Int.cast_neg, Int.cast_inj] ** case h.e'_7 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : w \u2208 sphere c |R| \u2192 -1 \u2264 n z : \u2102 hne : z \u2260 w \u2228 -1 \u2264 n hn' : \u2191n + 1 \u2260 0 \u22a2 (z - w) ^ n = \u2191(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (\u2191n + 1) ** simp [mul_assoc, mul_div_cancel_left _ hn'] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E n : \u2124 hn : n \u2260 -1 c w : \u2102 R : \u211d H : w \u2208 sphere c |R| \u2192 -1 \u2264 n z : \u2102 hne : z \u2260 w \u2228 -1 \u2264 n \u22a2 \u2191n + 1 \u2260 0 ** rwa [Ne, \u2190 eq_neg_iff_add_eq_zero, \u2190 Int.cast_one, \u2190 Int.cast_neg, Int.cast_inj] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hasFiniteIntegral_smul_iff ** \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 : MulActionWithZero \ud835\udd5c \u03b2 inst\u271d : BoundedSMul \ud835\udd5c \u03b2 c : \ud835\udd5c hc : IsUnit c f : \u03b1 \u2192 \u03b2 \u22a2 HasFiniteIntegral (c \u2022 f) \u2194 HasFiniteIntegral f ** obtain \u27e8c, rfl\u27e9 := hc ** case 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\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 : MulActionWithZero \ud835\udd5c \u03b2 inst\u271d : BoundedSMul \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 c : \ud835\udd5c\u02e3 \u22a2 HasFiniteIntegral (\u2191c \u2022 f) \u2194 HasFiniteIntegral f ** constructor ** case intro.mpr \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 : MulActionWithZero \ud835\udd5c \u03b2 inst\u271d : BoundedSMul \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 c : \ud835\udd5c\u02e3 \u22a2 HasFiniteIntegral f \u2192 HasFiniteIntegral (\u2191c \u2022 f) ** exact HasFiniteIntegral.smul _ ** case intro.mp \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 : MulActionWithZero \ud835\udd5c \u03b2 inst\u271d : BoundedSMul \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 c : \ud835\udd5c\u02e3 \u22a2 HasFiniteIntegral (\u2191c \u2022 f) \u2192 HasFiniteIntegral f ** intro h ** case intro.mp \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 : MulActionWithZero \ud835\udd5c \u03b2 inst\u271d : BoundedSMul \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 c : \ud835\udd5c\u02e3 h : HasFiniteIntegral (\u2191c \u2022 f) \u22a2 HasFiniteIntegral f ** simpa only [smul_smul, Units.inv_mul, one_smul] using h.smul ((c\u207b\u00b9 : \ud835\udd5c\u02e3) : \ud835\udd5c) ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_vector_zero ** \u03b1 : Type u_1 m n : \u2115 f : Vector3 \u03b1 0 \u2192 Prop x\u271d : Exists f v : Vector3 \u03b1 0 fv : f v \u22a2 f [] ** rw [\u2190 eq_nil v] ** \u03b1 : Type u_1 m n : \u2115 f : Vector3 \u03b1 0 \u2192 Prop x\u271d : Exists f v : Vector3 \u03b1 0 fv : f v \u22a2 f v ** exact fv ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p \u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4 ** have hp0_lt : 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 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p \u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4 ** have hp0 : 0 \u2264 p := le_of_lt hp0_lt ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p \u22a2 \u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p \u2202\u03bc ** refine' lintegral_mono fun a => _ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 \u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p ** have h_zero_lt_half_rpow : (0 : \u211d\u22650\u221e) < (1 / 2 : \u211d\u22650\u221e) ^ p := by\n  rw [\u2190 ENNReal.zero_rpow_of_pos hp0_lt]\n  exact ENNReal.rpow_lt_rpow (by simp [zero_lt_one]) hp0_lt ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p \u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p ** have h_rw : (1 / 2 : \u211d\u22650\u221e) ^ p * (2 : \u211d\u22650\u221e) ^ (p - 1) = 1 / 2 := by\n  rw [sub_eq_add_neg, ENNReal.rpow_add _ _ two_ne_zero ENNReal.coe_ne_top, \u2190 mul_assoc, \u2190\n    ENNReal.mul_rpow_of_nonneg _ _ hp0, one_div,\n    ENNReal.inv_mul_cancel two_ne_zero ENNReal.coe_ne_top, ENNReal.one_rpow, one_mul,\n    ENNReal.rpow_neg_one] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p h_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2 \u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p ** rw [\u2190 ENNReal.mul_le_mul_left (ne_of_lt h_zero_lt_half_rpow).symm _] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 \u22a2 0 < (1 / 2) ^ p ** rw [\u2190 ENNReal.zero_rpow_of_pos hp0_lt] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 \u22a2 0 ^ p < (1 / 2) ^ p ** exact ENNReal.rpow_lt_rpow (by simp [zero_lt_one]) hp0_lt ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 \u22a2 0 < 1 / 2 ** simp [zero_lt_one] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p \u22a2 (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2 ** rw [sub_eq_add_neg, ENNReal.rpow_add _ _ two_ne_zero ENNReal.coe_ne_top, \u2190 mul_assoc, \u2190\n  ENNReal.mul_rpow_of_nonneg _ _ hp0, one_div,\n  ENNReal.inv_mul_cancel two_ne_zero ENNReal.coe_ne_top, ENNReal.one_rpow, one_mul,\n  ENNReal.rpow_neg_one] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p h_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2 \u22a2 (1 / 2) ^ p * (f a + g a) ^ p \u2264 (1 / 2) ^ p * (2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p) ** rw [mul_add, \u2190 mul_assoc, \u2190 mul_assoc, h_rw, \u2190 ENNReal.mul_rpow_of_nonneg _ _ hp0, mul_add] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p h_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2 \u22a2 (1 / 2 * f a + 1 / 2 * g a) ^ p \u2264 1 / 2 * f a ^ p + 1 / 2 * g a ^ p ** refine'\n  ENNReal.rpow_arith_mean_le_arith_mean2_rpow (1 / 2 : \u211d\u22650\u221e) (1 / 2 : \u211d\u22650\u221e) (f a) (g a) _\n    hp1 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p h_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2 \u22a2 1 / 2 + 1 / 2 = 1 ** rw [ENNReal.div_add_div_same, one_add_one_eq_two,\n  ENNReal.div_self two_ne_zero ENNReal.coe_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 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p h_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2 \u22a2 (1 / 2) ^ p \u2260 \u22a4 ** rw [\u2190 lt_top_iff_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 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p h_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2 \u22a2 (1 / 2) ^ p < \u22a4 ** refine' ENNReal.rpow_lt_top_of_nonneg hp0 _ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p h_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2 \u22a2 1 / 2 \u2260 \u22a4 ** rw [one_div, ENNReal.inv_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 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p a : \u03b1 h_zero_lt_half_rpow : 0 < (1 / 2) ^ p h_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2 \u22a2 2 \u2260 0 ** exact two_ne_zero ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p \u22a2 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p \u2202\u03bc < \u22a4 ** have h_two : (2 : \u211d\u22650\u221e) ^ (p - 1) \u2260 \u22a4 :=\n  ENNReal.rpow_ne_top_of_nonneg (by simp [hp1]) ENNReal.coe_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 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p h_two : 2 ^ (p - 1) \u2260 \u22a4 \u22a2 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p \u2202\u03bc < \u22a4 ** rw [lintegral_add_left', lintegral_const_mul'' _ (hf.pow_const p),\n  lintegral_const_mul' _ _ h_two, ENNReal.add_lt_top] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p \u22a2 0 \u2264 p - 1 ** simp [hp1] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p h_two : 2 ^ (p - 1) \u2260 \u22a4 \u22a2 2 ^ (p - 1) * \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 \u2227 2 ^ (p - 1) * \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 ** exact \u27e8ENNReal.mul_lt_top h_two hf_top.ne, ENNReal.mul_lt_top h_two hg_top.ne\u27e9 ** case hf \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 hp1 : 1 \u2264 p hp0_lt : 0 < p hp0 : 0 \u2264 p h_two : 2 ^ (p - 1) \u2260 \u22a4 \u22a2 AEMeasurable fun a => 2 ^ (p - 1) * f a ^ p ** exact (hf.pow_const p).const_mul _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.iInf_apply' ** \u03b1 : Type u_1 \u03b9 : Sort u_2 m : \u03b9 \u2192 OuterMeasure \u03b1 s : Set \u03b1 hs : Set.Nonempty s \u22a2 \u2191(\u2a05 i, m i) s = \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (t n) ** rw [iInf, sInf_apply' hs] ** \u03b1 : Type u_1 \u03b9 : Sort u_2 m : \u03b9 \u2192 OuterMeasure \u03b1 s : Set \u03b1 hs : Set.Nonempty s \u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 \u03bc \u2208 range fun i => m i, \u2191\u03bc (t n) =     \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (t n) ** simp only [iInf_range] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.card_fintype_Ioo_of_lt ** a b : \u2124 h : a < b \u22a2 \u2191(Fintype.card \u2191(Set.Ioo a b)) = b - a - 1 ** rw [card_fintype_Ioo, sub_sub, toNat_sub_of_le h] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.getLast_eq_get ** \u03b1 : Type u_1 a : \u03b1 h : [a] \u2260 [] \u22a2 getLast [a] h = get [a] { val := length [a] - 1, isLt := (_ : length [a] - 1 < length [a]) } ** rw [getLast_singleton, get_singleton] ** \u03b1 : Type u_1 a b : \u03b1 l : List \u03b1 h : a :: b :: l \u2260 [] \u22a2 getLast (a :: b :: l) h =     get (a :: b :: l) { val := length (a :: b :: l) - 1, isLt := (_ : length (a :: b :: l) - 1 < length (a :: b :: l)) } ** rw [getLast_cons', getLast_eq_get (b :: l)] ** \u03b1 : Type u_1 a b : \u03b1 l : List \u03b1 h : a :: b :: l \u2260 [] \u22a2 get (b :: l) { val := length (b :: l) - 1, isLt := (_ : length (b :: l) - 1 < length (b :: l)) } =     get (a :: b :: l) { val := length (a :: b :: l) - 1, isLt := (_ : length (a :: b :: l) - 1 < length (a :: b :: l)) }  \u03b1 : Type u_1 a b : \u03b1 l : List \u03b1 h : a :: b :: l \u2260 [] \u22a2 b :: l \u2260 [] ** {rfl} ** \u03b1 : Type u_1 a b : \u03b1 l : List \u03b1 h : a :: b :: l \u2260 [] \u22a2 b :: l \u2260 [] ** exact cons_ne_nil b l ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.rnDeriv_sub ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t\u271d s t : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : HaveLebesgueDecomposition s \u03bc inst\u271d : HaveLebesgueDecomposition t \u03bc hst : HaveLebesgueDecomposition (s - t) \u03bc \u22a2 rnDeriv (s - t) \u03bc =\u1da0[ae \u03bc] rnDeriv s \u03bc - rnDeriv t \u03bc ** rw [sub_eq_add_neg] at hst ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t\u271d s t : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : HaveLebesgueDecomposition s \u03bc inst\u271d : HaveLebesgueDecomposition t \u03bc hst : HaveLebesgueDecomposition (s + -t) \u03bc \u22a2 rnDeriv (s - t) \u03bc =\u1da0[ae \u03bc] rnDeriv s \u03bc - rnDeriv t \u03bc ** rw [sub_eq_add_neg, sub_eq_add_neg] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t\u271d s t : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : HaveLebesgueDecomposition s \u03bc inst\u271d : HaveLebesgueDecomposition t \u03bc hst : HaveLebesgueDecomposition (s + -t) \u03bc \u22a2 rnDeriv (s + -t) \u03bc =\u1da0[ae \u03bc] rnDeriv s \u03bc + -rnDeriv t \u03bc ** exact ae_eq_trans (rnDeriv_add _ _ _) (Filter.EventuallyEq.add (ae_eq_refl _) (rnDeriv_neg _ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "LinearMap.exists_map_addHaar_eq_smul_addHaar' ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** have : ProperSpace E := properSpace_of_locallyCompactSpace \ud835\udd5c ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this : ProperSpace E \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** have : FiniteDimensional \ud835\udd5c E := finiteDimensional_of_locallyCompactSpace \ud835\udd5c ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** let S : Submodule \ud835\udd5c E := LinearMap.ker L ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** obtain \u27e8T, hT\u27e9 : \u2203 T : Submodule \ud835\udd5c E, IsCompl S T := Submodule.exists_isCompl S ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** let M : (S \u00d7 T) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** have M_cont : Continuous M.symm := LinearMap.continuous_of_finiteDimensional _ ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** let P : S \u00d7 T \u2192\u2097[\ud835\udd5c] T := LinearMap.snd \ud835\udd5c S T ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** have P_cont : Continuous P := LinearMap.continuous_of_finiteDimensional _ ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** have I : Function.Bijective (LinearMap.domRestrict L T) :=\n  \u27e8LinearMap.injective_domRestrict_iff.2 (IsCompl.inf_eq_bot hT.symm),\n  (LinearMap.surjective_domRestrict_iff h).2 hT.symm.sup_eq_top\u27e9 ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** let L' : T \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (LinearMap.domRestrict L T) I ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** have L'_cont : Continuous L' := LinearMap.continuous_of_finiteDimensional _ ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** have A : L = (L' : T \u2192\u2097[\ud835\udd5c] F).comp (P.comp (M.symm : E \u2192\u2097[\ud835\udd5c] (S \u00d7 T))) := by\n  ext x\n  obtain \u27e8y, z, hyz\u27e9 : \u2203 (y : S) (z : T), M.symm x = (y, z) := \u27e8_, _, rfl\u27e9\n  have : x = M (y, z) := by\n    rw [\u2190 hyz]; simp only [LinearEquiv.apply_symm_apply]\n  simp [this] ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** let \u03bcS : Measure S := addHaar ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** let \u03bcT : Measure T := addHaar ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** obtain \u27e8c\u2080, c\u2080_pos, c\u2080_fin, h\u2080\u27e9 :\n    \u2203 c\u2080 : \u211d\u22650\u221e, c\u2080 \u2260 0 \u2227 c\u2080 \u2260 \u221e \u2227 \u03bc.map M.symm = c\u2080 \u2022 \u03bcS.prod \u03bcT := by\n  have : IsAddHaarMeasure (\u03bc.map M.symm) :=\n    M.toContinuousLinearEquiv.symm.isAddHaarMeasure_map \u03bc\n  exact isAddHaarMeasure_eq_smul_isAddHaarMeasure _ _ ** case intro.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar c\u2080 : \u211d\u22650\u221e c\u2080_pos : c\u2080 \u2260 0 c\u2080_fin : c\u2080 \u2260 \u22a4 h\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** have J : (\u03bcS.prod \u03bcT).map P = (\u03bcS univ) \u2022 \u03bcT := map_snd_prod ** case intro.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar c\u2080 : \u211d\u22650\u221e c\u2080_pos : c\u2080 \u2260 0 c\u2080_fin : c\u2080 \u2260 \u22a4 h\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT J : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** obtain \u27e8c\u2081, c\u2081_pos, c\u2081_fin, h\u2081\u27e9 : \u2203 c\u2081 : \u211d\u22650\u221e, c\u2081 \u2260 0 \u2227 c\u2081 \u2260 \u221e \u2227 \u03bcT.map L' = c\u2081 \u2022 \u03bd := by\n  have : IsAddHaarMeasure (\u03bcT.map L') :=\n    L'.toContinuousLinearEquiv.isAddHaarMeasure_map \u03bcT\n  exact isAddHaarMeasure_eq_smul_isAddHaarMeasure _ _ ** case intro.intro.intro.intro.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar c\u2080 : \u211d\u22650\u221e c\u2080_pos : c\u2080 \u2260 0 c\u2080_fin : c\u2080 \u2260 \u22a4 h\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT J : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT c\u2081 : \u211d\u22650\u221e c\u2081_pos : c\u2081 \u2260 0 c\u2081_fin : c\u2081 \u2260 \u22a4 h\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd \u22a2 \u2203 c, 0 < c \u2227 c < \u22a4 \u2227 map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd ** refine \u27e8c\u2080 * c\u2081, by simp [pos_iff_ne_zero, c\u2080_pos, c\u2081_pos], ENNReal.mul_lt_top c\u2080_fin c\u2081_fin, ?_\u27e9 ** case intro.intro.intro.intro.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar c\u2080 : \u211d\u22650\u221e c\u2080_pos : c\u2080 \u2260 0 c\u2080_fin : c\u2080 \u2260 \u22a4 h\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT J : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT c\u2081 : \u211d\u22650\u221e c\u2081_pos : c\u2081 \u2260 0 c\u2081_fin : c\u2081 \u2260 \u22a4 h\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd \u22a2 map (\u2191L) \u03bc = (c\u2080 * c\u2081 * \u2191\u2191addHaar univ) \u2022 \u03bd ** simp only [I, h\u2080, Measure.map_smul, J, smul_smul, h\u2081] ** case intro.intro.intro.intro.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar c\u2080 : \u211d\u22650\u221e c\u2080_pos : c\u2080 \u2260 0 c\u2080_fin : c\u2080 \u2260 \u22a4 h\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT J : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT c\u2081 : \u211d\u22650\u221e c\u2081_pos : c\u2081 \u2260 0 c\u2081_fin : c\u2081 \u2260 \u22a4 h\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd \u22a2 (c\u2080 * \u2191\u2191addHaar univ * c\u2081) \u2022 \u03bd = (c\u2080 * c\u2081 * \u2191\u2191addHaar univ) \u2022 \u03bd ** rw [mul_assoc, mul_comm _ c\u2081, \u2190 mul_assoc] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d : ProperSpace E this : FiniteDimensional \ud835\udd5c E \u22a2 ProperSpace F ** rcases subsingleton_or_nontrivial E with hE|hE ** case inl \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d : ProperSpace E this : FiniteDimensional \ud835\udd5c E hE : Subsingleton E \u22a2 ProperSpace F ** have : Subsingleton F := Function.Surjective.subsingleton h ** case inl \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E hE : Subsingleton E this : Subsingleton F \u22a2 ProperSpace F ** infer_instance ** case inr \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d : ProperSpace E this : FiniteDimensional \ud835\udd5c E hE : Nontrivial E \u22a2 ProperSpace F ** have : ProperSpace \ud835\udd5c := properSpace_of_locallyCompact_module \ud835\udd5c E ** case inr \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E hE : Nontrivial E this : ProperSpace \ud835\udd5c \u22a2 ProperSpace F ** have : FiniteDimensional \ud835\udd5c F := Module.Finite.of_surjective L h ** case inr \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b2 : ProperSpace E this\u271d\u00b9 : FiniteDimensional \ud835\udd5c E hE : Nontrivial E this\u271d : ProperSpace \ud835\udd5c this : FiniteDimensional \ud835\udd5c F \u22a2 ProperSpace F ** exact FiniteDimensional.proper \ud835\udd5c F ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' \u22a2 L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) ** ext x ** case h \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' x : E \u22a2 \u2191L x = \u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))) x ** obtain \u27e8y, z, hyz\u27e9 : \u2203 (y : S) (z : T), M.symm x = (y, z) := \u27e8_, _, rfl\u27e9 ** case h.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' x : E y : { x // x \u2208 S } z : { x // x \u2208 T } hyz : \u2191(LinearEquiv.symm M) x = (y, z) \u22a2 \u2191L x = \u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))) x ** have : x = M (y, z) := by\n  rw [\u2190 hyz]; simp only [LinearEquiv.apply_symm_apply] ** case h.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b2 : ProperSpace E this\u271d\u00b9 : FiniteDimensional \ud835\udd5c E this\u271d : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' x : E y : { x // x \u2208 S } z : { x // x \u2208 T } hyz : \u2191(LinearEquiv.symm M) x = (y, z) this : x = \u2191M (y, z) \u22a2 \u2191L x = \u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M))) x ** simp [this] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' x : E y : { x // x \u2208 S } z : { x // x \u2208 T } hyz : \u2191(LinearEquiv.symm M) x = (y, z) \u22a2 x = \u2191M (y, z) ** rw [\u2190 hyz] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' x : E y : { x // x \u2208 S } z : { x // x \u2208 T } hyz : \u2191(LinearEquiv.symm M) x = (y, z) \u22a2 x = \u2191M (\u2191(LinearEquiv.symm M) x) ** simp only [LinearEquiv.apply_symm_apply] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) \u22a2 map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) ** rw [Measure.map_map, Measure.map_map, A] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) \u22a2 map (\u2191(comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)))) \u03bc = map ((\u2191L' \u2218 \u2191P) \u2218 \u2191(LinearEquiv.symm M)) \u03bc ** rfl ** case hg \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) \u22a2 Measurable (\u2191L' \u2218 \u2191P) ** exact L'_cont.measurable.comp P_cont.measurable ** case hf \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) \u22a2 Measurable \u2191(LinearEquiv.symm M) ** exact M_cont.measurable ** case hg \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) \u22a2 Measurable \u2191L' ** exact L'_cont.measurable ** case hf \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) \u22a2 Measurable \u2191P ** exact P_cont.measurable ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar \u22a2 \u2203 c\u2080, c\u2080 \u2260 0 \u2227 c\u2080 \u2260 \u22a4 \u2227 map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT ** have : IsAddHaarMeasure (\u03bc.map M.symm) :=\n  M.toContinuousLinearEquiv.symm.isAddHaarMeasure_map \u03bc ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b2 : ProperSpace E this\u271d\u00b9 : FiniteDimensional \ud835\udd5c E this\u271d : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar this : IsAddHaarMeasure (map (\u2191(LinearEquiv.symm M)) \u03bc) \u22a2 \u2203 c\u2080, c\u2080 \u2260 0 \u2227 c\u2080 \u2260 \u22a4 \u2227 map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT ** exact isAddHaarMeasure_eq_smul_isAddHaarMeasure _ _ ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar c\u2080 : \u211d\u22650\u221e c\u2080_pos : c\u2080 \u2260 0 c\u2080_fin : c\u2080 \u2260 \u22a4 h\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT J : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT \u22a2 \u2203 c\u2081, c\u2081 \u2260 0 \u2227 c\u2081 \u2260 \u22a4 \u2227 map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd ** have : IsAddHaarMeasure (\u03bcT.map L') :=\n  L'.toContinuousLinearEquiv.isAddHaarMeasure_map \u03bcT ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b2 : ProperSpace E this\u271d\u00b9 : FiniteDimensional \ud835\udd5c E this\u271d : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar c\u2080 : \u211d\u22650\u221e c\u2080_pos : c\u2080 \u2260 0 c\u2080_fin : c\u2080 \u2260 \u22a4 h\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT J : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT this : IsAddHaarMeasure (map (\u2191L') \u03bcT) \u22a2 \u2203 c\u2081, c\u2081 \u2260 0 \u2227 c\u2081 \u2260 \u22a4 \u2227 map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd ** exact isAddHaarMeasure_eq_smul_isAddHaarMeasure _ _ ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L this\u271d\u00b9 : ProperSpace E this\u271d : FiniteDimensional \ud835\udd5c E this : ProperSpace F S : Submodule \ud835\udd5c E := ker L T : Submodule \ud835\udd5c E hT : IsCompl S T M : ({ x // x \u2208 S } \u00d7 { x // x \u2208 T }) \u2243\u2097[\ud835\udd5c] E := Submodule.prodEquivOfIsCompl S T hT M_cont : Continuous \u2191(LinearEquiv.symm M) P : { x // x \u2208 S } \u00d7 { x // x \u2208 T } \u2192\u2097[\ud835\udd5c] { x // x \u2208 T } := snd \ud835\udd5c { x // x \u2208 S } { x // x \u2208 T } P_cont : Continuous \u2191P I\u271d : Function.Bijective \u2191(domRestrict L T) L' : { x // x \u2208 T } \u2243\u2097[\ud835\udd5c] F := LinearEquiv.ofBijective (domRestrict L T) I\u271d L'_cont : Continuous \u2191L' A : L = comp (\u2191L') (comp P \u2191(LinearEquiv.symm M)) I : map (\u2191L) \u03bc = map (\u2191L') (map (\u2191P) (map (\u2191(LinearEquiv.symm M)) \u03bc)) \u03bcS : Measure { x // x \u2208 S } := addHaar \u03bcT : Measure { x // x \u2208 T } := addHaar c\u2080 : \u211d\u22650\u221e c\u2080_pos : c\u2080 \u2260 0 c\u2080_fin : c\u2080 \u2260 \u22a4 h\u2080 : map (\u2191(LinearEquiv.symm M)) \u03bc = c\u2080 \u2022 Measure.prod \u03bcS \u03bcT J : map (\u2191P) (Measure.prod \u03bcS \u03bcT) = \u2191\u2191\u03bcS univ \u2022 \u03bcT c\u2081 : \u211d\u22650\u221e c\u2081_pos : c\u2081 \u2260 0 c\u2081_fin : c\u2081 \u2260 \u22a4 h\u2081 : map (\u2191L') \u03bcT = c\u2081 \u2022 \u03bd \u22a2 0 < c\u2080 * c\u2081 ** simp [pos_iff_ne_zero, c\u2080_pos, c\u2081_pos] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono ** \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 \u22a2 StrictMono (seqTendstoAeSeq hfg) ** refine' strictMono_nat_of_lt_succ fun n => _ ** \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 < seqTendstoAeSeq hfg (n + 1) ** rw [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 < max (seqTendstoAeSeqAux hfg (n + 1)) (seqTendstoAeSeq hfg n + 1) ** exact lt_of_lt_of_le (lt_add_one <| seqTendstoAeSeq hfg n) (le_max_right _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.fold_op_distrib ** \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\u271d : \u03b1 \u2192 \u03b2 b : \u03b2 s : Finset \u03b1 a : \u03b1 f g : \u03b1 \u2192 \u03b2 b\u2081 b\u2082 : \u03b2 \u22a2 fold op (op b\u2081 b\u2082) (fun x => op (f x) (g x)) s = op (fold op b\u2081 f s) (fold op b\u2082 g s) ** simp only [fold, fold_distrib] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpIndSMul_add ** \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\u2079 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2078 : NormedAddCommGroup E inst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2076 : CompleteSpace E inst\u271d\u00b9\u2075 : NormedAddCommGroup E' inst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b3 : CompleteSpace E' inst\u271d\u00b9\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedAddCommGroup G' inst\u271d\u2077 : NormedSpace \u211d G' inst\u271d\u2076 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2075 : IsROrC \ud835\udd5c' inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b2 : CompleteSpace E'' inst\u271d\u00b9 : NormedSpace \u211d E'' inst\u271d : NormedSpace \u211d G hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x y : G \u22a2 condexpIndSMul hm hs h\u03bcs (x + y) = condexpIndSMul hm hs h\u03bcs x + condexpIndSMul hm hs h\u03bcs y ** simp_rw [condexpIndSMul] ** \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\u2079 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2078 : NormedAddCommGroup E inst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2076 : CompleteSpace E inst\u271d\u00b9\u2075 : NormedAddCommGroup E' inst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b3 : CompleteSpace E' inst\u271d\u00b9\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedAddCommGroup G' inst\u271d\u2077 : NormedSpace \u211d G' inst\u271d\u2076 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2075 : IsROrC \ud835\udd5c' inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b2 : CompleteSpace E'' inst\u271d\u00b9 : NormedSpace \u211d E'' inst\u271d : NormedSpace \u211d G hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x y : G \u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d (x + y))) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =     \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +       \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d y)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) ** rw [toSpanSingleton_add, add_compLpL, add_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.iUnion_null_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 \u03b9 : Prop s : \u03b9 \u2192 Set \u03b1 \u22a2 (\u2200 (i : \u03b9), \u2191m (s i) = 0) \u2192 \u2191m (\u22c3 (i : \u03b9), s i) = 0 ** by_cases i : \u03b9 <;> simp [i] ** case pos \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 \u03b9 : Prop s : \u03b9 \u2192 Set \u03b1 i : \u03b9 \u22a2 (\u2200 (i : \u03b9), \u2191m (s i) = 0) \u2192 \u2191m (s (_ : \u03b9)) = 0 ** exact (fun h => h (Iff.mpr (Iff.of_eq (eq_true i)) trivial)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 \u03bc : Measure \u03a9 \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : CompleteSpace E f : \u03a9 \u2192 E inst\u271d\u2076 : IsCountablyGenerated atTop inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : SecondCountableTopology \u03b9 inst\u271d\u00b9 : BorelSpace \u03b9 h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 \u03bc[f|IsStoppingTime.measurableSpace         (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]     \u03bc[f|IsStoppingTime.measurableSpace h\u03c4] ** have : SigmaFinite (\u03bc.trim h\u03c4.measurableSpace_le) :=\n  haveI h_le : (h\u03c4.min h\u03c3).measurableSpace \u2264 h\u03c4.measurableSpace := by\n    rw [IsStoppingTime.measurableSpace_min]\n    exact inf_le_left; simp_all only\n  sigmaFiniteTrim_mono _ h_le ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 \u03bc : Measure \u03a9 \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : CompleteSpace E f : \u03a9 \u2192 E inst\u271d\u2076 : IsCountablyGenerated atTop inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : SecondCountableTopology \u03b9 inst\u271d\u00b9 : BorelSpace \u03b9 h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) \u22a2 \u03bc[f|IsStoppingTime.measurableSpace         (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9))] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}]     \u03bc[f|IsStoppingTime.measurableSpace h\u03c4] ** refine' (condexp_ae_eq_restrict_of_measurableSpace_eq_on h\u03c4.measurableSpace_le\n  (h\u03c4.min h\u03c3).measurableSpace_le (h\u03c4.measurableSet_le_stopping_time h\u03c3) fun t => _).symm ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 \u03bc : Measure \u03a9 \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : CompleteSpace E f : \u03a9 \u2192 E inst\u271d\u2076 : IsCountablyGenerated atTop inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : SecondCountableTopology \u03b9 inst\u271d\u00b9 : BorelSpace \u03b9 h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) t : Set \u03a9 \u22a2 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) \u2194 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) ** rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_le_iff] ** case h\u03c0 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 \u03bc : Measure \u03a9 \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : CompleteSpace E f : \u03a9 \u2192 E inst\u271d\u2076 : IsCountablyGenerated atTop inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : SecondCountableTopology \u03b9 inst\u271d\u00b9 : BorelSpace \u03b9 h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) this : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) t : Set \u03a9 \u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c3 \u03c9 ** simp_all only ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 \u03bc : Measure \u03a9 \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : CompleteSpace E f : \u03a9 \u2192 E inst\u271d\u2076 : IsCountablyGenerated atTop inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : SecondCountableTopology \u03b9 inst\u271d\u00b9 : BorelSpace \u03b9 h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 IsStoppingTime.measurableSpace h\u03c4 ** rw [IsStoppingTime.measurableSpace_min] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 \u03bc : Measure \u03a9 \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : CompleteSpace E f : \u03a9 \u2192 E inst\u271d\u2076 : IsCountablyGenerated atTop inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : SecondCountableTopology \u03b9 inst\u271d\u00b9 : BorelSpace \u03b9 h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 IsStoppingTime.measurableSpace ?h\u03c4 \u2293 IsStoppingTime.measurableSpace ?h\u03c0 \u2264 IsStoppingTime.measurableSpace h\u03c4  case h\u03c4 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 \u03bc : Measure \u03a9 \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : CompleteSpace E f : \u03a9 \u2192 E inst\u271d\u2076 : IsCountablyGenerated atTop inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : SecondCountableTopology \u03b9 inst\u271d\u00b9 : BorelSpace \u03b9 h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c4 \u03c9  case h\u03c0 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 \u03bc : Measure \u03a9 \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : CompleteSpace E f : \u03a9 \u2192 E inst\u271d\u2076 : IsCountablyGenerated atTop inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : SecondCountableTopology \u03b9 inst\u271d\u00b9 : BorelSpace \u03b9 h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c3 \u03c9 ** exact inf_le_left ** case h\u03c0 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 \u03bc : Measure \u03a9 \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : CompleteSpace E f : \u03a9 \u2192 E inst\u271d\u2076 : IsCountablyGenerated atTop inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : SecondCountableTopology \u03b9 inst\u271d\u00b9 : BorelSpace \u03b9 h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d :   SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime \u2131 fun \u03c9 => min (\u03c4 \u03c9) (\u03c3 \u03c9)) \u2264 m)) \u22a2 IsStoppingTime \u2131 fun \u03c9 => \u03c3 \u03c9 ** simp_all only ** Qed",
        "informal": ""
    },
    {
        "formal": "Orientation.measure_eq_volume ** \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) \u22a2 AlternatingMap.measure (volumeForm o) = volume ** have A : o.volumeForm.measure (stdOrthonormalBasis \u211d F).toBasis.parallelepiped = 1 :=\n  Orientation.measure_orthonormalBasis o (stdOrthonormalBasis \u211d F) ** \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) A :   \u2191\u2191(AlternatingMap.measure (volumeForm o))       \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) =     1 \u22a2 AlternatingMap.measure (volumeForm o) = volume ** rw [addHaarMeasure_unique o.volumeForm.measure\n  (stdOrthonormalBasis \u211d F).toBasis.parallelepiped, A, one_smul] ** \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) A :   \u2191\u2191(AlternatingMap.measure (volumeForm o))       \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) =     1 \u22a2 addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) = volume ** simp only [volume, Basis.addHaar] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.mkMetric'.trim_pre ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r\u271d : \u211d\u22650\u221e \u03bc : OuterMeasure X s : Set X inst\u271d\u00b9 : MeasurableSpace X inst\u271d : OpensMeasurableSpace X m : Set X \u2192 \u211d\u22650\u221e hcl : \u2200 (s : Set X), m (closure s) = m s r : \u211d\u22650\u221e \u22a2 trim (pre m r) = pre m r ** refine' le_antisymm (le_pre.2 fun s hs => _) (le_trim _) ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r\u271d : \u211d\u22650\u221e \u03bc : OuterMeasure X s\u271d : Set X inst\u271d\u00b9 : MeasurableSpace X inst\u271d : OpensMeasurableSpace X m : Set X \u2192 \u211d\u22650\u221e hcl : \u2200 (s : Set X), m (closure s) = m s r : \u211d\u22650\u221e s : Set X hs : diam s \u2264 r \u22a2 \u2191(trim (pre m r)) s \u2264 m s ** rw [trim_eq_iInf] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r\u271d : \u211d\u22650\u221e \u03bc : OuterMeasure X s\u271d : Set X inst\u271d\u00b9 : MeasurableSpace X inst\u271d : OpensMeasurableSpace X m : Set X \u2192 \u211d\u22650\u221e hcl : \u2200 (s : Set X), m (closure s) = m s r : \u211d\u22650\u221e s : Set X hs : diam s \u2264 r \u22a2 \u2a05 t, \u2a05 (_ : s \u2286 t), \u2a05 (_ : MeasurableSet t), \u2191(pre m r) t \u2264 m s ** refine' iInf_le_of_le (closure s) <| iInf_le_of_le subset_closure <|\n  iInf_le_of_le measurableSet_closure ((pre_le _).trans_eq (hcl _)) ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r\u271d : \u211d\u22650\u221e \u03bc : OuterMeasure X s\u271d : Set X inst\u271d\u00b9 : MeasurableSpace X inst\u271d : OpensMeasurableSpace X m : Set X \u2192 \u211d\u22650\u221e hcl : \u2200 (s : Set X), m (closure s) = m s r : \u211d\u22650\u221e s : Set X hs : diam s \u2264 r \u22a2 diam (closure s) \u2264 r ** rwa [diam_closure] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.replace_cons_self ** \u03b1 : Type u_1 as : List \u03b1 b : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : LawfulBEq \u03b1 a : \u03b1 \u22a2 replace (a :: as) a b = b :: as ** simp [replace_cons] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.rnDeriv_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bd s : Set \u03b1 hs : MeasurableSet s \u22a2 rnDeriv (restrict \u03bd s) \u03bd =\u1da0[ae \u03bd] indicator s 1 ** rw [\u2190 withDensity_indicator_one hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bd s : Set \u03b1 hs : MeasurableSet s \u22a2 rnDeriv (withDensity \u03bd (indicator s 1)) \u03bd =\u1da0[ae \u03bd] indicator s 1 ** exact rnDeriv_withDensity _ (measurable_one.indicator hs) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.OuterRegular.smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 inst\u271d : OuterRegular \u03bc x : \u211d\u22650\u221e hx : x \u2260 \u22a4 \u22a2 OuterRegular (x \u2022 \u03bc) ** rcases eq_or_ne x 0 with (rfl | h0) ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 inst\u271d : OuterRegular \u03bc hx : 0 \u2260 \u22a4 \u22a2 OuterRegular (0 \u2022 \u03bc) ** rw [zero_smul] ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 inst\u271d : OuterRegular \u03bc hx : 0 \u2260 \u22a4 \u22a2 OuterRegular 0 ** exact OuterRegular.zero ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 inst\u271d : OuterRegular \u03bc x : \u211d\u22650\u221e hx : x \u2260 \u22a4 h0 : x \u2260 0 \u22a2 OuterRegular (x \u2022 \u03bc) ** refine' \u27e8fun A _ r hr => _\u27e9 ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 inst\u271d : OuterRegular \u03bc x : \u211d\u22650\u221e hx : x \u2260 \u22a4 h0 : x \u2260 0 A : Set \u03b1 x\u271d : MeasurableSet A r : \u211d\u22650\u221e hr : r > \u2191\u2191(x \u2022 \u03bc) A \u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(x \u2022 \u03bc) U < r ** rw [smul_apply, A.measure_eq_iInf_isOpen, smul_eq_mul] at hr ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 inst\u271d : OuterRegular \u03bc x : \u211d\u22650\u221e hx : x \u2260 \u22a4 h0 : x \u2260 0 A : Set \u03b1 x\u271d : MeasurableSet A r : \u211d\u22650\u221e hr : r > x * \u2a05 U, \u2a05 (_ : A \u2286 U), \u2a05 (_ : IsOpen U), \u2191\u2191\u03bc U \u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(x \u2022 \u03bc) U < r ** simpa only [ENNReal.mul_iInf_of_ne h0 hx, gt_iff_lt, iInf_lt_iff, exists_prop] using hr ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.lintegral_mul_le_Lp_mul_Lq ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** by_cases hf_zero : \u222b\u207b a, f a ^ p \u2202\u03bc = 0 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** by_cases hg_zero : \u222b\u207b a, g a ^ q \u2202\u03bc = 0 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** by_cases hf_top : \u222b\u207b a, f a ^ p \u2202\u03bc = \u22a4 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** by_cases hg_top : \u222b\u207b a, g a ^ q \u2202\u03bc = \u22a4 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** exact ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top hpq hf hf_top hg_top hf_zero hg_zero ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** refine' Eq.trans_le _ (zero_le _) ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0 ** exact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.nonneg hf hf_zero ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** refine' Eq.trans_le _ (zero_le _) ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0 ** rw [mul_comm] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc = 0 ** exact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.symm.nonneg hg hg_zero ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** exact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.pos hpq.symm.nonneg hf_top hg_zero ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** rw [mul_comm, mul_comm ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p))] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0 hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4 \u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) ** exact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.symm.pos hpq.nonneg hg_top hf_zero ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.comap_compl ** \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' : MeasurableSpace \u03b2 inst\u271d : BooleanAlgebra \u03b2 h : Measurable compl f : \u03b1 \u2192 \u03b2 \u22a2 MeasurableSpace.comap (fun a => (f a)\u1d9c) inferInstance = MeasurableSpace.comap f inferInstance ** rw [\u2190Function.comp_def, \u2190MeasurableSpace.comap_comp] ** \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' : MeasurableSpace \u03b2 inst\u271d : BooleanAlgebra \u03b2 h : Measurable compl f : \u03b1 \u2192 \u03b2 \u22a2 MeasurableSpace.comap (fun a => f a) (MeasurableSpace.comap compl inferInstance) =     MeasurableSpace.comap f inferInstance ** congr ** case e_m \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' : MeasurableSpace \u03b2 inst\u271d : BooleanAlgebra \u03b2 h : Measurable compl f : \u03b1 \u2192 \u03b2 \u22a2 MeasurableSpace.comap compl inferInstance = inferInstance ** exact (MeasurableEquiv.ofInvolutive _ compl_involutive h).measurableEmbedding.comap_eq ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1.stmts_supportsStmt ** \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 \u03c3 : Type u_3 inst\u271d : Inhabited \u039b M : \u039b \u2192 Stmt\u2081 S : Finset \u039b q : Stmt\u2081 ss : Supports M S \u22a2 some q \u2208 stmts M S \u2192 SupportsStmt S q ** 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\u00b9 : Inhabited \u0393 \u039b : Type u_2 \u03c3 : Type u_3 inst\u271d : Inhabited \u039b M : \u039b \u2192 Stmt\u2081 S : Finset \u039b q : Stmt\u2081 ss : Supports M S \u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q ** exact fun l ls h \u21a6 stmts\u2081_supportsStmt_mono h (ss.2 _ ls) ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.zero_emod ** b : Int \u22a2 0 % b = 0 ** simp [mod_def', emod] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.sup_sdiff_left ** 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\u271d s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 a : \u03b1 \u22a2 (sup s fun b => a \\ f b) = a \\ inf s f ** refine' Finset.cons_induction_on s _ fun b t _ h => _ ** case refine'_1 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\u271d s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 a : \u03b1 \u22a2 (sup \u2205 fun b => a \\ f b) = a \\ inf \u2205 f ** rw [sup_empty, inf_empty, sdiff_top] ** case refine'_2 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\u271d s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 a : \u03b1 b : \u03b9 t : Finset \u03b9 x\u271d : \u00acb \u2208 t h : (sup t fun b => a \\ f b) = a \\ inf t f \u22a2 (sup (cons b t x\u271d) fun b => a \\ f b) = a \\ inf (cons b t x\u271d) f ** rw [sup_cons, inf_cons, h, sdiff_inf] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.StronglyMeasurable.integral_kernel_prod_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 : \u03b3 \u00d7 \u03b2 \u2192 E hf : StronglyMeasurable f \u22a2 StronglyMeasurable fun y => \u222b (x : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, y) ** change\n  StronglyMeasurable\n    ((fun y => \u222b x, (fun u : \u03b3 \u00d7 \u03b1 \u00d7 \u03b2 => f (u.1, u.2.2)) (x, y) \u2202\u03b7 y) \u2218 fun x => (a, 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 : \u03b3 \u00d7 \u03b2 \u2192 E hf : StronglyMeasurable f \u22a2 StronglyMeasurable ((fun y => \u222b (x : \u03b3), (fun u => f (u.1, u.2.2)) (x, y) \u2202\u2191\u03b7 y) \u2218 fun x => (a, x)) ** refine' StronglyMeasurable.comp_measurable _ measurable_prod_mk_left ** 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 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 : \u03b3 \u00d7 \u03b2 \u2192 E hf : StronglyMeasurable f \u22a2 StronglyMeasurable fun y => \u222b (x : \u03b3), (fun u => f (u.1, u.2.2)) (x, y) \u2202\u2191\u03b7 y  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 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 : \u03b3 \u00d7 \u03b2 \u2192 E hf : StronglyMeasurable f \u22a2 MeasurableSpace \u03b1 ** have := MeasureTheory.StronglyMeasurable.integral_kernel_prod_left' (\u03ba := \u03b7)\n  (hf.comp_measurable (measurable_fst.prod_mk measurable_snd.snd)) ** 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 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 : \u03b3 \u00d7 \u03b2 \u2192 E hf : StronglyMeasurable f this : StronglyMeasurable fun y => \u222b (x : \u03b3), (f \u2218 fun a => (a.1, a.2.2)) (x, y) \u2202\u2191\u03b7 y \u22a2 StronglyMeasurable fun y => \u222b (x : \u03b3), (fun u => f (u.1, u.2.2)) (x, y) \u2202\u2191\u03b7 y  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 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 : \u03b3 \u00d7 \u03b2 \u2192 E hf : StronglyMeasurable f \u22a2 MeasurableSpace \u03b1 ** simpa using this ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.injective_codRestrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 f : \u03b9 \u2192 \u03b1 s : Set \u03b1 h : \u2200 (x : \u03b9), f x \u2208 s \u22a2 Injective (codRestrict f s h) \u2194 Injective f ** simp only [Injective, Subtype.ext_iff, val_codRestrict_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.upcrossingsBefore_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 upcrossingsBefore 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N \u03c9 = upcrossingsBefore a b f N \u03c9 ** simp_rw [upcrossingsBefore, (crossing_pos_eq hab).1] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.smul_map_diagonal_volume_pi ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 \u22a2 ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = volume ** refine' (Measure.pi_eq fun s hs => _).symm ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 \u2191\u2191(ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume) (Set.pi univ s) =     \u220f i : \u03b9, \u2191\u2191volume (s i) ** simp only [det_diagonal, Measure.coe_smul, Algebra.id.smul_eq_mul, Pi.smul_apply] ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191(Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume) (Set.pi univ s) =     \u220f x : \u03b9, \u2191\u2191volume (s x) ** rw [Measure.map_apply _ (MeasurableSet.univ_pi hs)] ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =     \u220f x : \u03b9, \u2191\u2191volume (s x)  \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 Measurable \u2191(\u2191toLin' (Matrix.diagonal D)) ** swap ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i B : \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i) \u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =     \u220f x : \u03b9, \u2191\u2191volume (s x) ** rw [this, volume_pi_pi, Finset.abs_prod,\n  ENNReal.ofReal_prod_of_nonneg fun i _ => abs_nonneg (D i), \u2190 Finset.prod_mul_distrib] ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i B : \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i) \u22a2 \u220f x : \u03b9, ofReal |D x| * \u2191\u2191volume ((fun x_1 => D x * x_1) \u207b\u00b9' s x) = \u220f x : \u03b9, \u2191\u2191volume (s x) ** simp only [B] ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 Measurable \u2191(\u2191toLin' (Matrix.diagonal D)) ** exact Continuous.measurable (LinearMap.continuous_on_pi _) ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i ** ext f ** case h \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) f : \u03b9 \u2192 \u211d \u22a2 (f \u2208 \u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) \u2194     f \u2208 Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i ** simp only [LinearMap.coe_proj, Algebra.id.smul_eq_mul, LinearMap.smul_apply, mem_univ_pi,\n  mem_preimage, LinearMap.pi_apply, diagonal_toLin'] ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i \u22a2 \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i) ** intro i ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i i : \u03b9 \u22a2 ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i) ** have A : D i \u2260 0 := by\n  simp only [det_diagonal, Ne.def] at h\n  exact Finset.prod_ne_zero_iff.1 h i (Finset.mem_univ i) ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i i : \u03b9 A : D i \u2260 0 \u22a2 ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i) ** rw [volume_preimage_mul_left A, \u2190 mul_assoc, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul,\n  mul_inv_cancel A, abs_one, ENNReal.ofReal_one, one_mul] ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d h : det (Matrix.diagonal D) \u2260 0 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i i : \u03b9 \u22a2 D i \u2260 0 ** simp only [det_diagonal, Ne.def] at h ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 D : \u03b9 \u2192 \u211d s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i i : \u03b9 h : \u00ac\u220f i : \u03b9, D i = 0 \u22a2 D i \u2260 0 ** exact Finset.prod_ne_zero_iff.1 h i (Finset.mem_univ i) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_finset_sum_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 s : Finset \u03b9 hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable f \u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i = \u2211 i in s, \u222b (a : \u03b1), f a \u2202\u03bc i ** induction s using Finset.cons_induction_on with\n| h\u2081 => simp\n| h\u2082 h ih =>\n  rw [Finset.forall_mem_cons] at hf\n  rw [Finset.sum_cons, Finset.sum_cons, \u2190 ih hf.2]\n  exact integral_add_measure hf.1 (integrable_finset_sum_measure.2 hf.2) ** case h\u2081 \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 : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Integrable f \u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in \u2205, \u03bc i = \u2211 i in \u2205, \u222b (a : \u03b1), f a \u2202\u03bc i ** simp ** case h\u2082 \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 a\u271d : \u03b9 s\u271d : Finset \u03b9 h : \u00aca\u271d \u2208 s\u271d ih : (\u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable f) \u2192 \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i = \u2211 i in s\u271d, \u222b (a : \u03b1), f a \u2202\u03bc i hf : \u2200 (i : \u03b9), i \u2208 Finset.cons a\u271d s\u271d h \u2192 Integrable f \u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in Finset.cons a\u271d s\u271d h, \u03bc i = \u2211 i in Finset.cons a\u271d s\u271d h, \u222b (a : \u03b1), f a \u2202\u03bc i ** rw [Finset.forall_mem_cons] at hf ** case h\u2082 \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 a\u271d : \u03b9 s\u271d : Finset \u03b9 h : \u00aca\u271d \u2208 s\u271d ih : (\u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable f) \u2192 \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i = \u2211 i in s\u271d, \u222b (a : \u03b1), f a \u2202\u03bc i hf : Integrable f \u2227 \u2200 (x : \u03b9), x \u2208 s\u271d \u2192 Integrable f \u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in Finset.cons a\u271d s\u271d h, \u03bc i = \u2211 i in Finset.cons a\u271d s\u271d h, \u222b (a : \u03b1), f a \u2202\u03bc i ** rw [Finset.sum_cons, Finset.sum_cons, \u2190 ih hf.2] ** case h\u2082 \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 a\u271d : \u03b9 s\u271d : Finset \u03b9 h : \u00aca\u271d \u2208 s\u271d ih : (\u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable f) \u2192 \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i = \u2211 i in s\u271d, \u222b (a : \u03b1), f a \u2202\u03bc i hf : Integrable f \u2227 \u2200 (x : \u03b9), x \u2208 s\u271d \u2192 Integrable f \u22a2 \u222b (a : \u03b1), f a \u2202(\u03bc a\u271d + \u2211 x in s\u271d, \u03bc x) = \u222b (a : \u03b1), f a \u2202\u03bc a\u271d + \u222b (a : \u03b1), f a \u2202\u2211 i in s\u271d, \u03bc i ** exact integral_add_measure hf.1 (integrable_finset_sum_measure.2 hf.2) ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.ofReal_cinfi ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 f : \u03b1 \u2192 \u211d inst\u271d : Nonempty \u03b1 \u22a2 ENNReal.ofReal (\u2a05 i, f i) = \u2a05 i, ENNReal.ofReal (f i) ** by_cases hf : BddBelow (range f) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 f : \u03b1 \u2192 \u211d inst\u271d : Nonempty \u03b1 hf : BddBelow (range f) \u22a2 ENNReal.ofReal (\u2a05 i, f i) = \u2a05 i, ENNReal.ofReal (f i) ** exact\n  Monotone.map_ciInf_of_continuousAt ENNReal.continuous_ofReal.continuousAt\n    (fun i j hij => ENNReal.ofReal_le_ofReal hij) hf ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 f : \u03b1 \u2192 \u211d inst\u271d : Nonempty \u03b1 hf : \u00acBddBelow (range f) \u22a2 ENNReal.ofReal (\u2a05 i, f i) = \u2a05 i, ENNReal.ofReal (f i) ** symm ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 f : \u03b1 \u2192 \u211d inst\u271d : Nonempty \u03b1 hf : \u00acBddBelow (range f) \u22a2 \u2a05 i, ENNReal.ofReal (f i) = ENNReal.ofReal (\u2a05 i, f i) ** rw [Real.iInf_of_not_bddBelow hf, ENNReal.ofReal_zero, \u2190 ENNReal.bot_eq_zero, iInf_eq_bot] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 f : \u03b1 \u2192 \u211d inst\u271d : Nonempty \u03b1 hf : \u00acBddBelow (range f) \u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b ** obtain \u27e8y, hy_mem, hy_neg\u27e9 := not_bddBelow_iff.mp hf 0 ** case neg.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 f : \u03b1 \u2192 \u211d inst\u271d : Nonempty \u03b1 hf : \u00acBddBelow (range f) y : \u211d hy_mem : y \u2208 range f hy_neg : y < 0 \u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b ** obtain \u27e8i, rfl\u27e9 := mem_range.mpr hy_mem ** case neg.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 f : \u03b1 \u2192 \u211d inst\u271d : Nonempty \u03b1 hf : \u00acBddBelow (range f) i : \u03b1 hy_mem : (fun y => f y) i \u2208 range f hy_neg : (fun y => f y) i < 0 \u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b ** refine' fun x hx => \u27e8i, _\u27e9 ** case neg.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 f : \u03b1 \u2192 \u211d inst\u271d : Nonempty \u03b1 hf : \u00acBddBelow (range f) i : \u03b1 hy_mem : (fun y => f y) i \u2208 range f hy_neg : (fun y => f y) i < 0 x : \u211d\u22650\u221e hx : x > \u22a5 \u22a2 ENNReal.ofReal (f i) < x ** rwa [ENNReal.ofReal_of_nonpos hy_neg.le] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.IicSnd_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) r : \u211d s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(IicSnd \u03c1 r) s = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic r) ** rw [IicSnd, fst_apply hs,\n  restrict_apply' (MeasurableSet.univ.prod (measurableSet_Iic : MeasurableSet (Iic r))), \u2190\n  prod_univ, prod_inter_prod, inter_univ, univ_inter] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_const_lt_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 : IsFiniteMeasure \u03bc c : \u211d\u22650\u221e hc : c \u2260 \u22a4 \u22a2 \u222b\u207b (x : \u03b1), c \u2202\u03bc < \u22a4 ** simpa only [Measure.restrict_univ] using set_lintegral_const_lt_top (univ : Set \u03b1) hc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.martingale_martingalePart ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 \u22a2 Martingale (martingalePart f \u2131 \u03bc) \u2131 \u03bc ** refine' \u27e8adapted_martingalePart hf, fun i j hij => _\u27e9 ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) \u22a2 \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i ** refine' h_eq_sum.trans _ ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) \u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i ** have h_ge : \u2200 k, i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] 0 := by\n  intro k hk\n  have : \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2131 i] :=\n    condexp_condexp_of_le (\u2131.mono hk) (\u2131.le k)\n  filter_upwards [this] with x hx\n  rw [Pi.sub_apply, Pi.zero_apply, hx, sub_self] ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 h_lt :   \u2200 (k : \u2115),     k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k] \u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i ** rw [martingalePart_eq_sum] ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 h_lt :   \u2200 (k : \u2115),     k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k] \u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\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])) i ** refine' EventuallyEq.add EventuallyEq.rfl _ ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 h_lt :   \u2200 (k : \u2115),     k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k] \u22a2 (fun x => Finset.sum (Finset.range j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc]     fun x => Finset.sum (Finset.range i) (fun i => f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) x ** rw [\u2190 Finset.sum_range_add_sum_Ico _ hij, \u2190\n  add_zero (\u2211 i in Finset.range i, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2131 i]))] ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 h_lt :   \u2200 (k : \u2115),     k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k] \u22a2 (fun x =>       (\u2211 k in Finset.range i, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) +           \u2211 k in Finset.Ico i j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]))         x) =\u1d50[\u03bc]     fun x => (\u2211 i in Finset.range i, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) + 0) x ** refine' (eventuallyEq_sum fun k hk => h_lt k (Finset.mem_range.mp hk)).add _ ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 h_lt :   \u2200 (k : \u2115),     k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k] \u22a2 (fun x => Finset.sum (Finset.Ico i j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc]     fun x => OfNat.ofNat 0 x ** refine' (eventuallyEq_sum fun k hk => h_ge k (Finset.mem_Ico.mp hk).1).trans _ ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 h_lt :   \u2200 (k : \u2115),     k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k] \u22a2 \u2211 i in Finset.Ico i j, 0 =\u1d50[\u03bc] fun x => OfNat.ofNat 0 x ** simp only [Finset.sum_const_zero, Pi.zero_apply] ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 h_lt :   \u2200 (k : \u2115),     k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k] \u22a2 0 =\u1d50[\u03bc] fun x => 0 ** rfl ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j \u22a2 \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) ** rw [martingalePart_eq_sum] ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j \u22a2 \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])) j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) ** refine' (condexp_add (hf_int 0) _).trans _ ** case refine'_2 \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j \u22a2 \u03bc[f 0|\u2191\u2131 i] + \u03bc[\u2211 i in Finset.range j, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) ** refine' (EventuallyEq.add EventuallyEq.rfl (condexp_finset_sum fun i _ => _)).trans _ ** case refine'_2.refine'_2 \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j \u22a2 (fun x =>       (\u03bc[f 0|\u2191\u2131 i]) x +         Finset.sum (Finset.range j) (fun i_1 => \u03bc[f (i_1 + 1) - f i_1 - \u03bc[f (i_1 + 1) - f i_1|\u2191\u2131 i_1]|\u2191\u2131 i]) x) =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) ** refine' EventuallyEq.add _ _ ** case refine'_1 \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j \u22a2 Integrable (\u2211 i in Finset.range j, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])) ** exact integrable_finset_sum' _ fun i _ => ((hf_int _).sub (hf_int _)).sub integrable_condexp ** case refine'_2.refine'_1 \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i\u271d j : \u2115 hij : i\u271d \u2264 j i : \u2115 x\u271d : i \u2208 Finset.range j \u22a2 Integrable (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) ** exact ((hf_int _).sub (hf_int _)).sub integrable_condexp ** case refine'_2.refine'_2.refine'_1 \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j \u22a2 (fun x => (\u03bc[f 0|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => f 0 x ** rw [condexp_of_stronglyMeasurable (\u2131.le _) _ (hf_int 0)] ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j \u22a2 StronglyMeasurable (f 0) ** exact (hf 0).mono (\u2131.mono (zero_le i)) ** case refine'_2.refine'_2.refine'_2 \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j \u22a2 (fun x =>       Finset.sum (Finset.range j) (fun i_1 => \u03bc[f (i_1 + 1) - f i_1 - \u03bc[f (i_1 + 1) - f i_1|\u2191\u2131 i_1]|\u2191\u2131 i]) x) =\u1d50[\u03bc]     fun x => Finset.sum (Finset.range j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x ** exact eventuallyEq_sum fun k _ => condexp_sub ((hf_int _).sub (hf_int _)) integrable_condexp ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) \u22a2 \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 ** intro k hk ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) k : \u2115 hk : i \u2264 k \u22a2 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 ** have : \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2131 i] :=\n  condexp_condexp_of_le (\u2131.mono hk) (\u2131.le k) ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) k : \u2115 hk : i \u2264 k this : \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2191\u2131 i] \u22a2 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 ** filter_upwards [this] with x hx ** 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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) k : \u2115 hk : i \u2264 k this : \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2191\u2131 i] x : \u03a9 hx : (\u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x = (\u03bc[f (k + 1) - f k|\u2191\u2131 i]) x \u22a2 (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x = OfNat.ofNat 0 x ** rw [Pi.sub_apply, Pi.zero_apply, hx, sub_self] ** \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 \u22a2 \u2200 (k : \u2115),     k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k] ** refine' fun k hk => EventuallyEq.sub _ _ ** case refine'_1 \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 k : \u2115 hk : k < i \u22a2 (fun x => (\u03bc[f (k + 1) - f k|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => (f (k + 1) - f k) x ** rw [condexp_of_stronglyMeasurable] ** case refine'_1.hf \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 k : \u2115 hk : k < i \u22a2 StronglyMeasurable (f (k + 1) - f k) ** exact ((hf (k + 1)).mono (\u2131.mono (Nat.succ_le_of_lt hk))).sub ((hf k).mono (\u2131.mono hk.le)) ** case refine'_1.hfi \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 k : \u2115 hk : k < i \u22a2 Integrable (f (k + 1) - f k) ** exact (hf_int _).sub (hf_int _) ** case refine'_2 \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 k : \u2115 hk : k < i \u22a2 (fun x => (\u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => (\u03bc[f (k + 1) - f k|\u2191\u2131 k]) x ** rw [condexp_of_stronglyMeasurable] ** case refine'_2.hf \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 k : \u2115 hk : k < i \u22a2 StronglyMeasurable (\u03bc[f (k + 1) - f k|\u2191\u2131 k]) ** exact stronglyMeasurable_condexp.mono (\u2131.mono hk.le) ** case refine'_2.hfi \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 : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 hf : Adapted \u2131 f hf_int : \u2200 (n : \u2115), Integrable (f n) inst\u271d : SigmaFiniteFiltration \u03bc \u2131 i j : \u2115 hij : i \u2264 j h_eq_sum :   \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]     f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) h_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0 k : \u2115 hk : k < i \u22a2 Integrable (\u03bc[f (k + 1) - f k|\u2191\u2131 k]) ** exact integrable_condexp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Icc_ssubset_Icc_right ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 hI : a\u2082 \u2264 b\u2082 ha : a\u2082 \u2264 a\u2081 hb : b\u2081 < b\u2082 \u22a2 Icc a\u2081 b\u2081 \u2282 Icc a\u2082 b\u2082 ** rw [\u2190 coe_ssubset, coe_Icc, coe_Icc] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 hI : a\u2082 \u2264 b\u2082 ha : a\u2082 \u2264 a\u2081 hb : b\u2081 < b\u2082 \u22a2 Set.Icc a\u2081 b\u2081 \u2282 Set.Icc a\u2082 b\u2082 ** exact Set.Icc_ssubset_Icc_right hI ha hb ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_upperSemicontinuous_le_integral_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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 f x) \u2227       UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc ** lift \u03b5 to \u211d\u22650 using \u03b5pos.le ** case 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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 f x) \u2227       UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc ** have If : (\u222b\u207b x, f x \u2202\u03bc) < \u221e := hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral ** case 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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 f x) \u2227       UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc ** rcases exists_upperSemicontinuous_le_lintegral_le f If.ne \u03b5pos.ne' with \u27e8g, gf, gcont, gint\u27e9 ** case 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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 f x) \u2227       UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc ** have Ig : (\u222b\u207b x, g x \u2202\u03bc) < \u221e := by\n  refine' lt_of_le_of_lt (lintegral_mono fun x => _) If\n  simpa using gf x ** case 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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 f x) \u2227       UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc ** refine' \u27e8g, gf, gcont, _, _\u27e9 ** \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 \u22a2 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 ** refine' lt_of_le_of_lt (lintegral_mono fun x => _) If ** \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 x : \u03b1 \u22a2 \u2191(g x) \u2264 \u2191(f x) ** simpa using gf x ** case intro.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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 Integrable fun x => \u2191(g x) ** refine'\n  Integrable.mono fint gcont.measurable.coe_nnreal_real.aemeasurable.aestronglyMeasurable _ ** case intro.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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016\u2191(g a)\u2016 \u2264 \u2016\u2191(f a)\u2016 ** exact Filter.eventually_of_forall fun x => by simp [gf x] ** \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 x : \u03b1 \u22a2 \u2016\u2191(g x)\u2016 \u2264 \u2016\u2191(f x)\u2016 ** simp [gf x] ** case intro.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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc ** rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae] ** case intro.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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) - \u2191\u03b5 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) ** rw [sub_le_iff_le_add] ** case intro.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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5 ** convert ENNReal.toReal_mono _ gint ** case h.e'_3.h.e'_1.h.e'_4.h \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 x\u271d : \u03b1 \u22a2 ENNReal.ofReal \u2191(f x\u271d) = \u2191(f x\u271d) ** simp ** 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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5 = ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5) ** rw [ENNReal.toReal_add Ig.ne ENNReal.coe_ne_top] ** 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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5 =     ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) + ENNReal.toReal \u2191\u03b5 ** simp ** case intro.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 \u211d\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 \u2260 \u22a4 ** simpa using Ig.ne ** case intro.intro.intro.intro.refine'_2.hf \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(g x) ** apply Filter.eventually_of_forall ** case intro.intro.intro.intro.refine'_2.hf.hp \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(g x)) x ** simp ** case intro.intro.intro.intro.refine'_2.hfm \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 AEStronglyMeasurable (fun x => \u2191(g x)) \u03bc ** exact gcont.measurable.coe_nnreal_real.aemeasurable.aestronglyMeasurable ** case intro.intro.intro.intro.refine'_2.hf \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x) ** apply Filter.eventually_of_forall ** case intro.intro.intro.intro.refine'_2.hf.hp \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x ** simp ** case intro.intro.intro.intro.refine'_2.hfm \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 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 If : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4 g : \u03b1 \u2192 \u211d\u22650 gf : \u2200 (x : \u03b1), g x \u2264 f x gcont : UpperSemicontinuous g gint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 Ig : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4 \u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc ** exact fint.aestronglyMeasurable ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.eval\u2082Hom_X ** R\u271d : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R\u271d e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommRing R\u271d p q : MvPolynomial \u03c3 R\u271d inst\u271d : CommRing S f\u271d : R\u271d \u2192+* S g : \u03c3 \u2192 S R : Type u c : \u2124 \u2192+* S f : MvPolynomial R \u2124 \u2192+* S x : MvPolynomial R \u2124 \u22a2 eval\u2082 c (\u2191f \u2218 X) x = \u2191f x ** apply MvPolynomial.induction_on x\n  (fun n => by\n    rw [hom_C f, eval\u2082_C]\n    exact eq_intCast c n)\n  (fun p q hp hq => by\n    rw [eval\u2082_add, hp, hq]\n    exact (f.map_add _ _).symm)\n  (fun p n hp => by\n    rw [eval\u2082_mul, eval\u2082_X, hp]\n    exact (f.map_mul _ _).symm) ** R\u271d : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R\u271d e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommRing R\u271d p q : MvPolynomial \u03c3 R\u271d inst\u271d : CommRing S f\u271d : R\u271d \u2192+* S g : \u03c3 \u2192 S R : Type u c : \u2124 \u2192+* S f : MvPolynomial R \u2124 \u2192+* S x : MvPolynomial R \u2124 n : \u2124 \u22a2 eval\u2082 c (\u2191f \u2218 X) (\u2191C n) = \u2191f (\u2191C n) ** rw [hom_C f, eval\u2082_C] ** R\u271d : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R\u271d e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommRing R\u271d p q : MvPolynomial \u03c3 R\u271d inst\u271d : CommRing S f\u271d : R\u271d \u2192+* S g : \u03c3 \u2192 S R : Type u c : \u2124 \u2192+* S f : MvPolynomial R \u2124 \u2192+* S x : MvPolynomial R \u2124 n : \u2124 \u22a2 \u2191c n = \u2191n ** exact eq_intCast c n ** R\u271d : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R\u271d e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommRing R\u271d p\u271d q\u271d : MvPolynomial \u03c3 R\u271d inst\u271d : CommRing S f\u271d : R\u271d \u2192+* S g : \u03c3 \u2192 S R : Type u c : \u2124 \u2192+* S f : MvPolynomial R \u2124 \u2192+* S x p q : MvPolynomial R \u2124 hp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p hq : eval\u2082 c (\u2191f \u2218 X) q = \u2191f q \u22a2 eval\u2082 c (\u2191f \u2218 X) (p + q) = \u2191f (p + q) ** rw [eval\u2082_add, hp, hq] ** R\u271d : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R\u271d e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommRing R\u271d p\u271d q\u271d : MvPolynomial \u03c3 R\u271d inst\u271d : CommRing S f\u271d : R\u271d \u2192+* S g : \u03c3 \u2192 S R : Type u c : \u2124 \u2192+* S f : MvPolynomial R \u2124 \u2192+* S x p q : MvPolynomial R \u2124 hp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p hq : eval\u2082 c (\u2191f \u2218 X) q = \u2191f q \u22a2 \u2191f p + \u2191f q = \u2191f (p + q) ** exact (f.map_add _ _).symm ** R\u271d : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R\u271d e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommRing R\u271d p\u271d q : MvPolynomial \u03c3 R\u271d inst\u271d : CommRing S f\u271d : R\u271d \u2192+* S g : \u03c3 \u2192 S R : Type u c : \u2124 \u2192+* S f : MvPolynomial R \u2124 \u2192+* S x p : MvPolynomial R \u2124 n : R hp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p \u22a2 eval\u2082 c (\u2191f \u2218 X) (p * X n) = \u2191f (p * X n) ** rw [eval\u2082_mul, eval\u2082_X, hp] ** R\u271d : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R\u271d e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommRing R\u271d p\u271d q : MvPolynomial \u03c3 R\u271d inst\u271d : CommRing S f\u271d : R\u271d \u2192+* S g : \u03c3 \u2192 S R : Type u c : \u2124 \u2192+* S f : MvPolynomial R \u2124 \u2192+* S x p : MvPolynomial R \u2124 n : R hp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p \u22a2 \u2191f p * (\u2191f \u2218 X) n = \u2191f (p * X n) ** exact (f.map_mul _ _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.VectorMeasure.AbsolutelyContinuous.map ** \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\u2076 : AddCommMonoid L inst\u271d\u2075 : TopologicalSpace L inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : AddCommMonoid N inst\u271d\u00b9 : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N inst\u271d : MeasureSpace \u03b2 h : v \u226a\u1d65 w f : \u03b1 \u2192 \u03b2 \u22a2 VectorMeasure.map v f \u226a\u1d65 VectorMeasure.map w f ** by_cases hf : Measurable f ** case pos \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\u2076 : AddCommMonoid L inst\u271d\u2075 : TopologicalSpace L inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : AddCommMonoid N inst\u271d\u00b9 : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N inst\u271d : MeasureSpace \u03b2 h : v \u226a\u1d65 w f : \u03b1 \u2192 \u03b2 hf : Measurable f \u22a2 VectorMeasure.map v f \u226a\u1d65 VectorMeasure.map w f ** refine' mk fun s hs hws => _ ** case pos \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\u2076 : AddCommMonoid L inst\u271d\u2075 : TopologicalSpace L inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : AddCommMonoid N inst\u271d\u00b9 : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N inst\u271d : MeasureSpace \u03b2 h : v \u226a\u1d65 w f : \u03b1 \u2192 \u03b2 hf : Measurable f s : Set \u03b2 hs : MeasurableSet s hws : \u2191(VectorMeasure.map w f) s = 0 \u22a2 \u2191(VectorMeasure.map v f) s = 0 ** rw [map_apply _ hf hs] at hws \u22a2 ** case pos \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\u2076 : AddCommMonoid L inst\u271d\u2075 : TopologicalSpace L inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : AddCommMonoid N inst\u271d\u00b9 : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N inst\u271d : MeasureSpace \u03b2 h : v \u226a\u1d65 w f : \u03b1 \u2192 \u03b2 hf : Measurable f s : Set \u03b2 hs : MeasurableSet s hws : \u2191w (f \u207b\u00b9' s) = 0 \u22a2 \u2191v (f \u207b\u00b9' s) = 0 ** exact h hws ** case neg \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\u2076 : AddCommMonoid L inst\u271d\u2075 : TopologicalSpace L inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : AddCommMonoid N inst\u271d\u00b9 : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N inst\u271d : MeasureSpace \u03b2 h : v \u226a\u1d65 w f : \u03b1 \u2192 \u03b2 hf : \u00acMeasurable f \u22a2 VectorMeasure.map v f \u226a\u1d65 VectorMeasure.map w f ** intro s _ ** case neg \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\u2076 : AddCommMonoid L inst\u271d\u2075 : TopologicalSpace L inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : TopologicalSpace M inst\u271d\u00b2 : AddCommMonoid N inst\u271d\u00b9 : TopologicalSpace N v : VectorMeasure \u03b1 M w : VectorMeasure \u03b1 N inst\u271d : MeasureSpace \u03b2 h : v \u226a\u1d65 w f : \u03b1 \u2192 \u03b2 hf : \u00acMeasurable f s : Set \u03b2 a\u271d : \u2191(VectorMeasure.map w f) s = 0 \u22a2 \u2191(VectorMeasure.map v f) s = 0 ** rw [map_not_measurable v hf, zero_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.mapAlgEquiv_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\u2076 : CommSemiring R A\u2081 : Type u_2 A\u2082 : Type u_3 A\u2083 : Type u_4 inst\u271d\u2075 : CommSemiring A\u2081 inst\u271d\u2074 : CommSemiring A\u2082 inst\u271d\u00b3 : CommSemiring A\u2083 inst\u271d\u00b2 : Algebra R A\u2081 inst\u271d\u00b9 : Algebra R A\u2082 inst\u271d : Algebra R A\u2083 e : A\u2081 \u2243\u2090[R] A\u2082 f : A\u2082 \u2243\u2090[R] A\u2083 \u22a2 AlgEquiv.trans (mapAlgEquiv \u03c3 e) (mapAlgEquiv \u03c3 f) = mapAlgEquiv \u03c3 (AlgEquiv.trans e f) ** ext ** case h.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\u271d : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u2076 : CommSemiring R A\u2081 : Type u_2 A\u2082 : Type u_3 A\u2083 : Type u_4 inst\u271d\u2075 : CommSemiring A\u2081 inst\u271d\u2074 : CommSemiring A\u2082 inst\u271d\u00b3 : CommSemiring A\u2083 inst\u271d\u00b2 : Algebra R A\u2081 inst\u271d\u00b9 : Algebra R A\u2082 inst\u271d : Algebra R A\u2083 e : A\u2081 \u2243\u2090[R] A\u2082 f : A\u2082 \u2243\u2090[R] A\u2083 a\u271d : MvPolynomial \u03c3 A\u2081 m\u271d : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff m\u271d (\u2191(AlgEquiv.trans (mapAlgEquiv \u03c3 e) (mapAlgEquiv \u03c3 f)) a\u271d) =     coeff m\u271d (\u2191(mapAlgEquiv \u03c3 (AlgEquiv.trans e f)) a\u271d) ** simp only [AlgEquiv.trans_apply, mapAlgEquiv_apply, map_map] ** case h.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\u271d : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u2076 : CommSemiring R A\u2081 : Type u_2 A\u2082 : Type u_3 A\u2083 : Type u_4 inst\u271d\u2075 : CommSemiring A\u2081 inst\u271d\u2074 : CommSemiring A\u2082 inst\u271d\u00b3 : CommSemiring A\u2083 inst\u271d\u00b2 : Algebra R A\u2081 inst\u271d\u00b9 : Algebra R A\u2082 inst\u271d : Algebra R A\u2083 e : A\u2081 \u2243\u2090[R] A\u2082 f : A\u2082 \u2243\u2090[R] A\u2083 a\u271d : MvPolynomial \u03c3 A\u2081 m\u271d : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff m\u271d (\u2191(map (RingHom.comp \u2191f \u2191e)) a\u271d) = coeff m\u271d (\u2191(map \u2191(AlgEquiv.trans e f)) a\u271d) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.ModEq.mul_left' ** m n a b c d : \u2124 h : a \u2261 b [ZMOD n] \u22a2 c * a \u2261 c * b [ZMOD c * n] ** obtain hc | rfl | hc := lt_trichotomy c 0 ** case inl m n a b c d : \u2124 h : a \u2261 b [ZMOD n] hc : c < 0 \u22a2 c * a \u2261 c * b [ZMOD c * n] ** rw [\u2190 neg_modEq_neg, \u2190 modEq_neg, \u2190 neg_mul, \u2190 neg_mul, \u2190 neg_mul] ** case inl m n a b c d : \u2124 h : a \u2261 b [ZMOD n] hc : c < 0 \u22a2 -c * a \u2261 -c * b [ZMOD -c * n] ** simp only [ModEq, mul_emod_mul_of_pos _ _ (neg_pos.2 hc), h.eq] ** case inr.inl m n a b d : \u2124 h : a \u2261 b [ZMOD n] \u22a2 0 * a \u2261 0 * b [ZMOD 0 * n] ** simp ** case inr.inr m n a b c d : \u2124 h : a \u2261 b [ZMOD n] hc : 0 < c \u22a2 c * a \u2261 c * b [ZMOD c * n] ** simp only [ModEq, mul_emod_mul_of_pos _ _ hc, h.eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_eq_succ ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard s = n + 1 \u2194 \u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n ** refine' \u27e8eq_insert_of_ncard_eq_succ, _\u27e9 ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 (\u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 ncard t = n) \u2192 ncard s = n + 1 ** rintro \u27e8a, t, hat, h, rfl\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 s t\u271d : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d a : \u03b1 t : Set \u03b1 hat : \u00aca \u2208 t h : insert a t = s \u22a2 ncard s = ncard t + 1 ** rw [\u2190 h, ncard_insert_of_not_mem hat (hs.subset ((subset_insert a t).trans_eq h))] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.fdiv_eq_ediv_of_dvd ** b c : Int \u22a2 fdiv (b * c) b = b * c / b ** if bz : b = 0 then simp [bz] else\nrw [mul_fdiv_cancel_left _ bz, mul_ediv_cancel_left _ bz] ** b c : Int bz : b = 0 \u22a2 fdiv (b * c) b = b * c / b ** simp [bz] ** b c : Int bz : \u00acb = 0 \u22a2 fdiv (b * c) b = b * c / b ** rw [mul_fdiv_cancel_left _ bz, mul_ediv_cancel_left _ bz] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.piecewise_univ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 f g : \u03b1 \u2192\u209b \u03b2 \u22a2 \u2191(piecewise univ (_ : MeasurableSet univ) f g) = \u2191f ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 f g : \u03b1 \u2192\u209b \u03b2 \u22a2 Set.piecewise univ \u2191f \u2191g = \u2191f ** convert Set.piecewise_univ f g ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_edist_lt_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 g : \u03b1 \u2192 \u03b2 hf : Integrable f hg : Integrable g \u22a2 \u222b\u207b (a : \u03b1), edist (f a) (OfNat.ofNat 0 a) \u2202\u03bc < \u22a4 \u2227 \u222b\u207b (a : \u03b1), edist (g a) (OfNat.ofNat 0 a) \u2202\u03bc < \u22a4 ** simp_rw [Pi.zero_apply, \u2190 hasFiniteIntegral_iff_edist] ** \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 \u22a2 (HasFiniteIntegral fun a => f a) \u2227 HasFiniteIntegral fun a => g a ** exact \u27e8hf.hasFiniteIntegral, hg.hasFiniteIntegral\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.mapAccumr\u2082_bisim ** \u03b1 : Type n : \u2115 xs : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type ys : Vector \u03b2 n f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 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) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 \u22a2 R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2 ** induction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082 ** case nil \u03b1 : Type n : \u2115 xs : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop hR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr\u2082 f\u2081 nil nil s\u2081).1 (mapAccumr\u2082 f\u2082 nil nil s\u2082).1 \u2227     (mapAccumr\u2082 f\u2081 nil nil s\u2081).2 = (mapAccumr\u2082 f\u2082 nil nil s\u2082).2  case snoc \u03b1 : Type n : \u2115 xs : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop hR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 n\u271d : \u2115 xs\u271d : Vector \u03b1 n\u271d ys\u271d : Vector \u03b2 n\u271d x\u271d : \u03b1 y\u271d : \u03b2 a\u271d :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192       R (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).1 (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).1 \u2227         (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).2 = (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).1 \u2227     (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).2 ** next => exact \u27e8h\u2080, rfl\u27e9 ** case snoc \u03b1 : Type n : \u2115 xs : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop hR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 n\u271d : \u2115 xs\u271d : Vector \u03b1 n\u271d ys\u271d : Vector \u03b2 n\u271d x\u271d : \u03b1 y\u271d : \u03b2 a\u271d :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192       R (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).1 (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).1 \u2227         (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).2 = (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).1 \u2227     (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).2 ** next xs ys x y ih =>\n  rcases (hR x y h\u2080) with \u27e8hR, _\u27e9\n  simp only [mapAccumr\u2082_snoc, ih hR, true_and]\n  congr 1 ** \u03b1 : Type n : \u2115 xs : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop hR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr\u2082 f\u2081 nil nil s\u2081).1 (mapAccumr\u2082 f\u2082 nil nil s\u2082).1 \u2227     (mapAccumr\u2082 f\u2081 nil nil s\u2081).2 = (mapAccumr\u2082 f\u2082 nil nil s\u2082).2 ** exact \u27e8h\u2080, rfl\u27e9 ** \u03b1 : Type n : \u2115 xs\u271d : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop hR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 n\u271d : \u2115 xs : Vector \u03b1 n\u271d ys : Vector \u03b2 n\u271d x : \u03b1 y : \u03b2 ih :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192       R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 \u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).1 \u2227     (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).2 ** rcases (hR x y h\u2080) with \u27e8hR, _\u27e9 ** case intro \u03b1 : Type n : \u2115 xs\u271d : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop hR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 n\u271d : \u2115 xs : Vector \u03b1 n\u271d ys : Vector \u03b2 n\u271d x : \u03b1 y : \u03b2 ih :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192       R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 hR : R (f\u2081 x y s\u2081).1 (f\u2082 x y s\u2082).1 right\u271d : (f\u2081 x y s\u2081).2 = (f\u2082 x y s\u2082).2 \u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).1 (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).1 \u2227     (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).2 = (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).2 ** simp only [mapAccumr\u2082_snoc, ih hR, true_and] ** case intro \u03b1 : Type n : \u2115 xs\u271d : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop hR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 n\u271d : \u2115 xs : Vector \u03b1 n\u271d ys : Vector \u03b2 n\u271d x : \u03b1 y : \u03b2 ih :   \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},     R s\u2081 s\u2082 \u2192       R (mapAccumr\u2082 f\u2081 xs ys s\u2081).1 (mapAccumr\u2082 f\u2082 xs ys s\u2082).1 \u2227 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h\u2080 : R s\u2081 s\u2082 hR : R (f\u2081 x y s\u2081).1 (f\u2082 x y s\u2082).1 right\u271d : (f\u2081 x y s\u2081).2 = (f\u2082 x y s\u2082).2 \u22a2 snoc (mapAccumr\u2082 f\u2082 xs ys (f\u2082 x y s\u2082).1).2 (f\u2081 x y s\u2081).2 = snoc (mapAccumr\u2082 f\u2082 xs ys (f\u2082 x y s\u2082).1).2 (f\u2082 x y s\u2082).2 ** congr 1 ** Qed",
        "informal": ""
    },
    {
        "formal": "NFA.to\u03b5NFA_correct ** \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S : Set \u03c3 x : List \u03b1 s : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 \u22a2 \u03b5NFA.accepts (to\u03b5NFA M) = accepts M ** rw [\u03b5NFA.accepts, \u03b5NFA.eval, to\u03b5NFA_evalFrom_match] ** \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S : Set \u03c3 x : List \u03b1 s : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 \u22a2 {x | \u2203 S, S \u2208 (to\u03b5NFA M).accept \u2227 S \u2208 evalFrom M (to\u03b5NFA M).start x} = accepts M ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.vars_sub_of_disjoint ** R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommRing R p q : MvPolynomial \u03c3 R inst\u271d : DecidableEq \u03c3 hpq : Disjoint (vars p) (vars q) \u22a2 vars (p - q) = vars p \u222a vars q ** rw [\u2190 vars_neg q] at hpq ** R : Type u S : Type v \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommRing R p q : MvPolynomial \u03c3 R inst\u271d : DecidableEq \u03c3 hpq : Disjoint (vars p) (vars (-q)) \u22a2 vars (p - q) = vars p \u222a vars q ** convert vars_add_of_disjoint hpq using 2 <;> simp [sub_eq_add_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_pi_le_diam_pow ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 s : Set (\u03b9 \u2192 \u211d) \u22a2 \u220f _i : \u03b9, \u21911 * EMetric.diam s = EMetric.diam s ^ Fintype.card \u03b9 ** simp only [ENNReal.coe_one, one_mul, Finset.prod_const, Fintype.card] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toMeasure_mono ** \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 \u2286 t \u22a2 \u2191\u2191(toMeasure p) s \u2264 \u2191\u2191(toMeasure p) t ** simpa only [p.toMeasure_apply_eq_toOuterMeasure_apply, hs, ht] using toOuterMeasure_mono p h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.setToL1_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\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 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 f : { x // x \u2208 Lp E 1 } \u22a2 \u2191(setToL1 hT) f = 0 ** suffices setToL1 hT = 0 by rw [this]; simp ** \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 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 f : { x // x \u2208 Lp E 1 } \u22a2 setToL1 hT = 0 ** 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 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 f : { x // x \u2208 Lp E 1 } \u22a2 ContinuousLinearMap.comp 0 (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 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(ContinuousLinearMap.comp 0 (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** rw [setToL1SCLM_zero_left' hT h_zero f, ContinuousLinearMap.zero_comp,\n  ContinuousLinearMap.zero_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 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 f : { x // x \u2208 Lp E 1 } this : setToL1 hT = 0 \u22a2 \u2191(setToL1 hT) f = 0 ** 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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 f : { x // x \u2208 Lp E 1 } this : setToL1 hT = 0 \u22a2 \u21910 f = 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_eq_two ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 encard s = 2 \u2194 \u2203 x y, x \u2260 y \u2227 s = {x, y} ** refine' \u27e8fun h \u21a6 _, fun \u27e8x, y, hne, hs\u27e9 \u21a6 by rw [hs, encard_pair hne]\u27e9 ** \u03b1 : Type u_1 s t : Set \u03b1 h : encard s = 2 \u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y} ** obtain \u27e8x, hx\u27e9 := nonempty_of_encard_ne_zero (s := s) (by rw [h]; simp) ** case intro \u03b1 : Type u_1 s t : Set \u03b1 h : encard s = 2 x : \u03b1 hx : x \u2208 s \u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y} ** rw [\u2190insert_eq_of_mem hx, \u2190insert_diff_singleton, encard_insert_of_not_mem (fun h \u21a6 h.2 rfl),\n  \u2190one_add_one_eq_two, WithTop.add_right_cancel_iff (WithTop.one_ne_top), encard_eq_one] at h ** case intro \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 h : \u2203 x_1, s \\ {x} = {x_1} hx : x \u2208 s \u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y} ** obtain \u27e8y, h\u27e9 := h ** case intro.intro \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 hx : x \u2208 s y : \u03b1 h : s \\ {x} = {y} \u22a2 \u2203 x y, x \u2260 y \u2227 s = {x, y} ** refine' \u27e8x, y, by rintro rfl; exact (h.symm.subset rfl).2 rfl, _\u27e9 ** case intro.intro \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 hx : x \u2208 s y : \u03b1 h : s \\ {x} = {y} \u22a2 s = {x, y} ** rw [\u2190h, insert_diff_singleton, insert_eq_of_mem hx] ** \u03b1 : Type u_1 s t : Set \u03b1 x\u271d : \u2203 x y, x \u2260 y \u2227 s = {x, y} x y : \u03b1 hne : x \u2260 y hs : s = {x, y} \u22a2 encard s = 2 ** rw [hs, encard_pair hne] ** \u03b1 : Type u_1 s t : Set \u03b1 h : encard s = 2 \u22a2 encard s \u2260 0 ** rw [h] ** \u03b1 : Type u_1 s t : Set \u03b1 h : encard s = 2 \u22a2 2 \u2260 0 ** simp ** \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 hx : x \u2208 s y : \u03b1 h : s \\ {x} = {y} \u22a2 x \u2260 y ** rintro rfl ** \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 hx : x \u2208 s h : s \\ {x} = {x} \u22a2 False ** exact (h.symm.subset rfl).2 rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L2.inner_indicatorConstLp_one ** \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 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \ud835\udd5c 2 } \u22a2 inner (indicatorConstLp 2 hs h\u03bcs 1) f = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** rw [L2.inner_indicatorConstLp_eq_inner_set_integral \ud835\udd5c hs h\u03bcs (1 : \ud835\udd5c) f] ** \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 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f : { x // x \u2208 Lp \ud835\udd5c 2 } \u22a2 inner 1 (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.univ_pi_Ioc_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 => Ioc (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 => Ioc (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Icc (f i) (g i) ** exact pi_Ioc_ae_eq_pi_Icc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.VectorMeasure.restrict_le_restrict_union ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : OrderedAddCommMonoid M inst\u271d : OrderClosedTopology M v w : VectorMeasure \u03b1 M i j : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : restrict v i \u2264 restrict w i hj\u2081 : MeasurableSet j hj\u2082 : restrict v j \u2264 restrict w j \u22a2 restrict v (i \u222a j) \u2264 restrict w (i \u222a j) ** rw [Set.union_eq_iUnion] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : OrderedAddCommMonoid M inst\u271d : OrderClosedTopology M v w : VectorMeasure \u03b1 M i j : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : restrict v i \u2264 restrict w i hj\u2081 : MeasurableSet j hj\u2082 : restrict v j \u2264 restrict w j \u22a2 restrict v (\u22c3 b, bif b then i else j) \u2264 restrict w (\u22c3 b, bif b then i else j) ** refine' restrict_le_restrict_countable_iUnion v w _ _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : OrderedAddCommMonoid M inst\u271d : OrderClosedTopology M v w : VectorMeasure \u03b1 M i j : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : restrict v i \u2264 restrict w i hj\u2081 : MeasurableSet j hj\u2082 : restrict v j \u2264 restrict w j \u22a2 \u2200 (b : Bool), MeasurableSet (bif b then i else j) ** measurability ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : OrderedAddCommMonoid M inst\u271d : OrderClosedTopology M v w : VectorMeasure \u03b1 M i j : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : restrict v i \u2264 restrict w i hj\u2081 : MeasurableSet j hj\u2082 : restrict v j \u2264 restrict w j \u22a2 \u2200 (b : Bool), restrict v (bif b then i else j) \u2264 restrict w (bif b then i else j) ** rintro (_ | _) <;> simpa ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Infinite.meas_eq_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\u271d s' t : Set \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 s : Set \u03b1 hs : Set.Infinite s h' : \u2203 \u03b5, \u03b5 \u2260 0 \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u03b5 \u2264 \u2191\u2191\u03bc {x} \u03b5 : \u211d\u22650\u221e hne\u271d : \u03b5 \u2260 0 h\u03b5 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u03b5 \u2264 \u2191\u2191\u03bc {x} this : Infinite \u2191s x y : \u2191s hne : x \u2260 y \u22a2 (Disjoint on fun x => {\u2191x}) x y ** simpa [Subtype.val_inj] ** \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\u271d s' t : Set \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 s : Set \u03b1 hs : Set.Infinite s h' : \u2203 \u03b5, \u03b5 \u2260 0 \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u03b5 \u2264 \u2191\u2191\u03bc {x} \u03b5 : \u211d\u22650\u221e hne : \u03b5 \u2260 0 h\u03b5 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u03b5 \u2264 \u2191\u2191\u03bc {x} this : Infinite \u2191s \u22a2 \u2191\u2191\u03bc (\u22c3 x, {\u2191x}) = \u2191\u2191\u03bc s ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_Icc_eq_integral_Ioo ** \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 t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : PartialOrder \u03b1 a b : \u03b1 inst\u271d : NoAtoms \u03bc \u22a2 \u222b (t : \u03b1) in Icc a b, f t \u2202\u03bc = \u222b (t : \u03b1) in Ico a b, f t \u2202\u03bc ** rw [integral_Icc_eq_integral_Ico, integral_Ico_eq_integral_Ioo] ** Qed",
        "informal": ""
    },
    {
        "formal": "Equiv.piCongrLeft_preimage_univ_pi ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 f : \u03b9' \u2243 \u03b9 t : (i : \u03b9) \u2192 Set (\u03b1 i) \u22a2 \u2191(piCongrLeft \u03b1 f) \u207b\u00b9' pi univ t = pi univ fun i => t (\u2191f i) ** simpa [f.surjective.range_eq] using piCongrLeft_preimage_pi f univ t ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_null_frontiers_thickening ** \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 \u22a2 \u2203 rs, Tendsto rs atTop (\ud835\udcdd 0) \u2227 \u2200 (n : \u2115), 0 < rs n \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening (rs n) s)) = 0 ** rcases exists_seq_strictAnti_tendsto (0 : \u211d) with \u27e8Rs, \u27e8_, \u27e8Rs_pos, Rs_lim\u27e9\u27e9\u27e9 ** case intro.intro.intro \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 Rs : \u2115 \u2192 \u211d left\u271d : StrictAnti Rs Rs_pos : \u2200 (n : \u2115), 0 < Rs n Rs_lim : Tendsto Rs atTop (\ud835\udcdd 0) \u22a2 \u2203 rs, Tendsto rs atTop (\ud835\udcdd 0) \u2227 \u2200 (n : \u2115), 0 < rs n \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening (rs n) s)) = 0 ** have obs := fun n : \u2115 => exists_null_frontier_thickening \u03bc s (Rs_pos n) ** case intro.intro.intro \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 Rs : \u2115 \u2192 \u211d left\u271d : StrictAnti Rs Rs_pos : \u2200 (n : \u2115), 0 < Rs n Rs_lim : Tendsto Rs atTop (\ud835\udcdd 0) obs : \u2200 (n : \u2115), \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 \u22a2 \u2203 rs, Tendsto rs atTop (\ud835\udcdd 0) \u2227 \u2200 (n : \u2115), 0 < rs n \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening (rs n) s)) = 0 ** refine' \u27e8fun n : \u2115 => (obs n).choose, \u27e8_, _\u27e9\u27e9 ** case intro.intro.intro.refine'_1 \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 Rs : \u2115 \u2192 \u211d left\u271d : StrictAnti Rs Rs_pos : \u2200 (n : \u2115), 0 < Rs n Rs_lim : Tendsto Rs atTop (\ud835\udcdd 0) obs : \u2200 (n : \u2115), \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 \u22a2 Tendsto (fun n => Exists.choose (_ : \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0)) atTop (\ud835\udcdd 0) ** exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds Rs_lim\n  (fun n => (obs n).choose_spec.1.1.le) fun n => (obs n).choose_spec.1.2.le ** case intro.intro.intro.refine'_2 \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 Rs : \u2115 \u2192 \u211d left\u271d : StrictAnti Rs Rs_pos : \u2200 (n : \u2115), 0 < Rs n Rs_lim : Tendsto Rs atTop (\ud835\udcdd 0) obs : \u2200 (n : \u2115), \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 \u22a2 \u2200 (n : \u2115),     0 < (fun n => Exists.choose (_ : \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0)) n \u2227       \u2191\u2191\u03bc           (frontier             (Metric.thickening               ((fun n => Exists.choose (_ : \u2203 r, r \u2208 Ioo 0 (Rs n) \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0)) n)               s)) =         0 ** exact fun n => \u27e8(obs n).choose_spec.1.1, (obs n).choose_spec.2\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image\u2082_curry ** \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t\u271d t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3 s : Finset \u03b1 t : Finset \u03b2 \u22a2 image\u2082 (curry f) s t = image f (s \u00d7\u02e2 t) ** classical rw [\u2190 image\u2082_mk_eq_product, image_image\u2082]; dsimp [curry] ** \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t\u271d t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3 s : Finset \u03b1 t : Finset \u03b2 \u22a2 image\u2082 (curry f) s t = image f (s \u00d7\u02e2 t) ** rw [\u2190 image\u2082_mk_eq_product, image_image\u2082] ** \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t\u271d t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3 s : Finset \u03b1 t : Finset \u03b2 \u22a2 image\u2082 (curry f) s t = image\u2082 (fun a b => f (a, b)) s t ** dsimp [curry] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.prod_prod_Ioi_mul_eq_prod_prod_off_diag ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u2074 : Fintype \u03b9 inst\u271d\u00b3 : LinearOrder \u03b9 inst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9 inst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9 inst\u271d : CommMonoid \u03b1 f : \u03b9 \u2192 \u03b9 \u2192 \u03b1 \u22a2 \u220f i : \u03b9, \u220f j in Ioi i, f j i * f i j = \u220f i : \u03b9, \u220f j in {i}\u1d9c, f j i ** simp_rw [\u2190 Ioi_disjUnion_Iio, prod_disjUnion, prod_mul_distrib] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u2074 : Fintype \u03b9 inst\u271d\u00b3 : LinearOrder \u03b9 inst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9 inst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9 inst\u271d : CommMonoid \u03b1 f : \u03b9 \u2192 \u03b9 \u2192 \u03b1 \u22a2 (\u220f x : \u03b9, \u220f x_1 in Ioi x, f x_1 x) * \u220f x : \u03b9, \u220f x_1 in Ioi x, f x x_1 =     (\u220f x : \u03b9, \u220f x_1 in Ioi x, f x_1 x) * \u220f x : \u03b9, \u220f x_1 in Iio x, f x_1 x ** congr 1 ** case e_a \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u2074 : Fintype \u03b9 inst\u271d\u00b3 : LinearOrder \u03b9 inst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9 inst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9 inst\u271d : CommMonoid \u03b1 f : \u03b9 \u2192 \u03b9 \u2192 \u03b1 \u22a2 \u220f x : \u03b9, \u220f x_1 in Ioi x, f x x_1 = \u220f x : \u03b9, \u220f x_1 in Iio x, f x_1 x ** rw [prod_sigma', prod_sigma'] ** case e_a \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u2074 : Fintype \u03b9 inst\u271d\u00b3 : LinearOrder \u03b9 inst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9 inst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9 inst\u271d : CommMonoid \u03b1 f : \u03b9 \u2192 \u03b9 \u2192 \u03b1 \u22a2 \u220f x in Finset.sigma univ fun x => Ioi x, f x.fst x.snd = \u220f x in Finset.sigma univ fun x => Iio x, f x.snd x.fst ** refine' prod_bij' (fun i _ => \u27e8i.2, i.1\u27e9) _ _ (fun i _ => \u27e8i.2, i.1\u27e9) _ _ _ <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "List.get_set ** \u03b1 : Type u_1 a : \u03b1 m n : Nat l : List \u03b1 h : n < length (set l m a) \u22a2 get (set l m a) { val := n, isLt := h } = if m = n then a else get l { val := n, isLt := (_ : n < length l) } ** if h : m = n then subst m; simp else simp [h] ** \u03b1 : Type u_1 a : \u03b1 m n : Nat l : List \u03b1 h\u271d : n < length (set l m a) h : m = n \u22a2 get (set l m a) { val := n, isLt := h\u271d } = if m = n then a else get l { val := n, isLt := (_ : n < length l) } ** subst m ** \u03b1 : Type u_1 a : \u03b1 n : Nat l : List \u03b1 h : n < length (set l n a) \u22a2 get (set l n a) { val := n, isLt := h } = if n = n then a else get l { val := n, isLt := (_ : n < length l) } ** simp ** \u03b1 : Type u_1 a : \u03b1 m n : Nat l : List \u03b1 h\u271d : n < length (set l m a) h : \u00acm = n \u22a2 get (set l m a) { val := n, isLt := h\u271d } = if m = n then a else get l { val := n, isLt := (_ : n < length l) } ** simp [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.cauchy_complete_\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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), Mem\u2112p (f n) 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 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** obtain \u27e8f_lim, h_f_lim_meas, h_lim\u27e9 :\n    \u2203 (f_lim : \u03b1 \u2192 E) (_ : StronglyMeasurable f_lim),\n      \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => f n x) atTop (nhds (f_lim x)) :=\n  exists_stronglyMeasurable_limit_of_tendsto_ae (fun n => (hf n).1)\n    (ae_tendsto_of_cauchy_snorm (fun n => (hf n).1) hp hB h_cau) ** case intro.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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), Mem\u2112p (f n) 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 f_lim : \u03b1 \u2192 E h_f_lim_meas : StronglyMeasurable f_lim h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) \u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** have h_tendsto' : atTop.Tendsto (fun n => snorm (f n - f_lim) p \u03bc) (\ud835\udcdd 0) :=\n  cauchy_tendsto_of_tendsto (fun m => (hf m).1) f_lim hB h_cau h_lim ** case intro.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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), Mem\u2112p (f n) 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 f_lim : \u03b1 \u2192 E h_f_lim_meas : StronglyMeasurable f_lim h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) h_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) \u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** have h_\u2112p_lim : Mem\u2112p f_lim p \u03bc :=\n  mem\u2112p_of_cauchy_tendsto hp hf f_lim h_f_lim_meas.aestronglyMeasurable h_tendsto' ** case intro.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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), Mem\u2112p (f n) 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 f_lim : \u03b1 \u2192 E h_f_lim_meas : StronglyMeasurable f_lim h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) h_tendsto' : Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) h_\u2112p_lim : Mem\u2112p f_lim p \u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** exact \u27e8f_lim, h_\u2112p_lim, h_tendsto'\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.sym_eq_empty ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n : \u2115 m : Sym \u03b1 n \u22a2 Finset.sym s n = \u2205 \u2194 n \u2260 0 \u2227 s = \u2205 ** cases n ** case zero \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 m : Sym \u03b1 Nat.zero \u22a2 Finset.sym s Nat.zero = \u2205 \u2194 Nat.zero \u2260 0 \u2227 s = \u2205 ** exact iff_of_false (singleton_ne_empty _) fun h \u21a6 (h.1 rfl).elim ** case succ \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n\u271d : \u2115 m : Sym \u03b1 (Nat.succ n\u271d) \u22a2 Finset.sym s (Nat.succ n\u271d) = \u2205 \u2194 Nat.succ n\u271d \u2260 0 \u2227 s = \u2205 ** refine \u27e8fun h \u21a6 \u27e8Nat.succ_ne_zero _, eq_empty_of_sym_eq_empty h\u27e9, ?_\u27e9 ** case succ \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n\u271d : \u2115 m : Sym \u03b1 (Nat.succ n\u271d) \u22a2 Nat.succ n\u271d \u2260 0 \u2227 s = \u2205 \u2192 Finset.sym s (Nat.succ n\u271d) = \u2205 ** rintro \u27e8_, rfl\u27e9 ** case succ.intro \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 t : Finset \u03b1 a b : \u03b1 n\u271d : \u2115 m : Sym \u03b1 (Nat.succ n\u271d) left\u271d : Nat.succ n\u271d \u2260 0 \u22a2 Finset.sym \u2205 (Nat.succ n\u271d) = \u2205 ** exact sym_empty _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.add_ofNat' ** \u03b1 : Type u_1 m n : \u2115 \u22a2 ofNat' (m + n) = ofNat' m + ofNat' n ** have : \u2200 {n}, ofNat' n.succ = ofNat' n + 1 := ofNat'_succ ** \u03b1 : Type u_1 m n : \u2115 this : \u2200 {n : \u2115}, ofNat' (Nat.succ n) = ofNat' n + 1 \u22a2 ofNat' (m + n) = ofNat' m + ofNat' n ** induction n <;> simp [Nat.add_zero, this, add_zero, Nat.add_succ, add_one, add_succ, *] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.mem_image\u2082_iff ** \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : 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 hf : Injective2 f \u22a2 f a b \u2208 image\u2082 f s t \u2194 a \u2208 s \u2227 b \u2208 t ** rw [\u2190 mem_coe, coe_image\u2082, mem_image2_iff hf, mem_coe, mem_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.min_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.min (erase s x) \u2260 \u2191x ** convert @max_erase_ne_self \u03b1\u1d52\u1d48 _ (toDual x) (s.map toDual.toEmbedding) using 1 ** case h.e'_2.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\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 e_1\u271d : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48 \u22a2 Finset.min (erase s x) = Finset.max (erase (map (Equiv.toEmbedding toDual) s) (\u2191toDual x)) ** apply congr_arg ** case h.e'_2.h.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\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 e_1\u271d : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48 \u22a2 erase s x = erase (map (Equiv.toEmbedding toDual) s) (\u2191toDual x) ** congr! ** case h.e'_2.h.h.h.e'_3.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\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 e_1\u271d\u00b9 : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48 e_1\u271d : \u03b1 = \u03b1\u1d52\u1d48 \u22a2 s = map (Equiv.toEmbedding toDual) s ** ext ** case h.e'_2.h.h.h.e'_3.h.a 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 e_1\u271d\u00b9 : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48 e_1\u271d : \u03b1 = \u03b1\u1d52\u1d48 a\u271d : \u03b1 \u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 map (Equiv.toEmbedding toDual) s ** simp only [mem_map_equiv] ** case h.e'_2.h.h.h.e'_3.h.a 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 e_1\u271d\u00b9 : WithTop \u03b1 = WithBot \u03b1\u1d52\u1d48 e_1\u271d : \u03b1 = \u03b1\u1d52\u1d48 a\u271d : \u03b1 \u22a2 a\u271d \u2208 s \u2194 \u2191toDual.symm a\u271d \u2208 s ** exact Iff.rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.vector_ofFn ** \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 : Fin n \u2192 \u03b1 \u2192 \u03c3 hf : \u2200 (i : Fin n), Primrec (f i) \u22a2 Primrec fun a => Vector.toList (Vector.ofFn fun i => f i a) ** simp [list_ofFn hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "volume_regionBetween_eq_lintegral ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc ** have h\u2081 :\n  (fun y => ENNReal.ofReal ((g - f) y)) =\u1d50[\u03bc.restrict s] fun y =>\n    ENNReal.ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) :=\n  (hg.ae_eq_mk.sub hf.ae_eq_mk).fun_comp ENNReal.ofReal ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc ** have h\u2082 :\n  (\u03bc.restrict s).prod volume (regionBetween f g s) =\n    (\u03bc.restrict s).prod volume\n      (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) := by\n  apply measure_congr\n  apply EventuallyEq.rfl.inter\n  exact\n    ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n      (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk) ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) h\u2082 :   \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =     \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc ** rw [lintegral_congr_ae h\u2081, \u2190\n  volume_regionBetween_eq_lintegral' hf.measurable_mk hg.measurable_mk hs] ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) h\u2082 :   \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =     \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =     \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) ** convert h\u2082 using 1 ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) \u22a2 \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =     \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) ** apply measure_congr ** case H \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) \u22a2 regionBetween f g s =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)]     regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s ** apply EventuallyEq.rfl.inter ** case H \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) \u22a2 (fun p => Ioo (f p.1) (g p.1) p.2) =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)] fun p =>     Ioo (AEMeasurable.mk f hf p.1) (AEMeasurable.mk g hg p.1) p.2 ** exact\n  ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n    (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk) ** case h.e'_2 \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) h\u2082 :   \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =     \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) ** rw [Measure.restrict_prod_eq_prod_univ] ** case h.e'_2 \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) h\u2082 :   \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =     \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =     \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ)) (regionBetween f g s) ** exact (Measure.restrict_eq_self _ (regionBetween_subset f g s)).symm ** case h.e'_3 \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) h\u2082 :   \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =     \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =     \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) ** rw [Measure.restrict_prod_eq_prod_univ] ** case h.e'_3 \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc hf : AEMeasurable f hg : AEMeasurable g hs : MeasurableSet s h\u2081 :   (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>     ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) h\u2082 :   \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =     \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =     \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ))       (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) ** exact\n  (Measure.restrict_eq_self _\n      (regionBetween_subset (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.add_mul_ediv_right ** a b c : Int H : c \u2260 0 this : \u2200 \u2983a b c : Int\u2984, 0 < c \u2192 ediv (a + b * c) c = ediv a c + b hlt : c < 0 \u22a2 (a + b * c) / c = a / c + b ** rw [\u2190 Int.neg_inj, \u2190 Int.ediv_neg, Int.neg_add, \u2190 Int.ediv_neg, \u2190 Int.neg_mul_neg] ** a b c : Int H : c \u2260 0 this : \u2200 \u2983a b c : Int\u2984, 0 < c \u2192 ediv (a + b * c) c = ediv a c + b hlt : c < 0 \u22a2 (a + -b * -c) / -c = a / -c + -b ** exact this (Int.neg_pos_of_neg hlt) ** a\u271d b\u271d c\u271d : Int H\u271d : c\u271d \u2260 0 this : \u2200 {k n : Nat} {a : Int}, ediv (a + \u2191n * \u2191(succ k)) \u2191(succ k) = ediv a \u2191(succ k) + \u2191n a b c : Int k n : Nat H : 0 < \u2191(succ k) \u22a2 ediv (a - \u2191(succ n) * \u2191(succ k)) \u2191(succ k) = ediv a \u2191(succ k) - \u2191(succ n) ** rw [\u2190 Int.add_sub_cancel (ediv ..), \u2190 this, Int.sub_add_cancel] ** a b c : Int H : c \u2260 0 k n m : Nat \u22a2 ediv (-[m+1] + \u2191n * \u2191(succ k)) \u2191(succ k) = ediv -[m+1] \u2191(succ k) + \u2191n ** show ((n * k.succ : Nat) - m.succ : Int).ediv k.succ = n - (m / k.succ + 1 : Nat) ** a b c : Int H : c \u2260 0 k n m : Nat \u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1) ** if h : m < n * k.succ then\n  rw [\u2190 Int.ofNat_sub h, \u2190 Int.ofNat_sub ((Nat.div_lt_iff_lt_mul k.succ_pos).2 h)]\n  apply congrArg ofNat\n  rw [Nat.mul_comm, Nat.mul_sub_div]; rwa [Nat.mul_comm]\nelse\n  have h := Nat.not_lt.1 h\n  have H {a b : Nat} (h : a \u2264 b) : (a : Int) + -((b : Int) + 1) = -[b - a +1] := by\n    rw [negSucc_eq, Int.ofNat_sub h]\n    simp only [Int.sub_eq_add_neg, Int.neg_add, Int.neg_neg, Int.add_left_comm, Int.add_assoc]\n  show ediv (\u2191(n * succ k) + -((m : Int) + 1)) (succ k) = n + -(\u2191(m / succ k) + 1 : Int)\n  rw [H h, H ((Nat.le_div_iff_mul_le k.succ_pos).2 h)]\n  apply congrArg negSucc\n  rw [Nat.mul_comm, Nat.sub_mul_div]; rwa [Nat.mul_comm] ** a b c : Int H : c \u2260 0 k n m : Nat h : m < n * succ k \u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1) ** rw [\u2190 Int.ofNat_sub h, \u2190 Int.ofNat_sub ((Nat.div_lt_iff_lt_mul k.succ_pos).2 h)] ** a b c : Int H : c \u2260 0 k n m : Nat h : m < n * succ k \u22a2 ediv \u2191(n * succ k - succ m) \u2191(succ k) = \u2191(n - succ (m / succ k)) ** apply congrArg ofNat ** a b c : Int H : c \u2260 0 k n m : Nat h : m < n * succ k \u22a2 (n * succ k - succ m) / succ k = n - succ (m / succ k) ** rw [Nat.mul_comm, Nat.mul_sub_div] ** case h\u2081 a b c : Int H : c \u2260 0 k n m : Nat h : m < n * succ k \u22a2 m < succ k * n ** rwa [Nat.mul_comm] ** a b c : Int H : c \u2260 0 k n m : Nat h : \u00acm < n * succ k \u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1) ** have h := Nat.not_lt.1 h ** a b c : Int H : c \u2260 0 k n m : Nat h\u271d : \u00acm < n * succ k h : n * succ k \u2264 m \u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1) ** have H {a b : Nat} (h : a \u2264 b) : (a : Int) + -((b : Int) + 1) = -[b - a +1] := by\n  rw [negSucc_eq, Int.ofNat_sub h]\n  simp only [Int.sub_eq_add_neg, Int.neg_add, Int.neg_neg, Int.add_left_comm, Int.add_assoc] ** a b c : Int H\u271d : c \u2260 0 k n m : Nat h\u271d : \u00acm < n * succ k h : n * succ k \u2264 m H : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1] \u22a2 ediv (\u2191(n * succ k) - \u2191(succ m)) \u2191(succ k) = \u2191n - \u2191(m / succ k + 1) ** show ediv (\u2191(n * succ k) + -((m : Int) + 1)) (succ k) = n + -(\u2191(m / succ k) + 1 : Int) ** a b c : Int H\u271d : c \u2260 0 k n m : Nat h\u271d : \u00acm < n * succ k h : n * succ k \u2264 m H : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1] \u22a2 ediv (\u2191(n * succ k) + -(\u2191m + 1)) \u2191(succ k) = \u2191n + -(\u2191(m / succ k) + 1) ** rw [H h, H ((Nat.le_div_iff_mul_le k.succ_pos).2 h)] ** a b c : Int H\u271d : c \u2260 0 k n m : Nat h\u271d : \u00acm < n * succ k h : n * succ k \u2264 m H : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1] \u22a2 ediv -[m - n * succ k+1] \u2191(succ k) = -[m / succ k - n+1] ** apply congrArg negSucc ** a b c : Int H\u271d : c \u2260 0 k n m : Nat h\u271d : \u00acm < n * succ k h : n * succ k \u2264 m H : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1] \u22a2 (m - n * succ k) / succ k = m / succ k - n ** rw [Nat.mul_comm, Nat.sub_mul_div] ** case h\u2081 a b c : Int H\u271d : c \u2260 0 k n m : Nat h\u271d : \u00acm < n * succ k h : n * succ k \u2264 m H : \u2200 {a b : Nat}, a \u2264 b \u2192 \u2191a + -(\u2191b + 1) = -[b - a+1] \u22a2 succ k * n \u2264 m ** rwa [Nat.mul_comm] ** a\u271d b\u271d c : Int H : c \u2260 0 k n m : Nat h\u271d\u00b9 : \u00acm < n * succ k h\u271d : n * succ k \u2264 m a b : Nat h : a \u2264 b \u22a2 \u2191a + -(\u2191b + 1) = -[b - a+1] ** rw [negSucc_eq, Int.ofNat_sub h] ** a\u271d b\u271d c : Int H : c \u2260 0 k n m : Nat h\u271d\u00b9 : \u00acm < n * succ k h\u271d : n * succ k \u2264 m a b : Nat h : a \u2264 b \u22a2 \u2191a + -(\u2191b + 1) = -(\u2191b - \u2191a + 1) ** simp only [Int.sub_eq_add_neg, Int.neg_add, Int.neg_neg, Int.add_left_comm, Int.add_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.insert_sigma ** \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 : \u03b1 i \u22a2 Set.Sigma (insert i s) t = Sigma.mk i '' t i \u222a Set.Sigma s t ** rw [insert_eq, union_sigma, singleton_sigma] ** \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 : \u03b1 i \u22a2 \u03b1 i ** exact a ** Qed",
        "informal": ""
    },
    {
        "formal": "tendsto_measure_thickening_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 (thickening R s) \u2260 \u22a4 h's : IsClosed s \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (thickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc s)) ** convert tendsto_measure_thickening 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 (thickening R s) \u2260 \u22a4 h's : IsClosed s \u22a2 s = closure s ** exact h's.closure_eq.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2081 a\u2082 b\u2081 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** wlog h\u2081 : a\u2081 \u2264 b\u2081 generalizing a\u2081 b\u2081 ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2082 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : a\u2081 \u2264 b\u2081 \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** wlog h\u2082 : a\u2082 \u2264 b\u2082 generalizing a\u2082 b\u2082 ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d h\u2081 : a\u2081 \u2264 b\u2081 a\u2082 b\u2082 : \u211d Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2082 : a\u2082 \u2264 b\u2082 \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** simp only [uIcc_of_le h\u2081, uIcc_of_le h\u2082, min_eq_left, max_eq_right, h\u2081, h\u2082] at Hcf Hcg Hdf Hdg Hi ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d h\u2081 : a\u2081 \u2264 b\u2081 a\u2082 b\u2082 : \u211d h\u2082 : a\u2082 \u2264 b\u2082 Hcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082) Hcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082) \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** calc\n  (\u222b x in a\u2081..b\u2081, \u222b y in a\u2082..b\u2082, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =\n      \u222b x in Icc a\u2081 b\u2081, \u222b y in Icc a\u2082 b\u2082, f' (x, y) (1, 0) + g' (x, y) (0, 1) := by\n    simp only [intervalIntegral.integral_of_le, h\u2081, h\u2082,\n      set_integral_congr_set_ae (Ioc_ae_eq_Icc (\u03b1 := \u211d) (\u03bc := volume))]\n  _ = \u222b x in Icc a\u2081 b\u2081 \u00d7\u02e2 Icc a\u2082 b\u2082, f' x (1, 0) + g' x (0, 1) := (set_integral_prod _ Hi).symm\n  _ = (((\u222b x in a\u2081..b\u2081, g (x, b\u2082)) - \u222b x in a\u2081..b\u2081, g (x, a\u2082)) + \u222b y in a\u2082..b\u2082, f (b\u2081, y)) -\n        \u222b y in a\u2082..b\u2082, f (a\u2081, y) := by\n    rw [Icc_prod_Icc] at *\n    apply integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le f g f' g'\n      (a\u2081, a\u2082) (b\u2081, b\u2082) \u27e8h\u2081, h\u2082\u27e9 s <;> assumption ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2081 a\u2082 b\u2081 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) this :   \u2200 (a\u2081 b\u2081 : \u211d),     ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192       ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192           (\u2200 (x : \u211d \u00d7 \u211d),               x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192             IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192               a\u2081 \u2264 b\u2081 \u2192                 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                   (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) +                       \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -                     \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) h\u2081 : \u00aca\u2081 \u2264 b\u2081 \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** specialize this b\u2081 a\u2081 ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2081 a\u2082 b\u2081 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : \u00aca\u2081 \u2264 b\u2081 this :   ContinuousOn f ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192     ContinuousOn g ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192       (\u2200 (x : \u211d \u00d7 \u211d),           x \u2208 Set.Ioo (min b\u2081 a\u2081) (max b\u2081 a\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min b\u2081 a\u2081) (max b\u2081 a\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192           IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192             b\u2081 \u2264 a\u2081 \u2192               \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                 (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -                   \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y) \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** rw [uIcc_comm b\u2081 a\u2081, min_comm b\u2081 a\u2081, max_comm b\u2081 a\u2081] at this ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2081 a\u2082 b\u2081 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : \u00aca\u2081 \u2264 b\u2081 this :   ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192     ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192       (\u2200 (x : \u211d \u00d7 \u211d),           x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192           IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192             b\u2081 \u2264 a\u2081 \u2192               \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                 (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -                   \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y) \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** simp only [intervalIntegral.integral_symm b\u2081 a\u2081] ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2081 a\u2082 b\u2081 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : \u00aca\u2081 \u2264 b\u2081 this :   ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192     ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192       (\u2200 (x : \u211d \u00d7 \u211d),           x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192           IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192             b\u2081 \u2264 a\u2081 \u2192               \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                 (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -                   \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y) \u22a2 -\u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((-\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - -\u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h\u2081))).trans _ ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2081 a\u2082 b\u2081 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : \u00aca\u2081 \u2264 b\u2081 this :   ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192     ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192       (\u2200 (x : \u211d \u00d7 \u211d),           x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192           IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192             b\u2081 \u2264 a\u2081 \u2192               \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                 (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -                   \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y) \u22a2 -((((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -         \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) =     (((-\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - -\u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** abel ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2082 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : a\u2081 \u2264 b\u2081 this :   \u2200 (a\u2082 b\u2082 : \u211d),     ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192       ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192           (\u2200 (x : \u211d \u00d7 \u211d),               x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192             IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192               a\u2082 \u2264 b\u2082 \u2192                 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                   (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) +                       \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -                     \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) h\u2082 : \u00aca\u2082 \u2264 b\u2082 \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** specialize this b\u2082 a\u2082 ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2082 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : a\u2081 \u2264 b\u2081 h\u2082 : \u00aca\u2082 \u2264 b\u2082 this :   ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192     ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192       (\u2200 (x : \u211d \u00d7 \u211d),           x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min b\u2082 a\u2082) (max b\u2082 a\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min b\u2082 a\u2082) (max b\u2082 a\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192           IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192             b\u2082 \u2264 a\u2082 \u2192               \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                 (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -                   \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y) \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** rw [uIcc_comm b\u2082 a\u2082, min_comm b\u2082 a\u2082, max_comm b\u2082 a\u2082] at this ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2082 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : a\u2081 \u2264 b\u2081 h\u2082 : \u00aca\u2082 \u2264 b\u2082 this :   ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192     ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192       (\u2200 (x : \u211d \u00d7 \u211d),           x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192           IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192             b\u2082 \u2264 a\u2082 \u2192               \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                 (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -                   \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y) \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** simp only [intervalIntegral.integral_symm b\u2082 a\u2082, intervalIntegral.integral_neg] ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2082 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : a\u2081 \u2264 b\u2081 h\u2082 : \u00aca\u2082 \u2264 b\u2082 this :   ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192     ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192       (\u2200 (x : \u211d \u00d7 \u211d),           x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192           IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192             b\u2082 \u2264 a\u2082 \u2192               \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                 (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -                   \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y) \u22a2 -\u222b (x : \u211d) in a\u2081..b\u2081, \u222b (x_1 : \u211d) in b\u2082..a\u2082, \u2191(f' (x, x_1)) (1, 0) + \u2191(g' (x, x_1)) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + -\u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -       -\u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y) ** refine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h\u2082))).trans _ ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u2082 b\u2082 : \u211d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d Hcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) h\u2081 : a\u2081 \u2264 b\u2081 h\u2082 : \u00aca\u2082 \u2264 b\u2082 this :   ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192     ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192       (\u2200 (x : \u211d \u00d7 \u211d),           x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192         (\u2200 (x : \u211d \u00d7 \u211d),             x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192           IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192             b\u2082 \u2264 a\u2082 \u2192               \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =                 (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -                   \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y) \u22a2 -((((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -         \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + -\u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -       -\u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y) ** abel ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d h\u2081 : a\u2081 \u2264 b\u2081 a\u2082 b\u2082 : \u211d h\u2082 : a\u2082 \u2264 b\u2082 Hcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082) Hcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082) \u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =     \u222b (x : \u211d) in Set.Icc a\u2081 b\u2081, \u222b (y : \u211d) in Set.Icc a\u2082 b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) ** simp only [intervalIntegral.integral_of_le, h\u2081, h\u2082,\n  set_integral_congr_set_ae (Ioc_ae_eq_Icc (\u03b1 := \u211d) (\u03bc := volume))] ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d h\u2081 : a\u2081 \u2264 b\u2081 a\u2082 b\u2082 : \u211d h\u2082 : a\u2082 \u2264 b\u2082 Hcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082) Hcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082) \u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082, \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** rw [Icc_prod_Icc] at * ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s a\u2081 b\u2081 : \u211d h\u2081 : a\u2081 \u2264 b\u2081 a\u2082 b\u2082 : \u211d h\u2082 : a\u2082 \u2264 b\u2082 Hcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082)) Hcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082)) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082)) \u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082), \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =     (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -       \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y) ** apply integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le f g f' g'\n  (a\u2081, a\u2082) (b\u2081, b\u2082) \u27e8h\u2081, h\u2082\u27e9 s <;> assumption ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.restrict_union_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 s' t : Set \u03b1 \u22a2 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t) \u2194 restrict \u03bc s = restrict \u03bd s \u2227 restrict \u03bc t = restrict \u03bd t ** refine'\n  \u27e8fun h =>\n    \u27e8restrict_congr_mono (subset_union_left _ _) h,\n      restrict_congr_mono (subset_union_right _ _) h\u27e9,\n    _\u27e9 ** \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 \u22a2 restrict \u03bc s = restrict \u03bd s \u2227 restrict \u03bc t = restrict \u03bd t \u2192 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t) ** rintro \u27e8hs, ht\u27e9 ** 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 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 hs : restrict \u03bc s = restrict \u03bd s ht : restrict \u03bc t = restrict \u03bd t \u22a2 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t) ** ext1 u hu ** case intro.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 : Set \u03b1 hs : restrict \u03bc s = restrict \u03bd s ht : restrict \u03bc t = restrict \u03bd t u : Set \u03b1 hu : MeasurableSet u \u22a2 \u2191\u2191(restrict \u03bc (s \u222a t)) u = \u2191\u2191(restrict \u03bd (s \u222a t)) u ** simp only [restrict_apply hu, inter_union_distrib_left] ** case intro.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 : Set \u03b1 hs : restrict \u03bc s = restrict \u03bd s ht : restrict \u03bc t = restrict \u03bd t u : Set \u03b1 hu : MeasurableSet u \u22a2 \u2191\u2191\u03bc (u \u2229 s \u222a u \u2229 t) = \u2191\u2191\u03bd (u \u2229 s \u222a u \u2229 t) ** rcases exists_measurable_superset\u2082 \u03bc \u03bd (u \u2229 s) with \u27e8US, hsub, hm, h\u03bc, h\u03bd\u27e9 ** case intro.h.intro.intro.intro.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 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 hs : restrict \u03bc s = restrict \u03bd s ht : restrict \u03bc t = restrict \u03bd t u : Set \u03b1 hu : MeasurableSet u US : Set \u03b1 hsub : u \u2229 s \u2286 US hm : MeasurableSet US h\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s) h\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s) \u22a2 \u2191\u2191\u03bc (u \u2229 s \u222a u \u2229 t) = \u2191\u2191\u03bd (u \u2229 s \u222a u \u2229 t) ** calc\n  \u03bc (u \u2229 s \u222a u \u2229 t) = \u03bc (US \u222a u \u2229 t) :=\n    measure_union_congr_of_subset hsub h\u03bc.le Subset.rfl le_rfl\n  _ = \u03bc US + \u03bc ((u \u2229 t) \\ US) := (measure_add_diff hm _).symm\n  _ = restrict \u03bc s u + restrict \u03bc t (u \\ US) := by\n    simp only [restrict_apply, hu, hu.diff hm, h\u03bc, \u2190 inter_comm t, inter_diff_assoc]\n  _ = restrict \u03bd s u + restrict \u03bd t (u \\ US) := by rw [hs, ht]\n  _ = \u03bd US + \u03bd ((u \u2229 t) \\ US) := by\n    simp only [restrict_apply, hu, hu.diff hm, h\u03bd, \u2190 inter_comm t, inter_diff_assoc]\n  _ = \u03bd (US \u222a u \u2229 t) := (measure_add_diff hm _)\n  _ = \u03bd (u \u2229 s \u222a u \u2229 t) := Eq.symm <| measure_union_congr_of_subset hsub h\u03bd.le Subset.rfl le_rfl ** \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 hs : restrict \u03bc s = restrict \u03bd s ht : restrict \u03bc t = restrict \u03bd t u : Set \u03b1 hu : MeasurableSet u US : Set \u03b1 hsub : u \u2229 s \u2286 US hm : MeasurableSet US h\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s) h\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s) \u22a2 \u2191\u2191\u03bc US + \u2191\u2191\u03bc ((u \u2229 t) \\ US) = \u2191\u2191(restrict \u03bc s) u + \u2191\u2191(restrict \u03bc t) (u \\ US) ** simp only [restrict_apply, hu, hu.diff hm, h\u03bc, \u2190 inter_comm t, inter_diff_assoc] ** \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 hs : restrict \u03bc s = restrict \u03bd s ht : restrict \u03bc t = restrict \u03bd t u : Set \u03b1 hu : MeasurableSet u US : Set \u03b1 hsub : u \u2229 s \u2286 US hm : MeasurableSet US h\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s) h\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s) \u22a2 \u2191\u2191(restrict \u03bc s) u + \u2191\u2191(restrict \u03bc t) (u \\ US) = \u2191\u2191(restrict \u03bd s) u + \u2191\u2191(restrict \u03bd t) (u \\ US) ** rw [hs, 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 : Set \u03b1 hs : restrict \u03bc s = restrict \u03bd s ht : restrict \u03bc t = restrict \u03bd t u : Set \u03b1 hu : MeasurableSet u US : Set \u03b1 hsub : u \u2229 s \u2286 US hm : MeasurableSet US h\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s) h\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s) \u22a2 \u2191\u2191(restrict \u03bd s) u + \u2191\u2191(restrict \u03bd t) (u \\ US) = \u2191\u2191\u03bd US + \u2191\u2191\u03bd ((u \u2229 t) \\ US) ** simp only [restrict_apply, hu, hu.diff hm, h\u03bd, \u2190 inter_comm t, inter_diff_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "Decidable.not_iff_not ** a b : Prop inst\u271d\u00b9 : Decidable a inst\u271d : Decidable b \u22a2 (\u00aca \u2194 \u00acb) \u2194 (a \u2194 b) ** rw [@iff_def (\u00aca), @iff_def' a] ** a b : Prop inst\u271d\u00b9 : Decidable a inst\u271d : Decidable b \u22a2 (\u00aca \u2192 \u00acb) \u2227 (\u00acb \u2192 \u00aca) \u2194 (b \u2192 a) \u2227 (a \u2192 b) ** exact and_congr not_imp_not not_imp_not ** Qed",
        "informal": ""
    },
    {
        "formal": "Partrec.nat_rec ** \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 \u2115 g : \u03b1 \u2192. \u03c3 h : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3 hf : Computable f hg : Partrec g hh : Partrec\u2082 h n : \u2115 \u22a2 (Part.bind (Part.bind \u2191(decode n) fun a => Part.map encode (\u2191f a)) fun n_1 =>       Nat.rec (Part.bind \u2191(decode n) fun a => Part.map encode (g a))         (fun y IH => do           let i \u2190 IH           Part.bind \u2191(decode (Nat.pair n (Nat.pair y i))) fun a => Part.map encode ((fun p => h p.1 p.2) a))         n_1) =     Part.bind \u2191(decode n) fun a =>       Part.map encode ((fun a => Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (f a)) a) ** cases' e : decode (\u03b1 := \u03b1) n with a <;> simp [e] ** case some \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 \u2115 g : \u03b1 \u2192. \u03c3 h : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3 hf : Computable f hg : Partrec g hh : Partrec\u2082 h n : \u2115 a : \u03b1 e : decode n = Option.some a \u22a2 Nat.rec (Part.map encode (g a))       (fun y IH =>         Part.bind IH fun i =>           Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))       (f a) =     Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (f a)) ** induction' f a with m IH <;> simp ** case some.succ \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 \u2115 g : \u03b1 \u2192. \u03c3 h : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3 hf : Computable f hg : Partrec g hh : Partrec\u2082 h n : \u2115 a : \u03b1 e : decode n = Option.some a m : \u2115 IH :   Nat.rec (Part.map encode (g a))       (fun y IH =>         Part.bind IH fun i =>           Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))       m =     Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) \u22a2 (Part.bind       (Nat.rec (Part.map encode (g a))         (fun y IH =>           Part.bind IH fun i =>             Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))         m)       fun i => Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode i)) fun a => Part.map encode (h a.1 a.2)) =     Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y => Part.map encode (h a (m, y)) ** rw [IH, Part.bind_map] ** case some.succ \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 \u2115 g : \u03b1 \u2192. \u03c3 h : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3 hf : Computable f hg : Partrec g hh : Partrec\u2082 h n : \u2115 a : \u03b1 e : decode n = Option.some a m : \u2115 IH :   Nat.rec (Part.map encode (g a))       (fun y IH =>         Part.bind IH fun i =>           Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))       m =     Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) \u22a2 (Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y =>       Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode y))) fun a => Part.map encode (h a.1 a.2)) =     Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y => Part.map encode (h a (m, y)) ** congr ** case some.succ.e_g \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 \u2115 g : \u03b1 \u2192. \u03c3 h : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3 hf : Computable f hg : Partrec g hh : Partrec\u2082 h n : \u2115 a : \u03b1 e : decode n = Option.some a m : \u2115 IH :   Nat.rec (Part.map encode (g a))       (fun y IH =>         Part.bind IH fun i =>           Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))       m =     Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) \u22a2 (fun y => Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode y))) fun a => Part.map encode (h a.1 a.2)) =     fun y => Part.map encode (h a (m, y)) ** funext s ** case some.succ.e_g.h \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 \u2115 g : \u03b1 \u2192. \u03c3 h : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3 hf : Computable f hg : Partrec g hh : Partrec\u2082 h n : \u2115 a : \u03b1 e : decode n = Option.some a m : \u2115 IH :   Nat.rec (Part.map encode (g a))       (fun y IH =>         Part.bind IH fun i =>           Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.1 a.2))       m =     Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) s : \u03c3 \u22a2 (Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode s))) fun a => Part.map encode (h a.1 a.2)) =     Part.map encode (h a (m, s)) ** simp [encodek] ** Qed",
        "informal": ""
    },
    {
        "formal": "Measurable.nnreal_tsum ** \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 : MeasurableSpace \u03b1 \u03b9 : Type u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650 h : \u2200 (i : \u03b9), Measurable (f i) \u22a2 Measurable fun x => \u2211' (i : \u03b9), f i x ** simp_rw [NNReal.tsum_eq_toNNReal_tsum] ** \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 : MeasurableSpace \u03b1 \u03b9 : Type u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650 h : \u2200 (i : \u03b9), Measurable (f i) \u22a2 Measurable fun x => ENNReal.toNNReal (\u2211' (b : \u03b9), \u2191(f b x)) ** exact (Measurable.ennreal_tsum fun i => (h i).coe_nnreal_ennreal).ennreal_toNNReal ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.isEmpty_satelliteConfig_multiplicity ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E \u22a2 SatelliteConfig E (multiplicity E) (good\u03c4 E) \u2192 False ** intro a ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E a : SatelliteConfig E (multiplicity E) (good\u03c4 E) \u22a2 False ** let b := a.centerAndRescale ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E a : SatelliteConfig E (multiplicity E) (good\u03c4 E) b : SatelliteConfig E (multiplicity E) (good\u03c4 E) := SatelliteConfig.centerAndRescale a \u22a2 False ** rcases b.exists_normalized a.centerAndRescale_center a.centerAndRescale_radius\n    (one_lt_good\u03c4 E).le (good\u03b4 E) le_rfl (good\u03b4_lt_one E).le with\n  \u27e8c', c'_le_two, hc'\u27e9 ** case intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E a : SatelliteConfig E (multiplicity E) (good\u03c4 E) b : SatelliteConfig E (multiplicity E) (good\u03c4 E) := SatelliteConfig.centerAndRescale a c' : Fin (Nat.succ (multiplicity E)) \u2192 E c'_le_two : \u2200 (n : Fin (Nat.succ (multiplicity E))), \u2016c' n\u2016 \u2264 2 hc' : \u2200 (i j : Fin (Nat.succ (multiplicity E))), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016c' i - c' j\u2016 \u22a2 False ** exact\n  lt_irrefl _ ((Nat.lt_succ_self _).trans_le (le_multiplicity_of_\u03b4_of_fin c' c'_le_two hc')) ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.cast_mul ** \u03b1 : Type u_1 inst\u271d : Ring \u03b1 m n : ZNum \u22a2 \u2191(m * n) = \u2191m * \u2191n ** rw [\u2190 cast_to_int, mul_to_int, Int.cast_mul, cast_to_int, cast_to_int] ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.div_zero ** n : Num \u22a2 div n 0 = 0 ** cases n ** case zero  \u22a2 div zero 0 = 0 ** rfl ** case pos a\u271d : PosNum \u22a2 div (pos a\u271d) 0 = 0 ** simp [Num.div] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L2.norm_sq_eq_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 : { x // x \u2208 Lp E 2 } \u22a2 \u2016f\u2016 ^ 2 = \u2191IsROrC.re (inner f f) ** have h_two : (2 : \u211d\u22650\u221e).toReal = 2 := by simp ** \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 : { x // x \u2208 Lp E 2 } h_two : ENNReal.toReal 2 = 2 \u22a2 \u2016f\u2016 ^ 2 = \u2191IsROrC.re (inner f f) ** rw [inner_def, integral_inner_eq_sq_snorm, norm_def, \u2190 ENNReal.toReal_pow, IsROrC.ofReal_re,\n  ENNReal.toReal_eq_toReal (ENNReal.pow_ne_top (Lp.snorm_ne_top f)) _] ** \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 : { x // x \u2208 Lp E 2 } \u22a2 ENNReal.toReal 2 = 2 ** simp ** \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 : { x // x \u2208 Lp E 2 } h_two : ENNReal.toReal 2 = 2 \u22a2 snorm (\u2191\u2191f) 2 \u03bc ^ 2 = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc ** rw [\u2190 ENNReal.rpow_nat_cast, snorm_eq_snorm' two_ne_zero ENNReal.two_ne_top, snorm', \u2190\n  ENNReal.rpow_mul, one_div, h_two] ** \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 : { x // x \u2208 Lp E 2 } h_two : ENNReal.toReal 2 = 2 \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc) ^ (2\u207b\u00b9 * \u21912) = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc ** simp ** \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 : { x // x \u2208 Lp E 2 } h_two : ENNReal.toReal 2 = 2 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a ^ 2 \u2202\u03bc \u2260 \u22a4 ** refine' (lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top zero_lt_two _).ne ** \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 : { x // x \u2208 Lp E 2 } h_two : ENNReal.toReal 2 = 2 \u22a2 snorm' (fun a => \u2191\u2191f a) 2 \u03bc < \u22a4 ** rw [\u2190 h_two, \u2190 snorm_eq_snorm' two_ne_zero ENNReal.two_ne_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 : { x // x \u2208 Lp E 2 } h_two : ENNReal.toReal 2 = 2 \u22a2 snorm (fun a => \u2191\u2191f a) 2 \u03bc < \u22a4 ** exact Lp.snorm_lt_top f ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.predictablePart_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 predictablePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] g n ** filter_upwards [martingalePart_add_ae_eq hf hg hg0 hgint 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 \u03c9 : \u03a9 h\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 \u22a2 predictablePart (f + g) \u2131 \u03bc n \u03c9 = g n \u03c9 ** rw [\u2190 add_right_inj (f n \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 \u03c9 : \u03a9 h\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 \u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 + g n \u03c9 ** conv_rhs => rw [\u2190 Pi.add_apply, \u2190 Pi.add_apply, \u2190 martingalePart_add_predictablePart \u2131 \u03bc (f + g)] ** 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 \u03c9 : \u03a9 h\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 \u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = (martingalePart (f + g) \u2131 \u03bc + predictablePart (f + g) \u2131 \u03bc) n \u03c9 ** rw [Pi.add_apply, Pi.add_apply, h\u03c9] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_mul_Ioc_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' Ioc a b = Ico (b / c) (a / c) ** simpa only [mul_comm] using preimage_mul_const_Ioc_of_neg a b h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.average_mem_openSegment_compl_self ** \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\u271d t : Set \u03b1 f\u271d g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E s : Set \u03b1 hs : NullMeasurableSet s hs\u2080 : \u2191\u2191\u03bc s \u2260 0 hsc\u2080 : \u2191\u2191\u03bc s\u1d9c \u2260 0 hfi : Integrable f \u22a2 \u2a0d (x : \u03b1), f x \u2202\u03bc \u2208 openSegment \u211d (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d (x : \u03b1) in s\u1d9c, f x \u2202\u03bc) ** simpa only [union_compl_self, restrict_univ] using\n  average_union_mem_openSegment aedisjoint_compl_right hs.compl hs\u2080 hsc\u2080 (measure_ne_top _ _)\n    (measure_ne_top _ _) hfi.integrableOn hfi.integrableOn ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Content.outerMeasure_exists_open ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G A : Set G hA : \u2191(Content.outerMeasure \u03bc) A \u2260 \u22a4 \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 U, A \u2286 \u2191U \u2227 \u2191(Content.outerMeasure \u03bc) \u2191U \u2264 \u2191(Content.outerMeasure \u03bc) A + \u2191\u03b5 ** rcases inducedOuterMeasure_exists_set _ \u03bc.innerContent_iUnion_nat \u03bc.innerContent_mono hA\n    (ENNReal.coe_ne_zero.2 h\u03b5) with\n  \u27e8U, hU, h2U, h3U\u27e9 ** case intro.intro.intro G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G A : Set G hA : \u2191(Content.outerMeasure \u03bc) A \u2260 \u22a4 \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 U : Set G hU : IsOpen U h2U : A \u2286 U h3U :   \u2191(inducedOuterMeasure (fun s\u2081 hs\u2081 => innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 }) (_ : IsOpen \u2205)           (_ : innerContent \u03bc \u22a5 = 0))       U \u2264     \u2191(inducedOuterMeasure (fun s\u2081 hs\u2081 => innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 }) (_ : IsOpen \u2205)             (_ : innerContent \u03bc \u22a5 = 0))         A +       \u2191\u03b5 \u22a2 \u2203 U, A \u2286 \u2191U \u2227 \u2191(Content.outerMeasure \u03bc) \u2191U \u2264 \u2191(Content.outerMeasure \u03bc) A + \u2191\u03b5 ** exact \u27e8\u27e8U, hU\u27e9, h2U, h3U\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.shiftLeft_add ** m n k : \u2115 \u22a2 Nat.shiftLeft' false m (n + k) = Nat.shiftLeft' false (Nat.shiftLeft' false m n) k ** simp [Nat.pow_add, mul_assoc] ** m n\u271d k i n : \u2115 \u22a2 (fun n k i => \u2191m <<< i = \u2191(Nat.shiftLeft' false m n >>> k)) (n + i) n \u2191i ** dsimp ** m n\u271d k i n : \u2115 \u22a2 \u2191m <<< \u2191i = \u2191(Nat.shiftLeft' false m (n + i) >>> n) ** simp [- Nat.shiftLeft_eq, \u2190 Nat.shiftLeft_sub _ , add_tsub_cancel_left] ** m n\u271d k i n : \u2115 \u22a2 (fun n k i => \u2191m <<< i = \u2191(Nat.shiftLeft' false m n >>> k)) n (n + i + 1) -[i+1] ** dsimp ** m n\u271d k i n : \u2115 \u22a2 \u2191m <<< -[i+1] = \u2191(Nat.shiftLeft' false m n >>> (n + i)) / 2 ** simp [- Nat.shiftLeft_eq, Nat.shiftLeft_zero, Nat.shiftRight_add, \u2190 Nat.shiftLeft_sub] ** m n\u271d k i n : \u2115 \u22a2 \u2191m <<< -[i+1] = \u2191(m >>> i) / 2 ** rfl ** m n\u271d k i n : \u2115 \u22a2 Nat.shiftLeft' true m i = Nat.shiftLeft' true m (n + i) >>> n ** rw [\u2190 Nat.shiftLeft'_sub, add_tsub_cancel_left] ** case a m n\u271d k i n : \u2115 \u22a2 n \u2264 n + i ** apply Nat.le_add_right ** m n\u271d k i n : \u2115 \u22a2 m >>> Nat.succ i = Nat.shiftLeft' true m n >>> (n + i + 1) ** rw [add_assoc, Nat.shiftRight_add, \u2190 Nat.shiftLeft'_sub, tsub_self]\n<;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.pairwise_replaceF ** case nil \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 H : List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) [] \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) []) ** simp [H] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (a :: l) \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) (a :: l)) ** simp at H \u22a2 ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (match bif a.fst == k then some (k, f a) else none with     | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l     | some a => a :: l) ** generalize e : cond .. = z ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l z : Option (\u03b1 \u00d7 \u03b2) e : (bif a.fst == k then some (k, f a) else none) = z \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (match z with     | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l     | some a => a :: l) ** unfold cond at e ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l z : Option (\u03b1 \u00d7 \u03b2) e :   (match a.fst == k with     | true => some (k, f a)     | false => none) =     z \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (match z with     | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l     | some a => a :: l) ** revert e ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l z : Option (\u03b1 \u00d7 \u03b2) \u22a2 (match a.fst == k with       | true => some (k, f a)       | false => none) =       z \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (match z with       | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l       | some a => a :: l) ** split <;> (intro h; subst h; simp) ** case cons.h_2 \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l z : Option (\u03b1 \u00d7 \u03b2) c\u271d : Bool heq\u271d : (a.fst == k) = false \u22a2 none = z \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (match z with       | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l       | some a => a :: l) ** intro h ** case cons.h_2 \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l z : Option (\u03b1 \u00d7 \u03b2) c\u271d : Bool heq\u271d : (a.fst == k) = false h : none = z \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (match z with     | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l     | some a => a :: l) ** subst h ** case cons.h_2 \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l c\u271d : Bool heq\u271d : (a.fst == k) = false \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (match none with     | none => a :: List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l     | some a => a :: l) ** simp ** case cons.h_2 \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l c\u271d : Bool heq\u271d : (a.fst == k) = false \u22a2 (\u2200 (a' : \u03b1 \u00d7 \u03b2),       a' \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) ** next e =>\nrefine \u27e8fun a h => ?_, ih H.2\u27e9\nmatch mem_replaceF h with\n| .inl eq => exact eq \u25b8 ne_true_of_eq_false e\n| .inr h => exact H.1 a h ** \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l c\u271d : Bool e : (a.fst == k) = false \u22a2 (\u2200 (a' : \u03b1 \u00d7 \u03b2),       a' \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l \u2192 \u00ac(a.fst == a'.fst) = true) \u2227     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) ** refine \u27e8fun a h => ?_, ih H.2\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a\u271d : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a\u271d.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l c\u271d : Bool e : (a\u271d.fst == k) = false a : \u03b1 \u00d7 \u03b2 h : a \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l \u22a2 \u00ac(a\u271d.fst == a.fst) = true ** match mem_replaceF h with\n| .inl eq => exact eq \u25b8 ne_true_of_eq_false e\n| .inr h => exact H.1 a h ** \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a\u271d : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a\u271d.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l c\u271d : Bool e : (a\u271d.fst == k) = false a : \u03b1 \u00d7 \u03b2 h : a \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l eq : a.fst = k \u22a2 \u00ac(a\u271d.fst == a.fst) = true ** exact eq \u25b8 ne_true_of_eq_false e ** \u03b1 : Type u_1 \u03b2 : Type u_2 k : \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : PartialEquivBEq \u03b1 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 a\u271d : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l \u2192     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)       (List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l) H : (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 l \u2192 \u00ac(a\u271d.fst == a'.fst) = true) \u2227 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l c\u271d : Bool e : (a\u271d.fst == k) = false a : \u03b1 \u00d7 \u03b2 h\u271d : a \u2208 List.replaceF (fun a => bif a.fst == k then some (k, f a) else none) l h : a \u2208 l \u22a2 \u00ac(a\u271d.fst == a.fst) = true ** exact H.1 a h ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.list_join ** \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 l : List (List \u03b1) \u22a2 List.foldr (fun b s => (fun x x_1 => x ++ x_1) (l, b, s).2.1 (l, b, s).2.2) [] (id l) = List.join l ** dsimp ** \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 l : List (List \u03b1) \u22a2 List.foldr (fun b s => b ++ s) [] l = List.join l ** induction l <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Ioc_eq_empty_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 \u22a2 Ioc a b = \u2205 \u2194 \u00aca < b ** rw [\u2190 coe_eq_empty, coe_Ioc, Set.Ioc_eq_empty_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ordConnected_of_Ioo ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1\u271d inst\u271d\u00b9 : Preorder \u03b2 s\u271d t : Set \u03b1\u271d \u03b1 : Type u_3 inst\u271d : PartialOrder \u03b1 s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s \u22a2 OrdConnected s ** rw [ordConnected_iff] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1\u271d inst\u271d\u00b9 : Preorder \u03b2 s\u271d t : Set \u03b1\u271d \u03b1 : Type u_3 inst\u271d : PartialOrder \u03b1 s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s \u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2264 y \u2192 Icc x y \u2286 s ** intro x hx y hy hxy ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1\u271d inst\u271d\u00b9 : Preorder \u03b2 s\u271d t : Set \u03b1\u271d \u03b1 : Type u_3 inst\u271d : PartialOrder \u03b1 s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x \u2264 y \u22a2 Icc x y \u2286 s ** rcases eq_or_lt_of_le hxy with (rfl | hxy') ** case inr \u03b1\u271d : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1\u271d inst\u271d\u00b9 : Preorder \u03b2 s\u271d t : Set \u03b1\u271d \u03b1 : Type u_3 inst\u271d : PartialOrder \u03b1 s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x \u2264 y hxy' : x < y \u22a2 Icc x y \u2286 s ** rw [\u2190 Ioc_insert_left hxy, \u2190 Ioo_insert_right hxy'] ** case inr \u03b1\u271d : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1\u271d inst\u271d\u00b9 : Preorder \u03b2 s\u271d t : Set \u03b1\u271d \u03b1 : Type u_3 inst\u271d : PartialOrder \u03b1 s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x \u2264 y hxy' : x < y \u22a2 insert x (insert y (Ioo x y)) \u2286 s ** exact insert_subset_iff.2 \u27e8hx, insert_subset_iff.2 \u27e8hy, hs x hx y hy hxy'\u27e9\u27e9 ** case inl \u03b1\u271d : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1\u271d inst\u271d\u00b9 : Preorder \u03b2 s\u271d t : Set \u03b1\u271d \u03b1 : Type u_3 inst\u271d : PartialOrder \u03b1 s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s x : \u03b1 hx hy : x \u2208 s hxy : x \u2264 x \u22a2 Icc x x \u2286 s ** simpa ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.card_singleton_inter ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 \u22a2 card ({a} \u2229 s) \u2264 1 ** cases' Finset.decidableMem a s with h h ** case isFalse \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 h : \u00aca \u2208 s \u22a2 card ({a} \u2229 s) \u2264 1 ** simp [Finset.singleton_inter_of_not_mem h] ** case isTrue \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 h : a \u2208 s \u22a2 card ({a} \u2229 s) \u2264 1 ** simp [Finset.singleton_inter_of_mem h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.uniformIntegrable_of' ** \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 UniformIntegrable f p \u03bc ** refine' \u27e8fun i => (hf i).aestronglyMeasurable,\n  unifIntegrable_of \u03bc hp hp' (fun i => (hf i).aestronglyMeasurable) 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 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C ** obtain \u27e8C, hC\u27e9 := h 1 one_pos ** 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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C ** refine' \u27e8((C : \u211d\u22650\u221e) * \u03bc Set.univ ^ p.toReal\u207b\u00b9 + 1).toNNReal, fun i => _\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 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 \u22a2 snorm (f i) p \u03bc \u2264 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p \u03bc + snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc ** refine' le_trans (snorm_mono fun x => _) (snorm_add_le\n  (StronglyMeasurable.aestronglyMeasurable\n    ((hf i).indicator ((hf i).nnnorm.measurableSet_lt stronglyMeasurable_const)))\n  (StronglyMeasurable.aestronglyMeasurable\n    ((hf i).indicator (stronglyMeasurable_const.measurableSet_le (hf i).nnnorm))) 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 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 x : \u03b1 \u22a2 \u2016f i x\u2016 \u2264 \u2016(indicator {x | \u2016f i x\u2016\u208a < C} (f i) + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) x\u2016 ** rw [Pi.add_apply, Set.indicator_apply] ** \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 x : \u03b1 \u22a2 \u2016f i x\u2016 \u2264 \u2016(if x \u2208 {x | \u2016f i x\u2016\u208a < C} then f i x else 0) + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\u2016 ** split_ifs with hx ** 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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 x : \u03b1 hx : x \u2208 {x | \u2016f i x\u2016\u208a < C} \u22a2 \u2016f i x\u2016 \u2264 \u2016f i x + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\u2016 ** rw [Set.indicator_of_not_mem, add_zero] ** case pos.h \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 x : \u03b1 hx : x \u2208 {x | \u2016f i x\u2016\u208a < C} \u22a2 \u00acx \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} ** simpa using hx ** 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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 x : \u03b1 hx : \u00acx \u2208 {x | \u2016f i x\u2016\u208a < C} \u22a2 \u2016f i x\u2016 \u2264 \u20160 + indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x\u2016 ** rw [Set.indicator_of_mem, zero_add] ** case neg.h \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 x : \u03b1 hx : \u00acx \u2208 {x | \u2016f i x\u2016\u208a < C} \u22a2 x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} ** simpa using hx ** \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 \u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p \u03bc + snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264     \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1 ** have : \u2200\u1d50 x \u2202\u03bc, \u2016{ x : \u03b1 | \u2016f i x\u2016\u208a < C }.indicator (f i) x\u2016\u208a \u2264 C := by\n  refine' eventually_of_forall _\n  simp_rw [nnnorm_indicator_eq_indicator_nnnorm]\n  exact Set.indicator_le fun x (hx : _ < _) => hx.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 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 this : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C \u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p \u03bc + snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264     \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1 ** refine' add_le_add (le_trans (snorm_le_of_ae_bound this) _) (ENNReal.ofReal_one \u25b8 hC i) ** \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 this : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C \u22a2 \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 * ENNReal.ofReal ((fun a => \u2191a) C) \u2264 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 ** simp_rw [NNReal.val_eq_coe, ENNReal.ofReal_coe_nnreal, mul_comm] ** \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 this : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C \u22a2 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 \u2264 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 ** exact le_rfl ** \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C ** refine' eventually_of_forall _ ** \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 \u22a2 \u2200 (x : \u03b1), \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016\u208a \u2264 C ** simp_rw [nnnorm_indicator_eq_indicator_nnnorm] ** \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 \u22a2 \u2200 (x : \u03b1), indicator {x | \u2016f i x\u2016\u208a < C} (fun a => \u2016f i a\u2016\u208a) x \u2264 C ** exact Set.indicator_le fun x (hx : _ < _) => hx.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 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 \u22a2 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1 = \u2191(ENNReal.toNNReal (\u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1)) ** rw [ENNReal.coe_toNNReal] ** \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 : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) 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 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal 1 i : \u03b9 \u22a2 \u2191C * \u2191\u2191\u03bc univ ^ (ENNReal.toReal p)\u207b\u00b9 + 1 \u2260 \u22a4 ** exact ENNReal.add_ne_top.2\n  \u27e8ENNReal.mul_ne_top ENNReal.coe_ne_top (ENNReal.rpow_ne_top_of_nonneg\n    (inv_nonneg.2 ENNReal.toReal_nonneg) (measure_lt_top _ _).ne),\n  ENNReal.one_ne_top\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.card_fintype_Ioc_of_le ** a b : \u2124 h : a \u2264 b \u22a2 \u2191(Fintype.card \u2191(Set.Ioc a b)) = b - a ** rw [card_fintype_Ioc, toNat_sub_of_le h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u22a2 \u222b (x : \u03a9), stoppedValue f \u03c4 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f \u03c0 x \u2202\u03bc ** rw [\u2190 sub_nonneg, \u2190 integral_sub', stoppedValue_sub_eq_sum' hle hbdd] ** \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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u22a2 0 \u2264     \u222b (a : \u03a9),       (fun \u03c9 => Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i)) \u03c9)         a \u2202\u03bc ** simp only [Finset.sum_apply] ** \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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u22a2 0 \u2264 \u222b (a : \u03a9), Finset.sum (Finset.range (N + 1)) fun c => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 c \u2227 c < \u03c0 \u03c9} (f (c + 1) - f c) a \u2202\u03bc ** have : \u2200 i, MeasurableSet[\ud835\udca2 i] {\u03c9 : \u03a9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} := by\n  intro i\n  refine' (h\u03c4 i).inter _\n  convert (h\u03c0 i).compl using 1\n  ext x\n  simp; 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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N this : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} \u22a2 0 \u2264 \u222b (a : \u03a9), Finset.sum (Finset.range (N + 1)) fun c => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 c \u2227 c < \u03c0 \u03c9} (f (c + 1) - f c) a \u2202\u03bc ** rw [integral_finset_sum] ** case 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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N this : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} \u22a2 \u2200 (i : \u2115), i \u2208 Finset.range (N + 1) \u2192 Integrable fun a => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a ** intro i _ ** case 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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N this : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} i : \u2115 a\u271d : i \u2208 Finset.range (N + 1) \u22a2 Integrable fun a => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a ** exact Integrable.indicator (Integrable.sub (hf.integrable _) (hf.integrable _))\n  (\ud835\udca2.le _ _ (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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u22a2 \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} ** intro i ** \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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N i : \u2115 \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} ** refine' (h\u03c4 i).inter _ ** \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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N i : \u2115 \u22a2 MeasurableSet fun \u03c9 => Nat.le (Nat.succ i) (\u03c0 \u03c9) ** convert (h\u03c0 i).compl using 1 ** case h.e'_3 \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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N i : \u2115 \u22a2 (fun \u03c9 => Nat.le (Nat.succ i) (\u03c0 \u03c9)) = {\u03c9 | \u03c0 \u03c9 \u2264 i}\u1d9c ** ext x ** case h.e'_3.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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N i : \u2115 x : \u03a9 \u22a2 (x \u2208 fun \u03c9 => Nat.le (Nat.succ i) (\u03c0 \u03c9)) \u2194 x \u2208 {\u03c9 | \u03c0 \u03c9 \u2264 i}\u1d9c ** simp ** case h.e'_3.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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N i : \u2115 x : \u03a9 \u22a2 (x \u2208 fun \u03c9 => Nat.succ i \u2264 \u03c0 \u03c9) \u2194 i < \u03c0 x ** 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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N this : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} \u22a2 0 \u2264 Finset.sum (Finset.range (N + 1)) fun i => \u222b (a : \u03a9), Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a \u2202\u03bc ** refine' Finset.sum_nonneg fun i _ => _ ** \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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N this : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} i : \u2115 x\u271d : i \u2208 Finset.range (N + 1) \u22a2 0 \u2264 \u222b (a : \u03a9), Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (f (i + 1) - f i) a \u2202\u03bc ** rw [integral_indicator (\ud835\udca2.le _ _ (this _)), integral_sub', sub_nonneg] ** \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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N this : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} i : \u2115 x\u271d : i \u2208 Finset.range (N + 1) \u22a2 \u222b (a : \u03a9) in {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}, f i a \u2202\u03bc \u2264 \u222b (a : \u03a9) in {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}, f (i + 1) a \u2202\u03bc ** exact hf.set_integral_le (Nat.le_succ i) (this _) ** case 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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N this : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} i : \u2115 x\u271d : i \u2208 Finset.range (N + 1) \u22a2 Integrable (f (i + 1)) ** exact (hf.integrable _).integrableOn ** case hg \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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N this : \u2200 (i : \u2115), MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} i : \u2115 x\u271d : i \u2208 Finset.range (N + 1) \u22a2 Integrable (f i) ** exact (hf.integrable _).integrableOn ** case 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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u22a2 Integrable fun x => stoppedValue f \u03c0 x ** exact hf.integrable_stoppedValue h\u03c0 hbdd ** case hg \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 : SigmaFiniteFiltration \u03bc \ud835\udca2 hf : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u22a2 Integrable fun x => stoppedValue f \u03c4 x ** exact hf.integrable_stoppedValue h\u03c4 fun \u03c9 => le_trans (hle \u03c9) (hbdd \u03c9) ** Qed",
        "informal": ""
    },
    {
        "formal": "SignType.range_eq ** \u03b1 : Type u_1 f : SignType \u2192 \u03b1 \u22a2 Set.range f = {f zero, f neg, f pos} ** classical rw [\u2190 Fintype.coe_image_univ, univ_eq] ** \u03b1 : Type u_1 f : SignType \u2192 \u03b1 \u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos} ** classical simp [Finset.coe_insert] ** \u03b1 : Type u_1 f : SignType \u2192 \u03b1 \u22a2 Set.range f = {f zero, f neg, f pos} ** rw [\u2190 Fintype.coe_image_univ, univ_eq] ** \u03b1 : Type u_1 f : SignType \u2192 \u03b1 \u22a2 \u2191(Finset.image f {0, -1, 1}) = {f zero, f neg, f pos} ** simp [Finset.coe_insert] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_indep_eq ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 \u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc ** by_cases hfint : Integrable f \u03bc ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Integrable f \u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc  case neg \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : \u00acIntegrable f \u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc ** swap ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Integrable f \u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc ** refine' (ae_eq_condexp_of_forall_set_integral_eq hle\u2082 hfint\n  (fun s _ hs => integrableOn_const.2 (Or.inr hs)) (fun s hms hs => _)\n    stronglyMeasurable_const.aeStronglyMeasurable').symm ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Integrable f s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03a9) in s, \u222b (x : \u03a9), f x \u2202\u03bc \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc ** rw [set_integral_const] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Integrable f s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc ** rw [\u2190 mem\u2112p_one_iff_integrable] at hfint ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc ** refine' Mem\u2112p.induction_stronglyMeasurable hle\u2081 ENNReal.one_ne_top _ _ _ _ hfint _ ** case neg \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : \u00acIntegrable f \u22a2 \u03bc[f|m\u2082] =\u1d50[\u03bc] fun x => \u222b (x : \u03a9), f x \u2202\u03bc ** rw [condexp_undef hfint, integral_undef hfint] ** case neg \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : \u00acIntegrable f \u22a2 0 =\u1d50[\u03bc] fun x => 0 ** rfl ** case pos.refine'_1 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 \u22a2 AEStronglyMeasurable' m\u2081 f \u03bc ** exact \u27e8f, hf, EventuallyEq.rfl\u27e9 ** case pos.refine'_2 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2200 (c : E) \u2983s_1 : Set \u03a9\u2984,     MeasurableSet s_1 \u2192       \u2191\u2191\u03bc s_1 < \u22a4 \u2192         ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), Set.indicator s_1 (fun x => c) x \u2202\u03bc =           \u222b (x : \u03a9) in s, Set.indicator s_1 (fun x => c) x \u2202\u03bc ** intro c t hmt _ ** case pos.refine'_2 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 c : E t : Set \u03a9 hmt : MeasurableSet t a\u271d : \u2191\u2191\u03bc t < \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), Set.indicator t (fun x => c) x \u2202\u03bc =     \u222b (x : \u03a9) in s, Set.indicator t (fun x => c) x \u2202\u03bc ** rw [Indep_iff] at hindp ** case pos.refine'_2 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : \u2200 (t1 t2 : Set \u03a9), MeasurableSet t1 \u2192 MeasurableSet t2 \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 c : E t : Set \u03a9 hmt : MeasurableSet t a\u271d : \u2191\u2191\u03bc t < \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), Set.indicator t (fun x => c) x \u2202\u03bc =     \u222b (x : \u03a9) in s, Set.indicator t (fun x => c) x \u2202\u03bc ** rw [integral_indicator (hle\u2081 _ hmt), set_integral_const, smul_smul, \u2190 ENNReal.toReal_mul,\n  mul_comm, \u2190 hindp _ _ hmt hms, set_integral_indicator (hle\u2081 _ hmt), set_integral_const,\n  Set.inter_comm] ** case pos.refine'_3 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2200 \u2983f g : \u03a9 \u2192 E\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Mem\u2112p f 1 \u2192         Mem\u2112p g 1 \u2192           StronglyMeasurable f \u2192             StronglyMeasurable g \u2192               ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc \u2192                 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), g x \u2202\u03bc = \u222b (x : \u03a9) in s, g x \u2202\u03bc \u2192                   ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), (f + g) x \u2202\u03bc = \u222b (x : \u03a9) in s, (f + g) x \u2202\u03bc ** intro u v _ huint hvint hu hv hu_eq hv_eq ** case pos.refine'_3 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 u v : \u03a9 \u2192 E a\u271d : Disjoint (Function.support u) (Function.support v) huint : Mem\u2112p u 1 hvint : Mem\u2112p v 1 hu : StronglyMeasurable u hv : StronglyMeasurable v hu_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc hv_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), v x \u2202\u03bc = \u222b (x : \u03a9) in s, v x \u2202\u03bc \u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), (u + v) x \u2202\u03bc = \u222b (x : \u03a9) in s, (u + v) x \u2202\u03bc ** rw [mem\u2112p_one_iff_integrable] at huint hvint ** case pos.refine'_3 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 u v : \u03a9 \u2192 E a\u271d : Disjoint (Function.support u) (Function.support v) huint : Integrable u hvint : Integrable v hu : StronglyMeasurable u hv : StronglyMeasurable v hu_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc hv_eq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), v x \u2202\u03bc = \u222b (x : \u03a9) in s, v x \u2202\u03bc \u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), (u + v) x \u2202\u03bc = \u222b (x : \u03a9) in s, (u + v) x \u2202\u03bc ** rw [integral_add' huint hvint, smul_add, hu_eq, hv_eq,\n  integral_add' huint.integrableOn hvint.integrableOn] ** case pos.refine'_4 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 \u22a2 IsClosed {f | ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc = \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc} ** have heq\u2081 : (fun f : lpMeas E \u211d m\u2081 1 \u03bc => \u222b x, (f : \u03a9 \u2192 E) x \u2202\u03bc) =\n    (fun f : Lp E 1 \u03bc => \u222b x, f x \u2202\u03bc) \u2218 Submodule.subtypeL _ := by\n  refine' funext fun f => integral_congr_ae _\n  simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]; norm_cast ** case pos.refine'_4 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) \u22a2 IsClosed {f | ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc = \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc} ** have heq\u2082 : (fun f : lpMeas E \u211d m\u2081 1 \u03bc => \u222b x in s, (f : \u03a9 \u2192 E) x \u2202\u03bc) =\n    (fun f : Lp E 1 \u03bc => \u222b x in s, f x \u2202\u03bc) \u2218 Submodule.subtypeL _ := by\n  refine' funext fun f => integral_congr_ae (ae_restrict_of_ae _)\n  simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]\n  exact eventually_of_forall fun _ => (by trivial) ** case pos.refine'_4 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) heq\u2082 :   (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) \u22a2 IsClosed {f | ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc = \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc} ** refine' isClosed_eq (Continuous.const_smul _ _) _ ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 \u22a2 (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) ** refine' funext fun f => integral_congr_ae _ ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f\u271d hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f\u271d 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 f : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc } \u22a2 (fun x => \u2191\u2191\u2191f x) =\u1d50[\u03bc] fun x => \u2191\u2191(\u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) f) x ** simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f\u271d hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f\u271d 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 f : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc } \u22a2 (fun x => \u2191\u2191\u2191f x) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x ** norm_cast ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) \u22a2 (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) ** refine' funext fun f => integral_congr_ae (ae_restrict_of_ae _) ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f\u271d hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f\u271d 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) f : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc } \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, (fun x => \u2191\u2191\u2191f x) x = (fun x => \u2191\u2191(\u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) f) x) x ** simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f\u271d hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f\u271d 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) f : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc } \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, True ** exact eventually_of_forall fun _ => (by trivial) ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f\u271d hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f\u271d 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) f : { x // x \u2208 lpMeas E \u211d m\u2081 1 \u03bc } x\u271d : \u03a9 \u22a2 True ** trivial ** case pos.refine'_4.refine'_1 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) heq\u2082 :   (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) \u22a2 Continuous fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc ** rw [heq\u2081] ** case pos.refine'_4.refine'_1 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) heq\u2082 :   (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) \u22a2 Continuous ((fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))) ** exact continuous_integral.comp (ContinuousLinearMap.continuous _) ** case pos.refine'_4.refine'_2 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) heq\u2082 :   (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) \u22a2 Continuous fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc ** rw [heq\u2082] ** case pos.refine'_4.refine'_2 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 heq\u2081 : (fun f => \u222b (x : \u03a9), \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9), \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) heq\u2082 :   (fun f => \u222b (x : \u03a9) in s, \u2191\u2191\u2191f x \u2202\u03bc) = (fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc)) \u22a2 Continuous ((fun f => \u222b (x : \u03a9) in s, \u2191\u2191f x \u2202\u03bc) \u2218 \u2191(Submodule.subtypeL (lpMeas E \u211d m\u2081 1 \u03bc))) ** exact (continuous_set_integral _).comp (ContinuousLinearMap.continuous _) ** case pos.refine'_5 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2200 \u2983f g : \u03a9 \u2192 E\u2984,     f =\u1d50[\u03bc] g \u2192       Mem\u2112p f 1 \u2192         ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), f x \u2202\u03bc = \u222b (x : \u03a9) in s, f x \u2202\u03bc \u2192           ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), g x \u2202\u03bc = \u222b (x : \u03a9) in s, g x \u2202\u03bc ** intro u v huv _ hueq ** case pos.refine'_5 \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 u v : \u03a9 \u2192 E huv : u =\u1d50[\u03bc] v a\u271d : Mem\u2112p u 1 hueq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc \u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), v x \u2202\u03bc = \u222b (x : \u03a9) in s, v x \u2202\u03bc ** rwa [\u2190 integral_congr_ae huv, \u2190\n  (set_integral_congr_ae (hle\u2082 _ hms) _ : \u222b x in s, u x \u2202\u03bc = \u222b x in s, v x \u2202\u03bc)] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E m\u2081 m\u2082 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 E hle\u2081 : m\u2081 \u2264 m hle\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc hle\u2082) hf : StronglyMeasurable f hindp : Indep m\u2081 m\u2082 hfint : Mem\u2112p f 1 s : Set \u03a9 hms : MeasurableSet s hs : \u2191\u2191\u03bc s < \u22a4 u v : \u03a9 \u2192 E huv : u =\u1d50[\u03bc] v a\u271d : Mem\u2112p u 1 hueq : ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u222b (x : \u03a9), u x \u2202\u03bc = \u222b (x : \u03a9) in s, u x \u2202\u03bc \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 s \u2192 u x = v x ** filter_upwards [huv] with x hx _ using hx ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_le_of_forall_fin_meas_le_of_measurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C ** have : \u222b\u207b x in univ, f x \u2202\u03bc = \u222b\u207b x, f x \u2202\u03bc := by simp only [Measure.restrict_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C ** rw [\u2190 this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u22a2 \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc \u2264 C ** refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S \u22a2 \u222b\u207b (x : \u03b1) in \u22c3 n, S n, f x \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc ** rw [\u2190 lintegral_indicator] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S \u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc  case hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S \u22a2 MeasurableSet (\u22c3 n, S n) ** swap ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S \u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc ** have h_integral_indicator : \u2a06 n, \u222b\u207b x in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b x, (S n).indicator f x \u2202\u03bc := by\n  congr\n  ext1 n\n  rw [lintegral_indicator _ (hm _ (hS_meas n))] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc ** rw [h_integral_indicator, \u2190 lintegral_iSup] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C \u22a2 \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc ** simp only [Measure.restrict_univ] ** case hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S \u22a2 MeasurableSet (\u22c3 n, S n) ** exact hm (\u22c3 n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S \u22a2 \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc ** congr ** case e_s \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S \u22a2 (fun n => \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc) = fun n => \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc ** ext1 n ** case e_s.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S n : \u2115 \u22a2 \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc ** rw [lintegral_indicator _ (hm _ (hS_meas n))] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 n, S n) (fun x => f x) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2a06 n, indicator (S n) f a \u2202\u03bc ** refine' le_of_eq (lintegral_congr fun x => _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 \u22a2 indicator (\u22c3 n, S n) (fun x => f x) x = \u2a06 n, indicator (S n) f x ** simp_rw [indicator_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 \u22a2 (if x \u2208 \u22c3 n, S n then f x else 0) = \u2a06 n, if x \u2208 S n then f x else 0 ** by_cases hx_mem : x \u2208 iUnion S ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : x \u2208 iUnion S \u22a2 (if x \u2208 \u22c3 n, S n then f x else 0) = \u2a06 n, if x \u2208 S n then f x else 0 ** simp only [hx_mem, if_true] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : x \u2208 iUnion S \u22a2 f x = \u2a06 n, if x \u2208 S n then f x else 0 ** obtain \u27e8n, hxn\u27e9 := mem_iUnion.mp hx_mem ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : x \u2208 iUnion S n : \u2115 hxn : x \u2208 S n \u22a2 f x = \u2a06 n, if x \u2208 S n then f x else 0 ** refine' le_antisymm (_root_.trans _ (le_iSup _ n)) (iSup_le fun i => _) ** case pos.intro.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : x \u2208 iUnion S n : \u2115 hxn : x \u2208 S n \u22a2 f x \u2264 if x \u2208 S n then f x else 0 ** simp only [hxn, le_refl, if_true] ** case pos.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : x \u2208 iUnion S n : \u2115 hxn : x \u2208 S n i : \u2115 \u22a2 (if x \u2208 S i then f x else 0) \u2264 f x ** by_cases hxi : x \u2208 S i <;> simp [hxi] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : \u00acx \u2208 iUnion S \u22a2 (if x \u2208 \u22c3 n, S n then f x else 0) = \u2a06 n, if x \u2208 S n then f x else 0 ** simp only [hx_mem, if_false] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : \u00acx \u2208 iUnion S \u22a2 0 = \u2a06 n, if x \u2208 S n then f x else 0 ** rw [mem_iUnion] at hx_mem ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : \u00ac\u2203 i, x \u2208 S i \u22a2 0 = \u2a06 n, if x \u2208 S n then f x else 0 ** push_neg at hx_mem ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : \u2200 (i : \u2115), \u00acx \u2208 S i \u22a2 0 = \u2a06 n, if x \u2208 S n then f x else 0 ** refine' le_antisymm (zero_le _) (iSup_le fun n => _) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc x : \u03b1 hx_mem : \u2200 (i : \u2115), \u00acx \u2208 S i n : \u2115 \u22a2 (if x \u2208 S n then f x else 0) \u2264 0 ** simp only [hx_mem n, if_false, nonpos_iff_eq_zero] ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc \u22a2 \u2200 (n : \u2115), Measurable fun x => indicator (S n) f x ** exact fun n => hf_meas.indicator (hm _ (hS_meas n)) ** case h_mono \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc \u22a2 Monotone fun n x => indicator (S n) f x ** intro n\u2081 n\u2082 hn\u2081\u2082 a ** case h_mono \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc n\u2081 n\u2082 : \u2115 hn\u2081\u2082 : n\u2081 \u2264 n\u2082 a : \u03b1 \u22a2 (fun n x => indicator (S n) f x) n\u2081 a \u2264 (fun n x => indicator (S n) f x) n\u2082 a ** simp_rw [indicator_apply] ** case h_mono \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc n\u2081 n\u2082 : \u2115 hn\u2081\u2082 : n\u2081 \u2264 n\u2082 a : \u03b1 \u22a2 (if a \u2208 S n\u2081 then f a else 0) \u2264 if a \u2208 S n\u2082 then f a else 0 ** split_ifs with h h_1 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc n\u2081 n\u2082 : \u2115 hn\u2081\u2082 : n\u2081 \u2264 n\u2082 a : \u03b1 h : a \u2208 S n\u2081 h_1 : a \u2208 S n\u2082 \u22a2 f a \u2264 f a ** exact le_rfl ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc n\u2081 n\u2082 : \u2115 hn\u2081\u2082 : n\u2081 \u2264 n\u2082 a : \u03b1 h : a \u2208 S n\u2081 h_1 : \u00aca \u2208 S n\u2082 \u22a2 f a \u2264 0 ** exact absurd (mem_of_mem_of_subset h (hS_mono hn\u2081\u2082)) h_1 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc n\u2081 n\u2082 : \u2115 hn\u2081\u2082 : n\u2081 \u2264 n\u2082 a : \u03b1 h : \u00aca \u2208 S n\u2081 h\u271d : a \u2208 S n\u2082 \u22a2 0 \u2264 f a ** exact zero_le _ ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : Measurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C this : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc S : \u2115 \u2192 Set \u03b1 hS_meas : \u2200 (n : \u2115), MeasurableSet (S n) hS_mono : Monotone S h_integral_indicator : \u2a06 n, \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc n\u2081 n\u2082 : \u2115 hn\u2081\u2082 : n\u2081 \u2264 n\u2082 a : \u03b1 h : \u00aca \u2208 S n\u2081 h\u271d : \u00aca \u2208 S n\u2082 \u22a2 0 \u2264 0 ** exact le_rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.tendsto_condCdf_atBot ** \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)) atBot (\ud835\udcdd 0) ** have h_exists : \u2200 x : \u211d, \u2203 q : \u211a, x < q \u2227 \u2191q < x + 1 := fun x => exists_rat_btwn (lt_add_one 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 < \u2191q \u2227 \u2191q < x + 1 \u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0) ** 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 < \u2191q \u2227 \u2191q < x + 1 qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1) \u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0) ** have hqs_tendsto : Tendsto qs atBot atBot := by\n  rw [tendsto_atBot_atBot]\n  refine' fun q => \u27e8q - 1, fun y hy => _\u27e9\n  have h_le : \u2191(qs y) \u2264 (q : \u211d) - 1 + 1 :=\n    (h_exists y).choose_spec.2.le.trans (add_le_add hy le_rfl)\n  rw [sub_add_cancel] at h_le\n  exact_mod_cast 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 < \u2191q \u2227 \u2191q < x + 1 qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1) hqs_tendsto : Tendsto qs atBot atBot \u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0) ** refine'\n  tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds\n    ((tendsto_condCdfRat_atBot \u03c1 a).comp hqs_tendsto) (condCdf_nonneg \u03c1 a) fun 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 < \u2191q \u2227 \u2191q < x + 1 qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1) hqs_tendsto : Tendsto qs atBot atBot x : \u211d \u22a2 \u2191(condCdf \u03c1 a) x \u2264 (condCdfRat \u03c1 a \u2218 qs) 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 < \u2191q \u2227 \u2191q < x + 1 qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1) hqs_tendsto : Tendsto qs atBot atBot x : \u211d \u22a2 \u2191(condCdf \u03c1 a) x \u2264 \u2191(condCdf \u03c1 a) \u2191(qs x) ** exact (condCdf \u03c1 a).mono (h_exists x).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 < \u2191q \u2227 \u2191q < x + 1 qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1) \u22a2 Tendsto qs atBot atBot ** rw [tendsto_atBot_atBot] ** \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 < \u2191q \u2227 \u2191q < x + 1 qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1) \u22a2 \u2200 (b : \u211a), \u2203 i, \u2200 (a : \u211d), a \u2264 i \u2192 qs a \u2264 b ** 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 < \u2191q \u2227 \u2191q < x + 1 qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1) q : \u211a y : \u211d hy : y \u2264 \u2191q - 1 \u22a2 qs y \u2264 q ** have h_le : \u2191(qs y) \u2264 (q : \u211d) - 1 + 1 :=\n  (h_exists y).choose_spec.2.le.trans (add_le_add hy le_rfl) ** \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 < \u2191q \u2227 \u2191q < x + 1 qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1) q : \u211a y : \u211d hy : y \u2264 \u2191q - 1 h_le : \u2191(qs y) \u2264 \u2191q - 1 + 1 \u22a2 qs y \u2264 q ** rw [sub_add_cancel] 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 < \u2191q \u2227 \u2191q < x + 1 qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1) q : \u211a y : \u211d hy : y \u2264 \u2191q - 1 h_le : \u2191(qs y) \u2264 \u2191q \u22a2 qs y \u2264 q ** exact_mod_cast h_le ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.vars_map_of_injective ** 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 hf : Injective \u2191f \u22a2 vars (\u2191(map f) p) = vars p ** simp [vars, degrees_map_of_injective _ hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.prodAssoc_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\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 inst\u271d\u00b2 : SigmaFinite \u03bd inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : SigmaFinite \u03c4 \u22a2 map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4) = Measure.prod \u03bc (Measure.prod \u03bd \u03c4) ** refine' (prod_eq_generateFrom generateFrom_measurableSet generateFrom_prod\n  isPiSystem_measurableSet isPiSystem_prod \u03bc.toFiniteSpanningSetsIn\n  (\u03bd.toFiniteSpanningSetsIn.prod \u03c4.toFiniteSpanningSetsIn) _).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\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 inst\u271d\u00b2 : SigmaFinite \u03bd inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : SigmaFinite \u03c4 \u22a2 \u2200 (s : Set \u03b1),     s \u2208 {s | MeasurableSet s} \u2192       \u2200 (t : Set (\u03b2 \u00d7 \u03b3)),         t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192           \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 t) =             \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) t ** rintro s hs _ \u27e8t, u, ht, hu, rfl\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 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 inst\u271d\u00b2 : SigmaFinite \u03bd inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : SigmaFinite \u03c4 s : Set \u03b1 hs : s \u2208 {s | MeasurableSet s} t : Set \u03b2 u : Set \u03b3 ht : t \u2208 {s | MeasurableSet s} hu : u \u2208 {t | MeasurableSet t} \u22a2 \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 (fun x x_1 => x \u00d7\u02e2 x_1) t u) =     \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) ((fun x x_1 => x \u00d7\u02e2 x_1) t u) ** rw [mem_setOf_eq] at hs ht hu ** 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 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 inst\u271d\u00b2 : SigmaFinite \u03bd inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : SigmaFinite \u03c4 s : Set \u03b1 hs : MeasurableSet s t : Set \u03b2 u : Set \u03b3 ht : MeasurableSet t hu : MeasurableSet u \u22a2 \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 (fun x x_1 => x \u00d7\u02e2 x_1) t u) =     \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) ((fun x x_1 => x \u00d7\u02e2 x_1) t u) ** simp_rw [map_apply (MeasurableEquiv.measurable _) (hs.prod (ht.prod hu)),\n  MeasurableEquiv.prodAssoc, MeasurableEquiv.coe_mk, Equiv.prod_assoc_preimage, prod_prod,\n  mul_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.support_rename_of_injective ** \u03c3 : Type u_1 \u03c4 : Type u_2 \u03b1 : Type u_3 R : Type u_4 S : Type u_5 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 inst\u271d : DecidableEq \u03c4 h : Injective f \u22a2 support (\u2191(rename f) p) = Finset.image (Finsupp.mapDomain f) (support p) ** rw [rename_eq] ** \u03c3 : Type u_1 \u03c4 : Type u_2 \u03b1 : Type u_3 R : Type u_4 S : Type u_5 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 inst\u271d : DecidableEq \u03c4 h : Injective f \u22a2 support (Finsupp.mapDomain (Finsupp.mapDomain f) p) = Finset.image (Finsupp.mapDomain f) (support p) ** exact Finsupp.mapDomain_support_of_injective (mapDomain_injective h) _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.ofNat_dvd ** m n : Nat \u22a2 \u2191m \u2223 \u2191n \u2194 m \u2223 n ** refine \u27e8fun \u27e8a, ae\u27e9 => ?_, fun \u27e8k, e\u27e9 => \u27e8k, by rw [e, Int.ofNat_mul]\u27e9\u27e9 ** m n : Nat x\u271d : \u2191m \u2223 \u2191n a : Int ae : \u2191n = \u2191m * a \u22a2 m \u2223 n ** match Int.le_total a 0 with\n| .inl h =>\n  have := ae.symm \u25b8 Int.mul_nonpos_of_nonneg_of_nonpos (ofNat_zero_le _) h\n  rw [Nat.le_antisymm (ofNat_le.1 this) (Nat.zero_le _)]\n  apply Nat.dvd_zero\n| .inr h => match a, eq_ofNat_of_zero_le h with\n  | _, \u27e8k, rfl\u27e9 => exact \u27e8k, Int.ofNat.inj ae\u27e9 ** m n : Nat x\u271d : m \u2223 n k : Nat e : n = m * k \u22a2 \u2191n = \u2191m * \u2191k ** rw [e, Int.ofNat_mul] ** m n : Nat x\u271d : \u2191m \u2223 \u2191n a : Int ae : \u2191n = \u2191m * a h : a \u2264 0 \u22a2 m \u2223 n ** have := ae.symm \u25b8 Int.mul_nonpos_of_nonneg_of_nonpos (ofNat_zero_le _) h ** m n : Nat x\u271d : \u2191m \u2223 \u2191n a : Int ae : \u2191n = \u2191m * a h : a \u2264 0 this : \u2191n \u2264 0 \u22a2 m \u2223 n ** rw [Nat.le_antisymm (ofNat_le.1 this) (Nat.zero_le _)] ** m n : Nat x\u271d : \u2191m \u2223 \u2191n a : Int ae : \u2191n = \u2191m * a h : a \u2264 0 this : \u2191n \u2264 0 \u22a2 m \u2223 0 ** apply Nat.dvd_zero ** m n : Nat x\u271d : \u2191m \u2223 \u2191n a : Int ae : \u2191n = \u2191m * a h : 0 \u2264 a \u22a2 m \u2223 n ** match a, eq_ofNat_of_zero_le h with\n| _, \u27e8k, rfl\u27e9 => exact \u27e8k, Int.ofNat.inj ae\u27e9 ** m n : Nat x\u271d : \u2191m \u2223 \u2191n a : Int k : Nat ae : \u2191n = \u2191m * \u2191k h : 0 \u2264 \u2191k \u22a2 m \u2223 n ** exact \u27e8k, Int.ofNat.inj ae\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.bind\u2082_monomial_one ** \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 \u22a2 \u2191(bind\u2082 f) (\u2191(monomial d) 1) = \u2191(monomial d) 1 ** rw [bind\u2082_monomial, f.map_one, one_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_pi_Ioc_toReal ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211d h : a \u2264 b \u22a2 ENNReal.toReal (\u2191\u2191volume (Set.pi univ fun i => Ioc (a i) (b i))) = \u220f i : \u03b9, (b i - a i) ** simp only [volume_pi_Ioc, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.multiset_prod_subset_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 t : Multiset \u03b9 f\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1 hf : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i \u22a2 Multiset.prod (Multiset.map f\u2081 t) \u2286 Multiset.prod (Multiset.map f\u2082 t) ** induction t using Quotient.inductionOn ** case h \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\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1 a\u271d : List \u03b9 hf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i \u22a2 Multiset.prod (Multiset.map f\u2081 (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2286     Multiset.prod (Multiset.map f\u2082 (Quotient.mk (List.isSetoid \u03b9) a\u271d)) ** simp_rw [Multiset.quot_mk_to_coe, Multiset.coe_map, Multiset.coe_prod] ** case h \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\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1 a\u271d : List \u03b9 hf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i \u22a2 List.prod (List.map f\u2081 a\u271d) \u2286 List.prod (List.map f\u2082 a\u271d) ** exact list_prod_subset_list_prod _ _ _ hf ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.intervalIntegral_pos_of_pos_on ** \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 : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f : \u211d \u2192 \u211d a b : \u211d hfi : IntervalIntegrable f volume a b hpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x hab : a < b \u22a2 0 < \u222b (x : \u211d) in a..b, f x ** have hsupp : Ioo a b \u2286 support f \u2229 Ioc a b := fun x hx =>\n  \u27e8mem_support.mpr (hpos x hx).ne', Ioo_subset_Ioc_self hx\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\u271d g : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f : \u211d \u2192 \u211d a b : \u211d hfi : IntervalIntegrable f volume a b hpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x hab : a < b hsupp : Ioo a b \u2286 support f \u2229 Ioc a b \u22a2 0 < \u222b (x : \u211d) in a..b, f x ** have h\u2080 : 0 \u2264\u1d50[volume.restrict (uIoc a b)] f := by\n  rw [EventuallyLE, uIoc_of_le hab.le]\n  refine' ae_restrict_of_ae_eq_of_ae_restrict Ioo_ae_eq_Ioc _\n  exact (ae_restrict_iff' measurableSet_Ioo).mpr (ae_of_all _ fun x hx => (hpos x hx).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 g : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f : \u211d \u2192 \u211d a b : \u211d hfi : IntervalIntegrable f volume a b hpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x hab : a < b hsupp : Ioo a b \u2286 support f \u2229 Ioc a b h\u2080 : 0 \u2264\u1d50[Measure.restrict volume (\u0399 a b)] f \u22a2 0 < \u222b (x : \u211d) in a..b, f x ** rw [integral_pos_iff_support_of_nonneg_ae' h\u2080 hfi] ** \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 : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f : \u211d \u2192 \u211d a b : \u211d hfi : IntervalIntegrable f volume a b hpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x hab : a < b hsupp : Ioo a b \u2286 support f \u2229 Ioc a b h\u2080 : 0 \u2264\u1d50[Measure.restrict volume (\u0399 a b)] f \u22a2 a < b \u2227 0 < \u2191\u2191volume (support f \u2229 Ioc a b) ** exact \u27e8hab, ((Measure.measure_Ioo_pos _).mpr hab).trans_le (measure_mono hsupp)\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\u271d g : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f : \u211d \u2192 \u211d a b : \u211d hfi : IntervalIntegrable f volume a b hpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x hab : a < b hsupp : Ioo a b \u2286 support f \u2229 Ioc a b \u22a2 0 \u2264\u1d50[Measure.restrict volume (\u0399 a b)] f ** rw [EventuallyLE, uIoc_of_le hab.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 g : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f : \u211d \u2192 \u211d a b : \u211d hfi : IntervalIntegrable f volume a b hpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x hab : a < b hsupp : Ioo a b \u2286 support f \u2229 Ioc a b \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioc a b), OfNat.ofNat 0 x \u2264 f x ** refine' ae_restrict_of_ae_eq_of_ae_restrict Ioo_ae_eq_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\u271d g : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f : \u211d \u2192 \u211d a b : \u211d hfi : IntervalIntegrable f volume a b hpos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 < f x hab : a < b hsupp : Ioo a b \u2286 support f \u2229 Ioc a b \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioo a b), OfNat.ofNat 0 x \u2264 f x ** exact (ae_restrict_iff' measurableSet_Ioo).mpr (ae_of_all _ fun x hx => (hpos x hx).le) ** Qed",
        "informal": ""
    },
    {
        "formal": "List.erase_append_right ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List \u03b1 h : \u00aca \u2208 l\u2081 \u22a2 List.erase (l\u2081 ++ l\u2082) a = l\u2081 ++ List.erase l\u2082 a ** rw [erase_eq_eraseP, erase_eq_eraseP, eraseP_append_right] ** case a \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List \u03b1 h : \u00aca \u2208 l\u2081 \u22a2 \u2200 (b : \u03b1), b \u2208 l\u2081 \u2192 \u00acdecide (a = b) = true ** intros b h' h'' ** case a \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List \u03b1 h : \u00aca \u2208 l\u2081 b : \u03b1 h' : b \u2208 l\u2081 h'' : decide (a = b) = true \u22a2 False ** rw [of_decide_eq_true h''] at h ** case a \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List \u03b1 b : \u03b1 h : \u00acb \u2208 l\u2081 h' : b \u2208 l\u2081 h'' : decide (a = b) = true \u22a2 False ** exact h h' ** Qed",
        "informal": ""
    },
    {
        "formal": "List.aemeasurable_prod' ** M : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : MeasurableSpace M inst\u271d : MeasurableMul\u2082 M m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 l : List (\u03b1 \u2192 M) hl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f \u22a2 AEMeasurable (prod l) ** induction' l with f l ihl ** case cons M : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : MeasurableSpace M inst\u271d : MeasurableMul\u2082 M m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 l\u271d : List (\u03b1 \u2192 M) hl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f f : \u03b1 \u2192 M l : List (\u03b1 \u2192 M) ihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l) hl : \u2200 (f_1 : \u03b1 \u2192 M), f_1 \u2208 f :: l \u2192 AEMeasurable f_1 \u22a2 AEMeasurable (prod (f :: l)) ** rw [List.forall_mem_cons] at hl ** case cons M : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : MeasurableSpace M inst\u271d : MeasurableMul\u2082 M m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 l\u271d : List (\u03b1 \u2192 M) hl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f f : \u03b1 \u2192 M l : List (\u03b1 \u2192 M) ihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l) hl : AEMeasurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEMeasurable x \u22a2 AEMeasurable (prod (f :: l)) ** rw [List.prod_cons] ** case cons M : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : MeasurableSpace M inst\u271d : MeasurableMul\u2082 M m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 l\u271d : List (\u03b1 \u2192 M) hl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f f : \u03b1 \u2192 M l : List (\u03b1 \u2192 M) ihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l) hl : AEMeasurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEMeasurable x \u22a2 AEMeasurable (f * prod l) ** exact hl.1.mul (ihl hl.2) ** case nil M : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : MeasurableSpace M inst\u271d : MeasurableMul\u2082 M m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 l : List (\u03b1 \u2192 M) hl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f hl : \u2200 (f : \u03b1 \u2192 M), f \u2208 [] \u2192 AEMeasurable f \u22a2 AEMeasurable (prod []) ** exact aemeasurable_one ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.restrict_sub_eq_restrict_sub_restrict ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s \u22a2 restrict (\u03bc - \u03bd) s = restrict \u03bc s - restrict \u03bd s ** repeat' rw [sub_def] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s \u22a2 restrict (sInf {d | \u03bc \u2264 d + \u03bd}) s = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s} ** have h_nonempty : { d | \u03bc \u2264 d + \u03bd }.Nonempty := \u27e8\u03bc, Measure.le_add_right le_rfl\u27e9 ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u22a2 restrict (sInf {d | \u03bc \u2264 d + \u03bd}) s = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s} ** rw [restrict_sInf_eq_sInf_restrict h_nonempty h_meas_s] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u22a2 sInf ((fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd}) = sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s} ** apply le_antisymm ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s \u22a2 restrict (sInf {d | \u03bc \u2264 d + \u03bd}) s = restrict \u03bc s - restrict \u03bd s ** rw [sub_def] ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u22a2 sInf ((fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd}) \u2264 sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s} ** refine' sInf_le_sInf_of_forall_exists_le _ ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u22a2 \u2200 (x : Measure \u03b1),     x \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u2192 \u2203 y, y \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2227 y \u2264 x ** intro \u03bd' h_\u03bd'_in ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : \u03bd' \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u22a2 \u2203 y, y \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2227 y \u2264 \u03bd' ** rw [mem_setOf_eq] at h_\u03bd'_in ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s \u22a2 \u2203 y, y \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2227 y \u2264 \u03bd' ** refine' \u27e8\u03bd'.restrict s, _, restrict_le_self\u27e9 ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s \u22a2 restrict \u03bd' s \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} ** refine' \u27e8\u03bd' + (\u22a4 : Measure \u03b1).restrict s\u1d9c, _, _\u27e9 ** case a.refine'_1 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s \u22a2 \u03bd' + restrict \u22a4 s\u1d9c \u2208 {d | \u03bc \u2264 d + \u03bd} ** rw [mem_setOf_eq, add_right_comm, Measure.le_iff] ** case a.refine'_1 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s \u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191\u03bc s_1 \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) s_1 ** intro t h_meas_t ** case a.refine'_1 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) t ** repeat' rw [\u2190 measure_inter_add_diff t h_meas_s] ** case a.refine'_1 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (t \\ s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \u2229 s) + \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \\ s) ** refine' add_le_add _ _ ** case a.refine'_1 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (t \\ s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) t ** rw [\u2190 measure_inter_add_diff t h_meas_s] ** case a.refine'_1.refine'_1 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191\u03bc (t \u2229 s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \u2229 s) ** rw [add_apply, add_apply] ** case a.refine'_1.refine'_1 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191\u03bc (t \u2229 s) \u2264 \u2191\u2191\u03bd' (t \u2229 s) + \u2191\u2191\u03bd (t \u2229 s) + \u2191\u2191(restrict \u22a4 s\u1d9c) (t \u2229 s) ** apply le_add_right _ ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191\u03bc (t \u2229 s) \u2264 \u2191\u2191\u03bd' (t \u2229 s) + \u2191\u2191\u03bd (t \u2229 s) ** rw [\u2190 restrict_eq_self \u03bc (inter_subset_right _ _),\n  \u2190 restrict_eq_self \u03bd (inter_subset_right _ _)] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191(restrict \u03bc s) (t \u2229 s) \u2264 \u2191\u2191\u03bd' (t \u2229 s) + \u2191\u2191(restrict \u03bd s) (t \u2229 s) ** apply h_\u03bd'_in _ (h_meas_t.inter h_meas_s) ** case a.refine'_1.refine'_2 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191\u03bc (t \\ s) \u2264 \u2191\u2191(\u03bd' + \u03bd + restrict \u22a4 s\u1d9c) (t \\ s) ** rw [add_apply, restrict_apply (h_meas_t.diff h_meas_s), diff_eq, inter_assoc, inter_self,\n  \u2190 add_apply] ** case a.refine'_1.refine'_2 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191\u03bc (t \u2229 s\u1d9c) \u2264 \u2191\u2191(\u03bd' + \u03bd + \u22a4) (t \u2229 s\u1d9c) ** have h_mu_le_add_top : \u03bc \u2264 \u03bd' + \u03bd + \u22a4 := by simp only [add_top, le_top] ** case a.refine'_1.refine'_2 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t h_mu_le_add_top : \u03bc \u2264 \u03bd' + \u03bd + \u22a4 \u22a2 \u2191\u2191\u03bc (t \u2229 s\u1d9c) \u2264 \u2191\u2191(\u03bd' + \u03bd + \u22a4) (t \u2229 s\u1d9c) ** exact Measure.le_iff'.1 h_mu_le_add_top _ ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u03bc \u2264 \u03bd' + \u03bd + \u22a4 ** simp only [add_top, le_top] ** case a.refine'_2 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s \u22a2 (fun \u03bc => restrict \u03bc s) (\u03bd' + restrict \u22a4 s\u1d9c) = restrict \u03bd' s ** ext1 t h_meas_t ** case a.refine'_2.h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u03bd' : Measure \u03b1 h_\u03bd'_in : restrict \u03bc s \u2264 \u03bd' + restrict \u03bd s t : Set \u03b1 h_meas_t : MeasurableSet t \u22a2 \u2191\u2191((fun \u03bc => restrict \u03bc s) (\u03bd' + restrict \u22a4 s\u1d9c)) t = \u2191\u2191(restrict \u03bd' s) t ** simp [restrict_apply h_meas_t, restrict_apply (h_meas_t.inter h_meas_s), inter_assoc] ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u22a2 sInf {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u2264 sInf ((fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd}) ** refine' sInf_le_sInf_of_forall_exists_le _ ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} \u22a2 \u2200 (x : Measure \u03b1),     x \u2208 (fun \u03bc => restrict \u03bc s) '' {d | \u03bc \u2264 d + \u03bd} \u2192 \u2203 y, y \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s} \u2227 y \u2264 x ** refine' ball_image_iff.2 fun t h_t_in => \u27e8t.restrict s, _, le_rfl\u27e9 ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} t : Measure \u03b1 h_t_in : t \u2208 {d | \u03bc \u2264 d + \u03bd} \u22a2 restrict t s \u2208 {d | restrict \u03bc s \u2264 d + restrict \u03bd s} ** rw [Set.mem_setOf_eq, \u2190 restrict_add] ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_meas_s : MeasurableSet s h_nonempty : Set.Nonempty {d | \u03bc \u2264 d + \u03bd} t : Measure \u03b1 h_t_in : t \u2208 {d | \u03bc \u2264 d + \u03bd} \u22a2 restrict \u03bc s \u2264 restrict (t + \u03bd) s ** exact restrict_mono Subset.rfl h_t_in ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.find?_eq_zoom ** \u03b1 : Type u_1 cut : \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 x\u271d : optParam (Path \u03b1) Path.root \u22a2 find? cut (node c\u271d l\u271d v\u271d r\u271d) = root? (zoom cut (node c\u271d l\u271d v\u271d r\u271d) x\u271d).fst ** unfold find? zoom ** \u03b1 : Type u_1 cut : \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 x\u271d : optParam (Path \u03b1) Path.root \u22a2 (match cut v\u271d with     | Ordering.lt => find? cut l\u271d     | Ordering.gt => find? cut r\u271d     | Ordering.eq => some v\u271d) =     root?       (match cut v\u271d with         | Ordering.lt => zoom cut l\u271d (Path.left c\u271d x\u271d v\u271d r\u271d)         | Ordering.gt => zoom cut r\u271d (Path.right c\u271d l\u271d v\u271d x\u271d)         | Ordering.eq => (node c\u271d l\u271d v\u271d r\u271d, x\u271d)).fst ** split <;> [apply find?_eq_zoom; apply find?_eq_zoom; rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.pair_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 f : \u03b1 \u2192\u2098[\u03bc] \u03b2 g : \u03b1 \u2192\u2098[\u03bc] \u03b3 \u22a2 pair f g = mk (fun x => (\u2191f x, \u2191g x)) (_ : AEStronglyMeasurable (fun x => (\u2191f x, \u2191g x)) \u03bc) ** simp only [\u2190 pair_mk_mk, mk_coeFn, f.aestronglyMeasurable, g.aestronglyMeasurable] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.moment_truncation_eq_intervalIntegral_of_nonneg ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f \u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc ** have M : MeasurableSet (Set.Ioc 0 A) := measurableSet_Ioc ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) \u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc ** have M' : MeasurableSet (Set.Ioc A 0) := measurableSet_Ioc ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) \u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc ** rw [truncation_eq_of_nonneg h'f] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) \u22a2 \u222b (x : \u03b1), (indicator (Set.Ioc 0 A) id \u2218 fun x => f x) x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc ** change \u222b x, (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = _ ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) \u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc ** rcases le_or_lt 0 A with (hA | hA) ** case inl \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : 0 \u2264 A \u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc ** rw [\u2190 integral_map (f := fun z => _ ^ n) hf.aemeasurable, intervalIntegral.integral_of_le hA,\n  \u2190 integral_indicator M] ** case inl \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : 0 \u2264 A \u22a2 \u222b (y : \u211d), indicator (Set.Ioc 0 A) id y ^ n \u2202Measure.map f \u03bc =     \u222b (x : \u211d), indicator (Set.Ioc 0 A) (fun x => x ^ n) x \u2202Measure.map f \u03bc ** simp only [indicator, zero_pow' _ hn, id.def, ite_pow] ** case inl \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : 0 \u2264 A \u22a2 AEStronglyMeasurable (fun z => indicator (Set.Ioc 0 A) id z ^ n) (Measure.map f \u03bc) ** exact ((measurable_id.indicator M).pow_const n).aestronglyMeasurable ** case inr \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 \u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc ** rw [\u2190 integral_map (f := fun z => _ ^ n) hf.aemeasurable, intervalIntegral.integral_of_ge hA.le,\n  \u2190 integral_indicator M'] ** case inr \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 \u22a2 \u222b (y : \u211d), indicator (Set.Ioc 0 A) id y ^ n \u2202Measure.map f \u03bc =     -\u222b (x : \u211d), indicator (Set.Ioc A 0) (fun x => x ^ n) x \u2202Measure.map f \u03bc ** simp only [Set.Ioc_eq_empty_of_le hA.le, zero_pow' _ hn, Set.indicator_empty, integral_zero,\n  zero_eq_neg] ** case inr \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 \u22a2 \u222b (x : \u211d), indicator (Set.Ioc A 0) (fun x => x ^ n) x \u2202Measure.map f \u03bc = 0 ** apply integral_eq_zero_of_ae ** case inr.hf \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 \u22a2 (fun a => indicator (Set.Ioc A 0) (fun x => x ^ n) a) =\u1d50[Measure.map f \u03bc] 0 ** have : \u2200\u1d50 x \u2202Measure.map f \u03bc, (0 : \u211d) \u2264 x :=\n  (ae_map_iff hf.aemeasurable measurableSet_Ici).2 (eventually_of_forall h'f) ** case inr.hf \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 this : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x \u22a2 (fun a => indicator (Set.Ioc A 0) (fun x => x ^ n) a) =\u1d50[Measure.map f \u03bc] 0 ** filter_upwards [this] with x hx ** case h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 this : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x x : \u211d hx : 0 \u2264 x \u22a2 indicator (Set.Ioc A 0) (fun x => x ^ n) x = OfNat.ofNat 0 x ** simp only [indicator, Set.mem_Ioc, Pi.zero_apply, ite_eq_right_iff, and_imp] ** case h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 this : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x x : \u211d hx : 0 \u2264 x \u22a2 A < x \u2192 x \u2264 0 \u2192 x ^ n = 0 ** intro _ h''x ** case h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 this : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x x : \u211d hx : 0 \u2264 x a\u271d : A < x h''x : x \u2264 0 \u22a2 x ^ n = 0 ** have : x = 0 := by linarith ** case h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 this\u271d : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x x : \u211d hx : 0 \u2264 x a\u271d : A < x h''x : x \u2264 0 this : x = 0 \u22a2 x ^ n = 0 ** simp [this, zero_pow' _ hn] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 this : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x x : \u211d hx : 0 \u2264 x a\u271d : A < x h''x : x \u2264 0 \u22a2 x = 0 ** linarith ** case inr \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d n : \u2115 hn : n \u2260 0 h'f : 0 \u2264 f M : MeasurableSet (Set.Ioc 0 A) M' : MeasurableSet (Set.Ioc A 0) hA : A < 0 \u22a2 AEStronglyMeasurable (fun z => indicator (Set.Ioc 0 A) id z ^ n) (Measure.map f \u03bc) ** exact ((measurable_id.indicator M).pow_const n).aestronglyMeasurable ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm ** \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 l : Filter \u03b9 hp_ne_zero : p \u2260 0 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) \u22a2 TendstoInMeasure \u03bc f l g ** by_cases hp_ne_top : p = \u221e ** case pos \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 l : Filter \u03b9 hp_ne_zero : p \u2260 0 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) hp_ne_top : p = \u22a4 \u22a2 TendstoInMeasure \u03bc f l g ** subst hp_ne_top ** case pos \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc hp_ne_zero : \u22a4 \u2260 0 hfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0) \u22a2 TendstoInMeasure \u03bc f l g ** exact tendstoInMeasure_of_tendsto_snorm_top hfg ** case neg \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 l : Filter \u03b9 hp_ne_zero : p \u2260 0 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) hp_ne_top : \u00acp = \u22a4 \u22a2 TendstoInMeasure \u03bc f l g ** exact tendstoInMeasure_of_tendsto_snorm_of_ne_top hp_ne_zero hp_ne_top hf hg hfg ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AnalyticSet.measurablySeparable ** \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 s t : Set \u03b1 hs : AnalyticSet s ht : AnalyticSet t h : Disjoint s t \u22a2 MeasurablySeparable s t ** rw [AnalyticSet] at hs ht ** \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 s t : Set \u03b1 hs : s = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = s ht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t h : Disjoint s t \u22a2 MeasurablySeparable s t ** rcases hs with (rfl | \u27e8f, f_cont, rfl\u27e9) ** case inr.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 t : Set \u03b1 ht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t f : (\u2115 \u2192 \u2115) \u2192 \u03b1 f_cont : Continuous f h : Disjoint (range f) t \u22a2 MeasurablySeparable (range f) t ** rcases ht with (rfl | \u27e8g, g_cont, rfl\u27e9) ** case inr.intro.intro.inr.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 : (\u2115 \u2192 \u2115) \u2192 \u03b1 f_cont : Continuous f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 g_cont : Continuous g h : Disjoint (range f) (range g) \u22a2 MeasurablySeparable (range f) (range g) ** exact measurablySeparable_range_of_disjoint f_cont g_cont h ** case inl \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 t : Set \u03b1 ht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t h : Disjoint \u2205 t \u22a2 MeasurablySeparable \u2205 t ** refine' \u27e8\u2205, Subset.refl _, by simp, MeasurableSet.empty\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 t : Set \u03b1 ht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t h : Disjoint \u2205 t \u22a2 Disjoint t \u2205 ** simp ** case inr.intro.intro.inl \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 : (\u2115 \u2192 \u2115) \u2192 \u03b1 f_cont : Continuous f h : Disjoint (range f) \u2205 \u22a2 MeasurablySeparable (range f) \u2205 ** exact \u27e8univ, subset_univ _, by simp, MeasurableSet.univ\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 : (\u2115 \u2192 \u2115) \u2192 \u03b1 f_cont : Continuous f h : Disjoint (range f) \u2205 \u22a2 Disjoint \u2205 univ ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.add.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 ** intro den num ** 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad \u22a2 Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den ** have ae : ad * g = a.den := had \u25b8 Nat.div_mul_cancel (hg \u25b8 Nat.gcd_dvd_left ..) ** 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den \u22a2 Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den ** have be : bd * g = b.den := hbd \u25b8 Nat.div_mul_cancel (hg \u25b8 Nat.gcd_dvd_right ..) ** 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den \u22a2 Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den ** have hden : den = ad * bd * g := by rw [Nat.mul_assoc, be] ** 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g \u22a2 Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den ** rw [hden, Nat.Coprime.gcd_mul_left_cancel_right] ** case H 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g \u22a2 Nat.Coprime (ad * bd) (Int.natAbs num) ** have cop : ad.Coprime bd := had \u25b8 hbd \u25b8 hg \u25b8\n  Nat.coprime_div_gcd_div_gcd (Nat.gcd_pos_of_pos_left _ a.den_pos) ** case H 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd \u22a2 Nat.Coprime (ad * bd) (Int.natAbs num) ** have H1 (d : Nat) :\n    d.gcd num.natAbs \u2223 a.num.natAbs * bd \u2194 d.gcd num.natAbs \u2223 b.num.natAbs * ad := by\n  have := d.gcd_dvd_right num.natAbs\n  rw [\u2190 Int.ofNat_dvd, Int.dvd_natAbs] at this\n  have := Int.dvd_iff_dvd_of_dvd_add this\n  rwa [\u2190 Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul,\n    \u2190 Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul] at this ** case H 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd H1 :   \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad \u22a2 Nat.Coprime (ad * bd) (Int.natAbs num) ** apply Nat.Coprime.mul ** 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den \u22a2 den = ad * bd * g ** rw [Nat.mul_assoc, be] ** 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd d : Nat \u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad ** have := d.gcd_dvd_right num.natAbs ** 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd d : Nat this : Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs num \u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad ** rw [\u2190 Int.ofNat_dvd, Int.dvd_natAbs] 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd d : Nat this : \u2191(Nat.gcd d (Int.natAbs num)) \u2223 num \u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad ** have := Int.dvd_iff_dvd_of_dvd_add 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd d : Nat this\u271d : \u2191(Nat.gcd d (Int.natAbs num)) \u2223 num this : \u2191(Nat.gcd d (Int.natAbs num)) \u2223 a.num * \u2191bd \u2194 \u2191(Nat.gcd d (Int.natAbs num)) \u2223 b.num * \u2191ad \u22a2 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad ** rwa [\u2190 Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul,\n  \u2190 Int.dvd_natAbs, Int.ofNat_dvd, Int.natAbs_mul] at this ** case H.H1 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd H1 :   \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad \u22a2 Nat.Coprime ad (Int.natAbs num) ** have := (H1 ad).2 <| Nat.dvd_trans (Nat.gcd_dvd_left ..) (Nat.dvd_mul_left ..) ** case H.H1 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd H1 :   \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad this : Nat.gcd ad (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u22a2 Nat.Coprime ad (Int.natAbs num) ** have := (cop.coprime_dvd_left <| Nat.gcd_dvd_left ..).dvd_of_dvd_mul_right this ** case H.H1 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd H1 :   \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad this\u271d : Nat.gcd ad (Int.natAbs num) \u2223 Int.natAbs a.num * bd this : Nat.gcd ad (Int.natAbs num) \u2223 Int.natAbs a.num \u22a2 Nat.Coprime ad (Int.natAbs num) ** exact Nat.eq_one_of_dvd_one <| a.reduced.gcd_eq_one \u25b8 Nat.dvd_gcd this <|\n  Nat.dvd_trans (Nat.gcd_dvd_left ..) (ae \u25b8 Nat.dvd_mul_right ..) ** case H.H2 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd H1 :   \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad \u22a2 Nat.Coprime bd (Int.natAbs num) ** have := (H1 bd).1 <| Nat.dvd_trans (Nat.gcd_dvd_left ..) (Nat.dvd_mul_left ..) ** case H.H2 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd H1 :   \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad this : Nat.gcd bd (Int.natAbs num) \u2223 Int.natAbs b.num * ad \u22a2 Nat.Coprime bd (Int.natAbs num) ** have := (cop.symm.coprime_dvd_left <| Nat.gcd_dvd_left ..).dvd_of_dvd_mul_right this ** case H.H2 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 den : Nat := ad * b.den num : Int := a.num * \u2191bd + b.num * \u2191ad ae : ad * g = a.den be : bd * g = b.den hden : den = ad * bd * g cop : Nat.Coprime ad bd H1 :   \u2200 (d : Nat), Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs a.num * bd \u2194 Nat.gcd d (Int.natAbs num) \u2223 Int.natAbs b.num * ad this\u271d : Nat.gcd bd (Int.natAbs num) \u2223 Int.natAbs b.num * ad this : Nat.gcd bd (Int.natAbs num) \u2223 Int.natAbs b.num \u22a2 Nat.Coprime bd (Int.natAbs num) ** exact Nat.eq_one_of_dvd_one <| b.reduced.gcd_eq_one \u25b8 Nat.dvd_gcd this <|\n  Nat.dvd_trans (Nat.gcd_dvd_left ..) (be \u25b8 Nat.dvd_mul_right ..) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_const_lt_top_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 c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 \u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** have hp : 0 < p.toReal := ENNReal.toReal_pos hp_ne_zero hp_ne_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 : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p \u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** by_cases h\u03bc : \u03bc = 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 c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p h\u03bc : \u00ac\u03bc = 0 \u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** by_cases hc : c = 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 c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p h\u03bc : \u00ac\u03bc = 0 hc : \u00acc = 0 \u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** rw [snorm_const' c hp_ne_zero hp_ne_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 : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p h\u03bc : \u00ac\u03bc = 0 hc : \u00acc = 0 \u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** by_cases h\u03bc_top : \u03bc Set.univ = \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 c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p h\u03bc : \u00ac\u03bc = 0 hc : \u00acc = 0 h\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4 \u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** rw [ENNReal.mul_lt_top_iff] ** 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 c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p h\u03bc : \u00ac\u03bc = 0 hc : \u00acc = 0 h\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4 \u22a2 \u2191\u2016c\u2016\u208a < \u22a4 \u2227 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2228 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) = 0 \u2194     c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** simp only [true_and_iff, one_div, ENNReal.rpow_eq_zero_iff, h\u03bc, false_or_iff, or_false_iff,\n  ENNReal.coe_lt_top, nnnorm_eq_zero, ENNReal.coe_eq_zero,\n  MeasureTheory.Measure.measure_univ_eq_zero, hp, inv_lt_zero, hc, and_false_iff, false_and_iff,\n  _root_.inv_pos, or_self_iff, h\u03bc_top, Ne.lt_top h\u03bc_top, iff_true_iff] ** 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 c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p h\u03bc : \u00ac\u03bc = 0 hc : \u00acc = 0 h\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4 \u22a2 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9 < \u22a4 ** exact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.mpr hp.le) h\u03bc_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 c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p h\u03bc : \u03bc = 0 \u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** simp only [h\u03bc, Measure.coe_zero, Pi.zero_apply, or_true_iff, WithTop.zero_lt_top,\n  snorm_measure_zero] ** 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 c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p h\u03bc : \u00ac\u03bc = 0 hc : c = 0 \u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** simp only [hc, true_or_iff, eq_self_iff_true, WithTop.zero_lt_top, snorm_zero'] ** 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 c : F hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hp : 0 < ENNReal.toReal p h\u03bc : \u00ac\u03bc = 0 hc : \u00acc = 0 h\u03bc_top : \u2191\u2191\u03bc Set.univ = \u22a4 \u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** simp [hc, h\u03bc_top, hp] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.add_self_eq_zero_iff_eq_zero ** n : \u2115 hn : Odd n a : ZMod n \u22a2 a + a = 0 \u2194 a = 0 ** rw [Nat.odd_iff, \u2190 Nat.two_dvd_ne_zero, \u2190 Nat.prime_two.coprime_iff_not_dvd] at hn ** n : \u2115 hn : Nat.Coprime 2 n a : ZMod n \u22a2 a + a = 0 \u2194 a = 0 ** rw [\u2190mul_two, \u2190@Nat.cast_two (ZMod n), \u2190ZMod.coe_unitOfCoprime 2 hn, Units.mul_left_eq_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict ** 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 : OpensMeasurableSpace X hs : IsClosed s \u22a2 LocallyIntegrableOn f s \u2194 LocallyIntegrable f ** refine' \u27e8fun hf x => _, locallyIntegrableOn_of_locallyIntegrable_restrict\u27e9 ** 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 : OpensMeasurableSpace X hs : IsClosed s hf : LocallyIntegrableOn f s x : X \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** by_cases h : x \u2208 s ** case pos 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 : OpensMeasurableSpace X hs : IsClosed s hf : LocallyIntegrableOn f s x : X h : x \u2208 s \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** obtain \u27e8t, ht_nhds, ht_int\u27e9 := hf x h ** case pos.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 : OpensMeasurableSpace X hs : IsClosed s hf : LocallyIntegrableOn f s x : X h : x \u2208 s t : Set X ht_nhds : t \u2208 \ud835\udcdd[s] x ht_int : IntegrableOn f t \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** obtain \u27e8u, hu_o, hu_x, hu_sub\u27e9 := mem_nhdsWithin.mp ht_nhds ** case pos.intro.intro.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 : OpensMeasurableSpace X hs : IsClosed s hf : LocallyIntegrableOn f s x : X h : x \u2208 s t : Set X ht_nhds : t \u2208 \ud835\udcdd[s] x ht_int : IntegrableOn f t u : Set X hu_o : IsOpen u hu_x : x \u2208 u hu_sub : u \u2229 s \u2286 t \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** refine' \u27e8u, hu_o.mem_nhds hu_x, _\u27e9 ** case pos.intro.intro.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 : OpensMeasurableSpace X hs : IsClosed s hf : LocallyIntegrableOn f s x : X h : x \u2208 s t : Set X ht_nhds : t \u2208 \ud835\udcdd[s] x ht_int : IntegrableOn f t u : Set X hu_o : IsOpen u hu_x : x \u2208 u hu_sub : u \u2229 s \u2286 t \u22a2 IntegrableOn f u ** rw [IntegrableOn, restrict_restrict hu_o.measurableSet] ** case pos.intro.intro.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 : OpensMeasurableSpace X hs : IsClosed s hf : LocallyIntegrableOn f s x : X h : x \u2208 s t : Set X ht_nhds : t \u2208 \ud835\udcdd[s] x ht_int : IntegrableOn f t u : Set X hu_o : IsOpen u hu_x : x \u2208 u hu_sub : u \u2229 s \u2286 t \u22a2 Integrable f ** exact ht_int.mono_set hu_sub ** case neg 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 : OpensMeasurableSpace X hs : IsClosed s hf : LocallyIntegrableOn f s x : X h : \u00acx \u2208 s \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** rw [\u2190 isOpen_compl_iff] at hs ** case neg 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 : OpensMeasurableSpace X hs : IsOpen s\u1d9c hf : LocallyIntegrableOn f s x : X h : \u00acx \u2208 s \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** refine' \u27e8s\u1d9c, hs.mem_nhds h, _\u27e9 ** case neg 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 : OpensMeasurableSpace X hs : IsOpen s\u1d9c hf : LocallyIntegrableOn f s x : X h : \u00acx \u2208 s \u22a2 IntegrableOn f s\u1d9c ** rw [IntegrableOn, restrict_restrict, inter_comm, inter_compl_self, \u2190 IntegrableOn] ** case neg 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 : OpensMeasurableSpace X hs : IsOpen s\u1d9c hf : LocallyIntegrableOn f s x : X h : \u00acx \u2208 s \u22a2 IntegrableOn f \u2205  case neg 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 : OpensMeasurableSpace X hs : IsOpen s\u1d9c hf : LocallyIntegrableOn f s x : X h : \u00acx \u2208 s \u22a2 MeasurableSet s\u1d9c ** exacts [integrableOn_empty, hs.measurableSet] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.insert_entries_of_neg ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b : \u03b2 a s\u271d : Finmap \u03b2 s : AList \u03b2 h : \u00aca \u2208 \u27e6s\u27e7 \u22a2 (insert a b \u27e6s\u27e7).entries = { fst := a, snd := b } ::\u2098 \u27e6s\u27e7.entries ** simp [AList.insert_entries_of_neg (mt mem_toFinmap.1 h), -insert_entries] ** Qed",
        "informal": ""
    },
    {
        "formal": "Language.one_add_kstar_mul_self_eq_kstar ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x : List \u03b1 l : Language \u03b1 \u22a2 1 + l\u2217 * l = l\u2217 ** rw [mul_self_kstar_comm, one_add_self_mul_kstar_eq_kstar] ** Qed",
        "informal": ""
    },
    {
        "formal": "Multiset.noncommProd_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 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : Monoid \u03b2 l : List \u03b1 comm : Set.Pairwise {x | x \u2208 \u2191l} Commute \u22a2 noncommProd (\u2191l) comm = List.prod l ** rw [noncommProd] ** 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 l : List \u03b1 comm : Set.Pairwise {x | x \u2208 \u2191l} Commute \u22a2 noncommFold (fun x x_1 => x * x_1) (\u2191l) comm 1 = List.prod l ** simp only [noncommFold_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 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : Monoid \u03b2 l : List \u03b1 comm : Set.Pairwise {x | x \u2208 \u2191l} Commute \u22a2 List.foldr (fun x x_1 => x * x_1) 1 l = List.prod l ** induction' l with hd tl hl ** case nil 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 l : List \u03b1 comm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute comm : Set.Pairwise {x | x \u2208 \u2191[]} Commute \u22a2 List.foldr (fun x x_1 => x * x_1) 1 [] = List.prod [] ** simp ** case cons 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 l : List \u03b1 comm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute hd : \u03b1 tl : List \u03b1 hl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl comm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute \u22a2 List.foldr (fun x x_1 => x * x_1) 1 (hd :: tl) = List.prod (hd :: tl) ** rw [List.prod_cons, List.foldr, hl] ** case cons 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 l : List \u03b1 comm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute hd : \u03b1 tl : List \u03b1 hl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl comm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute \u22a2 Set.Pairwise {x | x \u2208 \u2191tl} Commute ** intro x hx y hy ** case cons 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 l : List \u03b1 comm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute hd : \u03b1 tl : List \u03b1 hl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl comm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute x : \u03b1 hx : x \u2208 {x | x \u2208 \u2191tl} y : \u03b1 hy : y \u2208 {x | x \u2208 \u2191tl} \u22a2 x \u2260 y \u2192 Commute x y ** exact comm (List.mem_cons_of_mem _ hx) (List.mem_cons_of_mem _ hy) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_lintegral_nn_filter_of_le_const ** \u03a9 : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : TopologicalSpace \u03a9 inst\u271d\u00b2 : OpensMeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 inst\u271d\u00b9 : IsCountablyGenerated L \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc fs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650 c : \u211d\u22650 fs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c f : \u03a9 \u2192 \u211d\u22650 fs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9)) \u22a2 Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(fs i) \u03c9) \u2202\u03bc) L (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(f \u03c9) \u2202\u03bc)) ** refine tendsto_lintegral_filter_of_dominated_convergence (fun _ => c)\n  (eventually_of_forall fun i => (ENNReal.continuous_coe.comp (fs i).continuous).measurable) ?_\n  (@lintegral_const_lt_top _ _ \u03bc _ _ (@ENNReal.coe_ne_top c)).ne ?_ ** case refine_1 \u03a9 : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : TopologicalSpace \u03a9 inst\u271d\u00b2 : OpensMeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 inst\u271d\u00b9 : IsCountablyGenerated L \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc fs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650 c : \u211d\u22650 fs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c f : \u03a9 \u2192 \u211d\u22650 fs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9)) \u22a2 \u2200\u1da0 (n : \u03b9) in L, \u2200\u1d50 (a : \u03a9) \u2202\u03bc, \u2191(\u2191(fs n) a) \u2264 (fun x => \u2191c) a ** simpa only [Function.comp_apply, ENNReal.coe_le_coe] using fs_le_const ** case refine_2 \u03a9 : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : TopologicalSpace \u03a9 inst\u271d\u00b2 : OpensMeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 inst\u271d\u00b9 : IsCountablyGenerated L \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc fs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650 c : \u211d\u22650 fs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c f : \u03a9 \u2192 \u211d\u22650 fs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9)) \u22a2 \u2200\u1d50 (a : \u03a9) \u2202\u03bc, Tendsto (fun n => \u2191(\u2191(fs n) a)) L (\ud835\udcdd \u2191(f a)) ** simpa only [Function.comp_apply, ENNReal.tendsto_coe] using fs_lim ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.isClopenable ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : PolishSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : BorelSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 IsClopenable s ** revert s ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : PolishSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : BorelSpace \u03b1 \u22a2 \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 IsClopenable s ** apply MeasurableSet.induction_on_open ** case h_open \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : PolishSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : BorelSpace \u03b1 \u22a2 \u2200 (U : Set \u03b1), IsOpen U \u2192 IsClopenable U ** exact fun u hu => hu.isClopenable ** case h_compl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : PolishSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : BorelSpace \u03b1 \u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 IsClopenable t \u2192 IsClopenable t\u1d9c ** exact fun u _ h'u => h'u.compl ** case h_union \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : PolishSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : BorelSpace \u03b1 \u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),     Pairwise (Disjoint on f) \u2192       (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), IsClopenable (f i)) \u2192 IsClopenable (\u22c3 i, f i) ** exact fun f _ _ hf => IsClopenable.iUnion hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.Heap.findMin_val ** \u03b1 : Type u_1 s\u271d : Heap \u03b1 le : \u03b1 \u2192 \u03b1 \u2192 Bool k : Heap \u03b1 \u2192 Heap \u03b1 res : FindMin \u03b1 r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 \u22a2 (findMin le k (cons r a c s) res).val = headD le res.val (cons r a c s) ** rw [findMin, headD] ** \u03b1 : Type u_1 s\u271d : Heap \u03b1 le : \u03b1 \u2192 \u03b1 \u2192 Bool k : Heap \u03b1 \u2192 Heap \u03b1 res : FindMin \u03b1 r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 \u22a2 (findMin le (k \u2218 cons r a c) s         (if le res.val a = true then res else { before := k, val := a, node := c, next := s })).val =     headD le (if le res.val a = true then res.val else a) s ** split <;> apply findMin_val ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.WF.filterMap ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) \u22a2 WF (Imp.filterMap f m) ** have H1 (l n acc) : filterMap.go f acc l n =\n    (((g\u2081 l).reverse ++ acc.toList).toAssocList, \u27e8n.1 + (g\u2081 l).length\u27e9) := by\n  induction l generalizing n acc with simp [filterMap.go, *]\n  | cons a b l => match f a b with\n    | none => rfl\n    | some c => simp; rw [Nat.add_right_comm]; rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) \u22a2 WF (Imp.filterMap f m) ** let g l := (g\u2081 l).reverse.toAssocList ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) \u22a2 WF (Imp.filterMap f m) ** let M := StateT (ULift Nat) Id ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) \u22a2 WF (Imp.filterMap f m) ** suffices \u2200 bk sz (h : 0 < bk.length),\n  m.buckets.val.mapM (m := M) (filterMap.go f .nil) \u27e80\u27e9 = (\u27e8bk\u27e9, \u27e8sz\u27e9) \u2192\n  WF \u27e8sz, \u27e8bk\u27e9, h\u27e9 from this _ _ _ rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) \u22a2 \u2200 (bk : List (AssocList \u03b1 \u03b3)) (sz : Nat) (h : 0 < List.length bk),     Array.mapM (filterMap.go f AssocList.nil) m.buckets.val { down := 0 } = ({ data := bk }, { down := sz }) \u2192       WF { size := sz, buckets := { val := { data := bk }, property := h } } ** simp [Array.mapM_eq_mapM_data, bind, StateT.bind, H2] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) \u22a2 \u2200 (bk : List (AssocList \u03b1 \u03b3)) (sz : Nat) (h : 0 < List.length bk),     pure           {             data :=               List.map                 (fun l =>                   List.toAssocList                     (List.reverse                       (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                         (AssocList.toList l))))                 m.buckets.val.data }           {             down :=               Nat.sum                 (List.map                   ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   m.buckets.val.data) } =         ({ data := bk }, { down := sz }) \u2192       WF { size := sz, buckets := { val := { data := bk }, property := h } } ** intro bk sz h e' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) bk : List (AssocList \u03b1 \u03b3) sz : Nat h : 0 < List.length bk e' :   pure       {         data :=           List.map             (fun l =>               List.toAssocList                 (List.reverse                   (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                     (AssocList.toList l))))             m.buckets.val.data }       {         down :=           Nat.sum             (List.map               ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                 List.toAssocList                   (List.reverse                     (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                       (AssocList.toList l))))               m.buckets.val.data) } =     ({ data := bk }, { down := sz }) \u22a2 WF { size := sz, buckets := { val := { data := bk }, property := h } } ** cases e' ** case refl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) \u22a2 WF     {       size :=         Nat.sum           (List.map             ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>               List.toAssocList                 (List.reverse                   (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                     (AssocList.toList l))))             m.buckets.val.data),       buckets :=         {           val :=             {               data :=                 List.map                   (fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   m.buckets.val.data },           property := h } } ** refine .mk (by simp [Buckets.size]) \u27e8?_, fun i h => ?_\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) l : AssocList \u03b1 \u03b2 n : ULift Nat acc : AssocList \u03b1 \u03b3 \u22a2 filterMap.go f acc l n =     (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) ** induction l generalizing n acc with simp [filterMap.go, *]\n| cons a b l => match f a b with\n  | none => rfl\n  | some c => simp; rw [Nat.add_right_comm]; rfl ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) a : \u03b1 b : \u03b2 l : AssocList \u03b1 \u03b2 tail_ih\u271d :   \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) n : ULift Nat acc : AssocList \u03b1 \u03b3 \u22a2 (match f a b with     | none =>       (List.toAssocList           (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++             AssocList.toList acc),         {           down :=             n.down +               List.length                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) })     | some c =>       (List.toAssocList           (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++             (a, c) :: AssocList.toList acc),         {           down :=             n.down + 1 +               List.length                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                   (AssocList.toList l)) })) =     (List.toAssocList         (List.reverse             (match Option.map (fun x => (a, x)) (f a b) with             | none =>               List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)             | some b =>               b ::                 List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++           AssocList.toList acc),       {         down :=           n.down +             List.length               (match Option.map (fun x => (a, x)) (f a b) with               | none =>                 List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)               | some b =>                 b ::                   List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                     (AssocList.toList l)) }) ** match f a b with\n| none => rfl\n| some c => simp; rw [Nat.add_right_comm]; rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) a : \u03b1 b : \u03b2 l : AssocList \u03b1 \u03b2 tail_ih\u271d :   \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) n : ULift Nat acc : AssocList \u03b1 \u03b3 \u22a2 (match none with     | none =>       (List.toAssocList           (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++             AssocList.toList acc),         {           down :=             n.down +               List.length                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) })     | some c =>       (List.toAssocList           (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++             (a, c) :: AssocList.toList acc),         {           down :=             n.down + 1 +               List.length                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                   (AssocList.toList l)) })) =     (List.toAssocList         (List.reverse             (match Option.map (fun x => (a, x)) none with             | none =>               List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)             | some b =>               b ::                 List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++           AssocList.toList acc),       {         down :=           n.down +             List.length               (match Option.map (fun x => (a, x)) none with               | none =>                 List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)               | some b =>                 b ::                   List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                     (AssocList.toList l)) }) ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) a : \u03b1 b : \u03b2 l : AssocList \u03b1 \u03b2 tail_ih\u271d :   \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) n : ULift Nat acc : AssocList \u03b1 \u03b3 c : \u03b3 \u22a2 (match some c with     | none =>       (List.toAssocList           (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++             AssocList.toList acc),         {           down :=             n.down +               List.length                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) })     | some c =>       (List.toAssocList           (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++             (a, c) :: AssocList.toList acc),         {           down :=             n.down + 1 +               List.length                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                   (AssocList.toList l)) })) =     (List.toAssocList         (List.reverse             (match Option.map (fun x => (a, x)) (some c) with             | none =>               List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)             | some b =>               b ::                 List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ++           AssocList.toList acc),       {         down :=           n.down +             List.length               (match Option.map (fun x => (a, x)) (some c) with               | none =>                 List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)               | some b =>                 b ::                   List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                     (AssocList.toList l)) }) ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) a : \u03b1 b : \u03b2 l : AssocList \u03b1 \u03b2 tail_ih\u271d :   \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) n : ULift Nat acc : AssocList \u03b1 \u03b3 c : \u03b3 \u22a2 {       down :=         n.down + 1 +           List.length             (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) } =     {       down :=         n.down +           Nat.succ             (List.length               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) } ** rw [Nat.add_right_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) a : \u03b1 b : \u03b2 l : AssocList \u03b1 \u03b2 tail_ih\u271d :   \u2200 (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) n : ULift Nat acc : AssocList \u03b1 \u03b3 c : \u03b3 \u22a2 {       down :=         n.down +             List.length               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) +           1 } =     {       down :=         n.down +           Nat.succ             (List.length               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) } ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id l : List (AssocList \u03b1 \u03b2) n : ULift Nat \u22a2 List.mapM (filterMap.go f AssocList.nil) l n =     (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) ** induction l generalizing n with\n| nil => rfl\n| cons l L IH => simp [bind, StateT.bind, IH, H1, Nat.add_assoc]; rfl ** case nil \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id n : ULift Nat \u22a2 List.mapM (filterMap.go f AssocList.nil) [] n =     (List.map g [], { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g [])) }) ** rfl ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id l : AssocList \u03b1 \u03b2 L : List (AssocList \u03b1 \u03b2) IH :   \u2200 (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) L n =       (List.map g L, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g L)) }) n : ULift Nat \u22a2 List.mapM (filterMap.go f AssocList.nil) (l :: L) n =     (List.map g (l :: L),       { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g (l :: L))) }) ** simp [bind, StateT.bind, IH, H1, Nat.add_assoc] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id l : AssocList \u03b1 \u03b2 L : List (AssocList \u03b1 \u03b2) IH :   \u2200 (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) L n =       (List.map g L, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g L)) }) n : ULift Nat \u22a2 pure       (List.toAssocList           (List.reverse             (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) ::         List.map           (fun l =>             List.toAssocList               (List.reverse                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))           L)       {         down :=           n.down +             (List.length                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) +               Nat.sum                 (List.map                   ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   L)) } =     (List.toAssocList           (List.reverse             (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) ::         List.map           (fun l =>             List.toAssocList               (List.reverse                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))           L,       {         down :=           n.down +             (List.length                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) +               Nat.sum                 (List.map                   ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   L)) }) ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) l : List (\u03b1 \u00d7 \u03b2) \u22a2 List.Sublist     (List.map (fun a => a.fst)       (List.filterMap         (fun x =>           match x with           | (a, b) => Option.map (fun x => (a, x)) (f a b))         l))     (List.map (fun x => x.fst) l) ** induction l with\n| nil => exact .slnil\n| cons a l ih =>\n  simp; exact match f a.1 a.2 with\n  | none => .cons _ ih\n  | some b => .cons\u2082 _ ih ** case nil \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) \u22a2 List.Sublist     (List.map (fun a => a.fst)       (List.filterMap         (fun x =>           match x with           | (a, b) => Option.map (fun x => (a, x)) (f a b))         []))     (List.map (fun x => x.fst) []) ** exact .slnil ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Sublist     (List.map (fun a => a.fst)       (List.filterMap         (fun x =>           match x with           | (a, b) => Option.map (fun x => (a, x)) (f a b))         l))     (List.map (fun x => x.fst) l) \u22a2 List.Sublist     (List.map (fun a => a.fst)       (List.filterMap         (fun x =>           match x with           | (a, b) => Option.map (fun x => (a, x)) (f a b))         (a :: l)))     (List.map (fun x => x.fst) (a :: l)) ** simp ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) a : \u03b1 \u00d7 \u03b2 l : List (\u03b1 \u00d7 \u03b2) ih :   List.Sublist     (List.map (fun a => a.fst)       (List.filterMap         (fun x =>           match x with           | (a, b) => Option.map (fun x => (a, x)) (f a b))         l))     (List.map (fun x => x.fst) l) \u22a2 List.Sublist     (List.map (fun a => a.fst)       (match Option.map (fun x => (a.fst, x)) (f a.fst a.snd) with       | none => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) l       | some b => b :: List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) l))     (a.fst :: List.map (fun x => x.fst) l) ** exact match f a.1 a.2 with\n| none => .cons _ ih\n| some b => .cons\u2082 _ ih ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) \u22a2 {         size :=           Nat.sum             (List.map               ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                 List.toAssocList                   (List.reverse                     (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                       (AssocList.toList l))))               m.buckets.val.data),         buckets :=           {             val :=               {                 data :=                   List.map                     (fun l =>                       List.toAssocList                         (List.reverse                           (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                             (AssocList.toList l))))                     m.buckets.val.data },             property := h } }.size =     Buckets.size       {           size :=             Nat.sum               (List.map                 ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                   List.toAssocList                     (List.reverse                       (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                         (AssocList.toList l))))                 m.buckets.val.data),           buckets :=             {               val :=                 {                   data :=                     List.map                       (fun l =>                         List.toAssocList                           (List.reverse                             (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                               (AssocList.toList l))))                       m.buckets.val.data },               property := h } }.buckets ** simp [Buckets.size] ** case refl.refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) \u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b3),     bucket \u2208         {                 size :=                   Nat.sum                     (List.map                       ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                         List.toAssocList                           (List.reverse                             (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                               (AssocList.toList l))))                       m.buckets.val.data),                 buckets :=                   {                     val :=                       {                         data :=                           List.map                             (fun l =>                               List.toAssocList                                 (List.reverse                                   (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                                     (AssocList.toList l))))                             m.buckets.val.data },                     property := h } }.buckets.val.data \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) ** simp only [List.forall_mem_map_iff, List.toAssocList_toList] ** case refl.refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) \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)         (List.reverse           (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList j))) ** refine fun l h => (List.pairwise_reverse.2 ?_).imp (mt PartialEquivBEq.symm) ** case refl.refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h\u271d :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) inst\u271d\u00b9 : LawfulHashable \u03b1 inst\u271d : PartialEquivBEq \u03b1 l : AssocList \u03b1 \u03b2 h : l \u2208 m.buckets.val.data \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l)) ** have := H.out.2.1 _ h ** case refl.refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h\u271d :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) inst\u271d\u00b9 : LawfulHashable \u03b1 inst\u271d : PartialEquivBEq \u03b1 l : AssocList \u03b1 \u03b2 h : l \u2208 m.buckets.val.data this : List.Pairwise (fun x x_1 => \u00ac(x == x_1) = true) (List.map (fun a => a.fst) (AssocList.toList l)) \u22a2 List.Pairwise (fun x x_1 => \u00ac(x == x_1) = true)     (List.map (fun a => a.fst)       (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))) ** exact this.sublist (H3 l.toList) ** case refl.refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h\u271d :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) i : Nat h :   i <     Array.size       {             size :=               Nat.sum                 (List.map                   ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   m.buckets.val.data),             buckets :=               {                 val :=                   {                     data :=                       List.map                         (fun l =>                           List.toAssocList                             (List.reverse                               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                                 (AssocList.toList l))))                         m.buckets.val.data },                 property := h\u271d } }.buckets.val \u22a2 AssocList.All     (fun k x =>       USize.toNat           (UInt64.toUSize (hash k) %             Array.size               {                     size :=                       Nat.sum                         (List.map                           ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                             List.toAssocList                               (List.reverse                                 (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                                   (AssocList.toList l))))                           m.buckets.val.data),                     buckets :=                       {                         val :=                           {                             data :=                               List.map                                 (fun l =>                                   List.toAssocList                                     (List.reverse                                       (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                                         (AssocList.toList l))))                                 m.buckets.val.data },                         property := h\u271d } }.buckets.val) =         i)     {             size :=               Nat.sum                 (List.map                   ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   m.buckets.val.data),             buckets :=               {                 val :=                   {                     data :=                       List.map                         (fun l =>                           List.toAssocList                             (List.reverse                               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                                 (AssocList.toList l))))                         m.buckets.val.data },                 property := h\u271d } }.buckets.val[i] ** simp [Array.getElem_eq_data_get] at h \u22a2 ** case refl.refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h\u271d\u00b9 :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) i : Nat h\u271d :   i <     Array.size       {             size :=               Nat.sum                 (List.map                   ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   m.buckets.val.data),             buckets :=               {                 val :=                   {                     data :=                       List.map                         (fun l =>                           List.toAssocList                             (List.reverse                               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                                 (AssocList.toList l))))                         m.buckets.val.data },                 property := h\u271d\u00b9 } }.buckets.val h : i < List.length m.buckets.val.data \u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % List.length m.buckets.val.data) = i)     (List.toAssocList       (List.reverse         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))           (AssocList.toList             (List.get m.buckets.val.data               { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) }))))) ** have := H.out.2.2 _ h ** case refl.refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h\u271d\u00b9 :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) i : Nat h\u271d :   i <     Array.size       {             size :=               Nat.sum                 (List.map                   ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   m.buckets.val.data),             buckets :=               {                 val :=                   {                     data :=                       List.map                         (fun l =>                           List.toAssocList                             (List.reverse                               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                                 (AssocList.toList l))))                         m.buckets.val.data },                 property := h\u271d\u00b9 } }.buckets.val h : i < List.length m.buckets.val.data this : AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size m.buckets.val) = i) m.buckets.val[i] \u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % List.length m.buckets.val.data) = i)     (List.toAssocList       (List.reverse         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))           (AssocList.toList             (List.get m.buckets.val.data               { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) }))))) ** simp [AssocList.All] at this \u22a2 ** case refl.refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h\u271d\u00b9 :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) i : Nat h\u271d :   i <     Array.size       {             size :=               Nat.sum                 (List.map                   ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   m.buckets.val.data),             buckets :=               {                 val :=                   {                     data :=                       List.map                         (fun l =>                           List.toAssocList                             (List.reverse                               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                                 (AssocList.toList l))))                         m.buckets.val.data },                 property := h\u271d\u00b9 } }.buckets.val h : i < List.length m.buckets.val.data this :   \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 \u22a2 \u2200 (a : \u03b1 \u00d7 \u03b3) (x : \u03b1 \u00d7 \u03b2),     x \u2208         AssocList.toList           (List.get m.buckets.val.data             { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) }) \u2192       \u2200 (x_1 : \u03b3),         f x.fst x.snd = some x_1 \u2192           (x.fst, x_1) = a \u2192 USize.toNat (UInt64.toUSize (hash a.fst) % List.length m.buckets.val.data) = i ** rintro _ _ h' _ _ rfl ** case refl.refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m g\u2081 : AssocList \u03b1 \u03b2 \u2192 List (\u03b1 \u00d7 \u03b3) :=   fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l) H1 :   \u2200 (l : AssocList \u03b1 \u03b2) (n : ULift Nat) (acc : AssocList \u03b1 \u03b3),     filterMap.go f acc l n =       (List.toAssocList (List.reverse (g\u2081 l) ++ AssocList.toList acc), { down := n.down + List.length (g\u2081 l) }) g : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b3 := fun l => List.toAssocList (List.reverse (g\u2081 l)) M : Type (max u_3 u_1) \u2192 Type (max u_3 u_1) := StateT (ULift Nat) Id H2 :   \u2200 (l : List (AssocList \u03b1 \u03b2)) (n : ULift Nat),     List.mapM (filterMap.go f AssocList.nil) l n =       (List.map g l, { down := n.down + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map g l)) }) H3 :   \u2200 (l : List (\u03b1 \u00d7 \u03b2)),     List.Sublist       (List.map (fun a => a.fst)         (List.filterMap           (fun x =>             match x with             | (a, b) => Option.map (fun x => (a, x)) (f a b))           l))       (List.map (fun x => x.fst) l) h\u271d\u00b9 :   0 <     List.length       (List.map         (fun l =>           List.toAssocList             (List.reverse               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) (AssocList.toList l))))         m.buckets.val.data) i : Nat h\u271d :   i <     Array.size       {             size :=               Nat.sum                 (List.map                   ((fun x => List.length (AssocList.toList x)) \u2218 fun l =>                     List.toAssocList                       (List.reverse                         (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                           (AssocList.toList l))))                   m.buckets.val.data),             buckets :=               {                 val :=                   {                     data :=                       List.map                         (fun l =>                           List.toAssocList                             (List.reverse                               (List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd))                                 (AssocList.toList l))))                         m.buckets.val.data },                 property := h\u271d\u00b9 } }.buckets.val h : i < List.length m.buckets.val.data this :   \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 x\u271d\u00b9 : \u03b1 \u00d7 \u03b2 h' :   x\u271d\u00b9 \u2208     AssocList.toList       (List.get m.buckets.val.data         { val := i, isLt := (_ : { val := i, isLt := h\u271d }.val < List.length m.buckets.val.data) }) x\u271d : \u03b3 a\u271d : f x\u271d\u00b9.fst x\u271d\u00b9.snd = some x\u271d \u22a2 USize.toNat (UInt64.toUSize (hash (x\u271d\u00b9.fst, x\u271d).fst) % List.length m.buckets.val.data) = i ** exact this _ h' ** Qed",
        "informal": ""
    },
    {
        "formal": "String.firstDiffPos_eq ** a b : String \u22a2 firstDiffPos a b = { byteIdx := utf8Len (List.takeWhile\u2082 (fun x x_1 => decide (x = x_1)) a.data b.data).fst } ** simpa [firstDiffPos] using\n  firstDiffPos_loop_eq [] [] a.1 b.1 ((utf8Len a.1).min (utf8Len b.1)) 0 rfl rfl (by simp) ** a b : String \u22a2 Nat.min (utf8Len a.data) (utf8Len b.data) = min (utf8Len [] + utf8Len a.data) (utf8Len [] + utf8Len b.data) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PartrecToTM2.trNormal_respects ** c : Code k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal c k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal c (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** induction c generalizing k v s ** case zero' k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.zero' k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.zero' (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082  case succ k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.succ k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.succ (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082  case tail k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.tail k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082  case cons a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case case a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case fix a\u271d : Code a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** case zero' => refine' \u27e8_, \u27e8s, rfl\u27e9, TransGen.single _\u27e9; simp ** case succ k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.succ k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.succ (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082  case tail k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.tail k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082  case cons a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case case a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case fix a\u271d : Code a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** case succ => refine' \u27e8_, \u27e8none, rfl\u27e9, head_main_ok.trans succ_ok\u27e9 ** case tail k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.tail k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082  case cons a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case case a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case fix a\u271d : Code a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** case tail =>\n  let o : Option \u0393' := List.casesOn v none fun _ _ => some \u0393'.cons\n  refine' \u27e8_, \u27e8o, rfl\u27e9, _\u27e9; convert clear_ok _ using 2; simp; rfl; swap\n  refine' splitAtPred_eq _ _ (trNat v.headI) _ _ (trNat_natEnd _) _\n  cases v <;> simp ** case cons a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.cons a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.cons a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case case a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case fix a\u271d : Code a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** case\n  cons f fs IHf _ =>\n  obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHf (Cont.cons\u2081 fs v k) v none\n  refine' \u27e8c, h\u2081, TransGen.head rfl <| (move_ok (by decide) (splitAtPred_false _)).trans _\u27e9\n  simp only [TM2.step, Option.mem_def, elim_stack, elim_update_stack, elim_update_main, ne_eq,\n    Function.update_noteq, elim_main, elim_rev, elim_update_rev]\n  refine' (copy_ok _ none [] (trList v).reverse _ _).trans _\n  convert h\u2082 using 2\n  simp [List.reverseAux_eq, trContStack] ** case comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.comp a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.comp a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case case a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d\u00b9 k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d\u00b9 (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.case a\u271d\u00b9 a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case a\u271d\u00b9 a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082  case fix a\u271d : Code a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** case comp f _ _ IHg => exact IHg (Cont.comp f k) v s ** case fix a\u271d : Code a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.fix a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.fix a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** case fix f IH => apply IH ** k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.zero' k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.zero' (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** refine' \u27e8_, \u27e8s, rfl\u27e9, TransGen.single _\u27e9 ** k : Cont v : List \u2115 s : Option \u0393' \u22a2 { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList (0 :: v)) [] [] (trContStack k) } \u2208     TM2.step tr { l := some (trNormal Code.zero' (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } ** simp ** k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.succ k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.succ (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** refine' \u27e8_, \u27e8none, rfl\u27e9, head_main_ok.trans succ_ok\u27e9 ** k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.tail k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** let o : Option \u0393' := List.casesOn v none fun _ _ => some \u0393'.cons ** k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 \u2203 b\u2082,     TrCfg (stepNormal Code.tail k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** refine' \u27e8_, \u27e8o, rfl\u27e9, _\u27e9 ** k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 Reaches\u2081 (TM2.step tr)     { l := some (trNormal Code.tail (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }     { l := some (\u039b'.ret (trCont k)), var := o, stk := elim (trList (List.tail v)) [] [] (trContStack k) } ** convert clear_ok _ using 2 ** case h.e'_4.h.e'_7 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 elim (trList (List.tail v)) [] [] (trContStack k) = update (elim (trList v) [] [] (trContStack k)) main ?convert_7  case convert_5 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 List \u0393'  case convert_7 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 List \u0393'  case convert_9 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, ?convert_7) ** simp ** case h.e'_4.h.e'_7 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 elim (trList (List.tail v)) [] [] (trContStack k) = elim ?convert_7 [] [] (trContStack k)  case convert_5 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 List \u0393'  case convert_7 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 List \u0393'  case convert_9 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, ?convert_7) ** rfl ** case convert_5 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 List \u0393'  case convert_9 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, trList (List.tail v)) ** swap ** case convert_9 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 splitAtPred natEnd (elim (trList v) [] [] (trContStack k) main) = (?convert_5, o, trList (List.tail v))  case convert_5 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 List \u0393' ** refine' splitAtPred_eq _ _ (trNat v.headI) _ _ (trNat_natEnd _) _ ** case convert_9 k : Cont v : List \u2115 s : Option \u0393' o : Option \u0393' := List.casesOn v none fun x x => some \u0393'.cons \u22a2 Option.elim' (elim (trList v) [] [] (trContStack k) main = trNat (List.headI v) \u2227 trList (List.tail v) = [])     (fun a =>       natEnd a = true \u2227 elim (trList v) [] [] (trContStack k) main = trNat (List.headI v) ++ a :: trList (List.tail v))     o ** cases v <;> simp ** f fs : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal fs k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.cons f fs) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.cons f fs) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHf (Cont.cons\u2081 fs v k) v none ** case intro.intro f fs : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal fs k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' c : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,       stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }     c \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.cons f fs) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.cons f fs) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** refine' \u27e8c, h\u2081, TransGen.head rfl <| (move_ok (by decide) (splitAtPred_false _)).trans _\u27e9 ** case intro.intro f fs : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal fs k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' c : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,       stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }     c \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     { l := some (\u039b'.copy (trNormal f (Cont'.cons\u2081 fs (trCont k)))), var := none,       stk :=         update           (update             (update (elim (trList v) [] [] (trContStack k)) stack               ((fun s => Option.iget ((fun x => some \u0393'.cons\u2097) s)) s :: elim (trList v) [] [] (trContStack k) stack))             main [])           rev           (List.reverseAux             (update (elim (trList v) [] [] (trContStack k)) stack               ((fun s => Option.iget ((fun x => some \u0393'.cons\u2097) s)) s :: elim (trList v) [] [] (trContStack k) stack)               main)             (update (elim (trList v) [] [] (trContStack k)) stack               ((fun s => Option.iget ((fun x => some \u0393'.cons\u2097) s)) s :: elim (trList v) [] [] (trContStack k) stack)               rev)) }     c ** simp only [TM2.step, Option.mem_def, elim_stack, elim_update_stack, elim_update_main, ne_eq,\n  Function.update_noteq, elim_main, elim_rev, elim_update_rev] ** case intro.intro f fs : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal fs k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' c : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,       stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }     c \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     { l := some (\u039b'.copy (trNormal f (Cont'.cons\u2081 fs (trCont k)))), var := none,       stk := elim [] (List.reverseAux (trList v) []) [] (Option.iget (some \u0393'.cons\u2097) :: trContStack k) }     c ** refine' (copy_ok _ none [] (trList v).reverse _ _).trans _ ** case intro.intro f fs : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal fs k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' c : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,       stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }     c \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     { l := some (trNormal f (Cont'.cons\u2081 fs (trCont k))), var := none,       stk :=         elim (List.reverseAux (List.reverse (trList v)) []) [] []           (List.reverseAux (List.reverse (trList v)) (Option.iget (some \u0393'.cons\u2097) :: trContStack k)) }     c ** convert h\u2082 using 2 ** case h.e'_1.h.e'_7 f fs : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal fs k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' c : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,       stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }     c \u22a2 elim (List.reverseAux (List.reverse (trList v)) []) [] []       (List.reverseAux (List.reverse (trList v)) (Option.iget (some \u0393'.cons\u2097) :: trContStack k)) =     elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) ** simp [List.reverseAux_eq, trContStack] ** f fs : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal fs k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal fs (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' c : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.cons\u2081 fs v k) v) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.cons\u2081 fs v k))), var := none,       stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs v k)) }     c \u22a2 main \u2260 rev ** decide ** f a\u271d : Code a_ih\u271d :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal a\u271d k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal a\u271d (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.comp f a\u271d) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.comp f a\u271d) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** exact IHg (Cont.comp f k) v s ** f g : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal g k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.case f g) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** rw [stepNormal] ** f g : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal g k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg         ((fun k v =>             Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v))           k v)         b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** simp only ** f g : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal g k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** obtain \u27e8s', h\u27e9 := pred_ok _ _ s v _ _ ** case intro f g : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal g k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s s' : Option \u0393' h :   Reaches\u2081 (TM2.step tr)     { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }     (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }       (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })       (List.headI v)) \u22a2 \u2203 b\u2082,     TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** revert h ** case intro f g : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal g k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s s' : Option \u0393' \u22a2 Reaches\u2081 (TM2.step tr)       { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }       (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }         (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })         (List.headI v)) \u2192     \u2203 b\u2082,       TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** cases' v.headI with n <;> intro h ** case intro.zero f g : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal g k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s s' : Option \u0393' h :   Reaches\u2081 (TM2.step tr)     { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }     (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }       (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })       Nat.zero) \u22a2 \u2203 b\u2082,     TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) Nat.zero) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHf k _ s' ** case intro.zero.intro.intro f g : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal g k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s s' : Option \u0393' h :   Reaches\u2081 (TM2.step tr)     { l := some (\u039b'.pred ?m.552562 ?m.552563), var := s, stk := elim (trList v) [] ?m.552564 ?m.552565 }     (Nat.rec { l := some ?m.552562, var := s', stk := elim (trList (List.tail v)) [] ?m.552564 ?m.552565 }       (fun n x => { l := some ?m.552563, var := s', stk := elim (trList (n :: List.tail v)) [] ?m.552564 ?m.552565 })       Nat.zero) c : Cfg' h\u2081 : TrCfg (stepNormal f k ?m.552695) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont k)), var := s', stk := elim (trList ?m.552695) [] [] (trContStack k) } c \u22a2 \u2203 b\u2082,     TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) Nat.zero) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** exact \u27e8_, h\u2081, h.trans h\u2082\u27e9 ** case intro.succ f g : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal g k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s s' : Option \u0393' n : \u2115 h :   Reaches\u2081 (TM2.step tr)     { l := some (\u039b'.pred (trNormal f (trCont k)) (trNormal g (trCont k))), var := s,       stk := elim (trList v) [] [] (trContStack k) }     (Nat.rec { l := some (trNormal f (trCont k)), var := s', stk := elim (trList (List.tail v)) [] [] (trContStack k) }       (fun n x =>         { l := some (trNormal g (trCont k)), var := s', stk := elim (trList (n :: List.tail v)) [] [] (trContStack k) })       (Nat.succ n)) \u22a2 \u2203 b\u2082,     TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (Nat.succ n)) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** obtain \u27e8c, h\u2081, h\u2082\u27e9 := IHg k _ s' ** case intro.succ.intro.intro f g : Code IHf :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 IHg :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal g k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s s' : Option \u0393' n : \u2115 h :   Reaches\u2081 (TM2.step tr)     { l := some (\u039b'.pred (trNormal f (trCont k)) (trNormal g (trCont k))), var := s,       stk := elim (trList v) [] [] (trContStack k) }     (Nat.rec { l := some (trNormal f (trCont k)), var := s', stk := elim (trList (List.tail v)) [] [] (trContStack k) }       (fun n x =>         { l := some (trNormal g (trCont k)), var := s', stk := elim (trList (n :: List.tail v)) [] [] (trContStack k) })       (Nat.succ n)) c : Cfg' h\u2081 : TrCfg (stepNormal g k ?m.552819) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal g (trCont k)), var := s', stk := elim (trList ?m.552819) [] [] (trContStack k) } c \u22a2 \u2203 b\u2082,     TrCfg (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (Nat.succ n)) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.case f g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** exact \u27e8_, h\u2081, h.trans h\u2082\u27e9 ** f : Code IH :   \u2200 (k : Cont) (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepNormal f k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal (Code.fix f) k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (trNormal (Code.fix f) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b\u2082 ** apply IH ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.assert_pos ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Prop f : p \u2192 Part \u03b1 h : p \u22a2 assert p f = f h ** dsimp [assert] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Prop f : p \u2192 Part \u03b1 h : p \u22a2 { Dom := \u2203 h, (f h).Dom, get := fun ha => get (f (_ : p)) (_ : (f (_ : p)).Dom) } = f h ** cases h' : f h ** case mk \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Prop f : p \u2192 Part \u03b1 h : p Dom\u271d : Prop get\u271d : Dom\u271d \u2192 \u03b1 h' : f h = { Dom := Dom\u271d, get := get\u271d } \u22a2 { Dom := \u2203 h, (f h).Dom, get := fun ha => get (f (_ : p)) (_ : (f (_ : p)).Dom) } = { Dom := Dom\u271d, get := get\u271d } ** simp only [h', mk.injEq, h, exists_prop_of_true, true_and] ** case mk \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Prop f : p \u2192 Part \u03b1 h : p Dom\u271d : Prop get\u271d : Dom\u271d \u2192 \u03b1 h' : f h = { Dom := Dom\u271d, get := get\u271d } \u22a2 HEq (fun ha => get\u271d (_ : { Dom := Dom\u271d, get := get\u271d }.Dom)) get\u271d ** apply Function.hfunext ** case mk.h\u03b1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Prop f : p \u2192 Part \u03b1 h : p Dom\u271d : Prop get\u271d : Dom\u271d \u2192 \u03b1 h' : f h = { Dom := Dom\u271d, get := get\u271d } \u22a2 (\u2203 h, (f h).Dom) = Dom\u271d ** simp only [h, h', exists_prop_of_true] ** case mk.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Prop f : p \u2192 Part \u03b1 h : p Dom\u271d : Prop get\u271d : Dom\u271d \u2192 \u03b1 h' : f h = { Dom := Dom\u271d, get := get\u271d } \u22a2 \u2200 (a : \u2203 h, (f h).Dom) (a' : Dom\u271d), HEq a a' \u2192 HEq (get\u271d (_ : { Dom := Dom\u271d, get := get\u271d }.Dom)) (get\u271d a') ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mul_le_addHaar_image_of_lt_det ** 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \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 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0 ** rcases eq_or_lt_of_le (zero_le m) with (rfl | mpos) ** case a.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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0 ** have hA : A.det \u2260 0 := by\n  intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm ** case a.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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0 ** let B := A.toContinuousLinearEquivOfDetNeZero hA ** case a.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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0 ** have I : ENNReal.ofReal |(B.symm : E \u2192L[\u211d] E).det| < (m\u207b\u00b9 : \u211d\u22650) := by\n  simp only [ENNReal.ofReal, abs_inv, Real.toNNReal_inv, ContinuousLinearEquiv.det_coe_symm,\n    ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero, ENNReal.coe_lt_coe] at hm \u22a2\n  exact NNReal.inv_lt_inv mpos.ne' hm ** case a.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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0 ** obtain \u27e8\u03b4\u2080, \u03b4\u2080pos, h\u03b4\u2080\u27e9 :\n  \u2203 \u03b4 : \u211d\u22650,\n    0 < \u03b4 \u2227\n      \u2200 (t : Set E) (g : E \u2192 E),\n        ApproximatesLinearOn g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t := by\n  have :\n    \u2200\u1da0 \u03b4 : \u211d\u22650 in \ud835\udcdd[>] 0,\n      \u2200 (t : Set E) (g : E \u2192 E),\n        ApproximatesLinearOn g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t :=\n    addHaar_image_le_mul_of_det_lt \u03bc B.symm I\n  rcases (this.and self_mem_nhdsWithin).exists with \u27e8\u03b4\u2080, h, h'\u27e9\n  exact \u27e8\u03b4\u2080, h', h\u27e9 ** case a.inr.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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0 ** filter_upwards [L1, L2] ** 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u22a2 \u2200 (a : \u211d\u22650),     Subsingleton E \u2228 a < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 \u2192       \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - a)\u207b\u00b9 * a < \u03b4\u2080 \u2192         \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) ** intro \u03b4 h1\u03b4 h2\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 \u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) ** have hf' : ApproximatesLinearOn f (B : E \u2192L[\u211d] E) s \u03b4 := by convert 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 hf' : ApproximatesLinearOn f (\u2191B) s \u03b4 \u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) ** let F := hf'.toLocalEquiv h1\u03b4 ** 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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 hf' : ApproximatesLinearOn f (\u2191B) s \u03b4 F : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4 \u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) ** suffices H : \u03bc (F.symm '' F.target) \u2264 (m\u207b\u00b9 : \u211d\u22650) * \u03bc F.target ** 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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 hf' : ApproximatesLinearOn f (\u2191B) s \u03b4 F : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4 \u22a2 \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target ** exact h\u03b4\u2080 _ _ ((hf'.to_inv h1\u03b4).mono_num h2\u03b4.le) ** case a.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 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 hm : \u21910 < ENNReal.ofReal |ContinuousLinearMap.det A| \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u21910 * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0 ** apply eventually_of_forall ** case a.inl.hp 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 hm : \u21910 < ENNReal.ofReal |ContinuousLinearMap.det A| \u22a2 \u2200 (x : \u211d\u22650) (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s x \u2192 \u21910 * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) ** simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m \u22a2 ContinuousLinearMap.det A \u2260 0 ** intro 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m h : ContinuousLinearMap.det A = 0 \u22a2 False ** simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA \u22a2 ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 ** simp only [ENNReal.ofReal, abs_inv, Real.toNNReal_inv, ContinuousLinearEquiv.det_coe_symm,\n  ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero, ENNReal.coe_lt_coe] at hm \u22a2 ** 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 mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA hm : m < Real.toNNReal |ContinuousLinearMap.det A| \u22a2 (Real.toNNReal |ContinuousLinearMap.det A|)\u207b\u00b9 < m\u207b\u00b9 ** exact NNReal.inv_lt_inv mpos.ne' 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       \u2200 (t : Set E) (g : E \u2192 E),         ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t ** have :\n  \u2200\u1da0 \u03b4 : \u211d\u22650 in \ud835\udcdd[>] 0,\n    \u2200 (t : Set E) (g : E \u2192 E),\n      ApproximatesLinearOn g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t :=\n  addHaar_image_le_mul_of_det_lt \u03bc B.symm 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 this :   \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,     \u2200 (t : Set E) (g : E \u2192 E),       ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       \u2200 (t : Set E) (g : E \u2192 E),         ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t ** rcases (this.and self_mem_nhdsWithin).exists with \u27e8\u03b4\u2080, 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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 this :   \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,     \u2200 (t : Set E) (g : E \u2192 E),       ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t \u03b4\u2080 : \u211d\u22650 h :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t h' : 0 < \u03b4\u2080 \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       \u2200 (t : Set E) (g : E \u2192 E),         ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t ** exact \u27e8\u03b4\u2080, h', h\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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t \u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 ** by_cases Subsingleton E ** case neg 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t h : \u00acSubsingleton E \u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 ** simp only [h, false_or_iff] ** case neg 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t h : \u00acSubsingleton E \u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0,     \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 ** apply Iio_mem_nhds ** case neg.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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t h : \u00acSubsingleton E \u22a2 0 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 ** simpa only [h, false_or_iff, inv_pos] using B.subsingleton_or_nnnorm_symm_pos ** case pos 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t h : Subsingleton E \u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 ** simp only [h, true_or_iff, eventually_const] ** 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 this :   Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)     (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0)) \u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 ** simp only [mul_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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 this :   Tendsto     (fun \u03b4 =>       \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *           (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *         \u03b4)     (\ud835\udcdd 0) (\ud835\udcdd 0) \u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 ** exact (tendsto_order.1 this).2 \u03b4\u2080 \u03b4\u2080pos ** 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 \u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)     (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0)) ** rcases eq_or_ne \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a 0 with (H | H) ** 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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 H : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0 \u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)     (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0)) ** refine' Tendsto.mul (tendsto_const_nhds.mul _) tendsto_id ** 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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 H : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0 \u22a2 Tendsto (fun \u03b4 => (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9) (\ud835\udcdd 0) (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9) ** refine' (Tendsto.sub tendsto_const_nhds tendsto_id).inv\u2080 _ ** 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 A : E \u2192L[\u211d] E m : \u211d\u22650 hm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 H : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0 \u22a2 \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0 \u2260 0 ** simpa only [tsub_zero, inv_eq_zero, Ne.def] using H ** 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 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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 H : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a = 0 \u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)     (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0)) ** simpa only [H, zero_mul] using tendsto_const_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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 \u22a2 ApproximatesLinearOn f (\u2191B) s \u03b4 ** convert hf ** 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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 hf' : ApproximatesLinearOn f (\u2191B) s \u03b4 F : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4 H : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target \u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s) ** change (m : \u211d\u22650\u221e) * \u03bc F.source \u2264 \u03bc F.target ** 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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 hf' : ApproximatesLinearOn f (\u2191B) s \u03b4 F : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4 H : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target \u22a2 \u2191m * \u2191\u2191\u03bc F.source \u2264 \u2191\u2191\u03bc F.target ** rwa [\u2190 F.symm_image_target_eq_source, mul_comm, \u2190 ENNReal.le_div_iff_mul_le, div_eq_mul_inv,\n  mul_comm, \u2190 ENNReal.coe_inv mpos.ne'] ** case h.h0 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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 hf' : ApproximatesLinearOn f (\u2191B) s \u03b4 F : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4 H : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target \u22a2 \u2191m \u2260 0 \u2228 \u2191\u2191\u03bc F.target \u2260 0 ** apply Or.inl ** case h.h0.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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 hf' : ApproximatesLinearOn f (\u2191B) s \u03b4 F : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4 H : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target \u22a2 \u2191m \u2260 0 ** simpa only [ENNReal.coe_eq_zero, Ne.def] using mpos.ne' ** case h.ht 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\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 : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A| mpos : 0 < m hA : ContinuousLinearMap.det A \u2260 0 B : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA I : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9 \u03b4\u2080 : \u211d\u22650 \u03b4\u2080pos : 0 < \u03b4\u2080 h\u03b4\u2080 :   \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t L1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 L2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 \u03b4 : \u211d\u22650 h1\u03b4 : Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 h2\u03b4 : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 hf' : ApproximatesLinearOn f (\u2191B) s \u03b4 F : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4 H : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target \u22a2 \u2191m \u2260 \u22a4 \u2228 \u2191\u2191\u03bc F.target \u2260 \u22a4 ** simp only [ENNReal.coe_ne_top, true_or_iff, Ne.def, not_false_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.X_pow_eq_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 \u22a2 X n ^ e = \u2191(monomial fun\u2080 | n => e) 1 ** simp [X, monomial_pow] ** Qed",
        "informal": ""
    },
    {
        "formal": "SatisfiesM_StateRefT_eq ** m : Type \u2192 Type \u03c9 \u03c3 \u03b1\u271d : Type p : \u03b1\u271d \u2192 Prop x : StateRefT' \u03c9 \u03c3 m \u03b1\u271d inst\u271d : Monad m \u22a2 SatisfiesM p x \u2194 \u2200 (s : ST.Ref \u03c9 \u03c3), SatisfiesM p (x s) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1to0.tr_supports ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S \u22a2 TM0.Supports (tr M) \u2191(trStmts M S) ** constructor ** case left \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S \u22a2 default \u2208 \u2191(trStmts M S) ** apply Finset.mem_product.2 ** case left \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S \u22a2 default.1 \u2208 TM1.stmts M S \u2227 default.2 \u2208 Finset.univ ** constructor ** case left.left \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S \u22a2 default.1 \u2208 TM1.stmts M S ** simp only [default, TM1.stmts, Finset.mem_insertNone, Option.mem_def, Option.some_inj,\n  forall_eq', Finset.mem_biUnion] ** case left.left \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S \u22a2 \u2203 a, a \u2208 S \u2227 M default \u2208 TM1.stmts\u2081 (M a) ** exact \u27e8_, ss.1, TM1.stmts\u2081_self\u27e9 ** case left.right \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S \u22a2 default.2 \u2208 Finset.univ ** apply Finset.mem_univ ** case right \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S \u22a2 \u2200 {q : \u039b'\u2081\u2080} {a : \u0393} {q' : \u039b'\u2081\u2080} {s : Stmt\u2080}, (q', s) \u2208 tr M q a \u2192 q \u2208 \u2191(trStmts M S) \u2192 q' \u2208 \u2191(trStmts M S) ** intro q a q' s h\u2081 h\u2082 ** case right \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S q : \u039b'\u2081\u2080 a : \u0393 q' : \u039b'\u2081\u2080 s : Stmt\u2080 h\u2081 : (q', s) \u2208 tr M q a h\u2082 : q \u2208 \u2191(trStmts M S) \u22a2 q' \u2208 \u2191(trStmts M S) ** rcases q with \u27e8_ | q, v\u27e9 ** case right.mk.some \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 q' : \u039b'\u2081\u2080 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 h\u2081 : (q', s) \u2208 tr M (some q, v) a h\u2082 : (some q, v) \u2208 \u2191(trStmts M S) \u22a2 q' \u2208 \u2191(trStmts M S) ** cases' q' with q' v' ** case right.mk.some.mk \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 h\u2082 : (some q, v) \u2208 \u2191(trStmts M S) q' : Option Stmt\u2081 v' : \u03c3 h\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a \u22a2 (q', v') \u2208 \u2191(trStmts M S) ** simp only [trStmts, Finset.mem_coe] at h\u2082 \u22a2 ** case right.mk.some.mk \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 q' : Option Stmt\u2081 v' : \u03c3 h\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a h\u2082 : (some q, v) \u2208 TM1.stmts M S \u00d7\u02e2 Finset.univ \u22a2 (q', v') \u2208 TM1.stmts M S \u00d7\u02e2 Finset.univ ** rw [Finset.mem_product] at h\u2082 \u22a2 ** case right.mk.some.mk \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 q' : Option Stmt\u2081 v' : \u03c3 h\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a h\u2082 : (some q, v).1 \u2208 TM1.stmts M S \u2227 (some q, v).2 \u2208 Finset.univ \u22a2 (q', v').1 \u2208 TM1.stmts M S \u2227 (q', v').2 \u2208 Finset.univ ** simp only [Finset.mem_univ, and_true_iff] at h\u2082 \u22a2 ** case right.mk.some.mk \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 q' : Option Stmt\u2081 v' : \u03c3 h\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a h\u2082 : some q \u2208 TM1.stmts M S \u22a2 q' \u2208 TM1.stmts M S ** cases q' ** case right.mk.some.mk.some \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 v' : \u03c3 h\u2082 : some q \u2208 TM1.stmts M S val\u271d : Stmt\u2081 h\u2081 : ((some val\u271d, v'), s) \u2208 tr M (some q, v) a \u22a2 some val\u271d \u2208 TM1.stmts M S ** simp only [tr, Option.mem_def] at h\u2081 ** case right.mk.some.mk.some \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 v' : \u03c3 h\u2082 : some q \u2208 TM1.stmts M S val\u271d : Stmt\u2081 h\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s) \u22a2 some val\u271d \u2208 TM1.stmts M S ** have := TM1.stmts_supportsStmt ss h\u2082 ** case right.mk.some.mk.some \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 v' : \u03c3 h\u2082 : some q \u2208 TM1.stmts M S val\u271d : Stmt\u2081 h\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s) this : TM1.SupportsStmt S q \u22a2 some val\u271d \u2208 TM1.stmts M S ** revert this ** case right.mk.some.mk.some \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 v' : \u03c3 h\u2082 : some q \u2208 TM1.stmts M S val\u271d : Stmt\u2081 h\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s) \u22a2 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S ** induction q generalizing v <;> intro hs ** case right.mk.some.mk.some.move \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d\u00b9 : Dir a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3),     some a\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.move a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.move a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d\u00b9 a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.write \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3),     some a\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.write a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.load \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3),     some a\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.branch \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3),     some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S a_ih\u271d :   \u2200 (v : \u03c3),     some a\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.goto \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 h\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 v : \u03c3 h\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S h\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S TM1.Stmt.halt \u22a2 some val\u271d \u2208 TM1.stmts M S ** case move d q =>\n  cases h\u2081; refine' TM1.stmts_trans _ h\u2082\n  unfold TM1.stmts\u2081\n  exact Finset.mem_insert_of_mem TM1.stmts\u2081_self ** case right.mk.some.mk.some.write \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3),     some a\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.write a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.load \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3),     some a\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.branch \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3),     some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S a_ih\u271d :   \u2200 (v : \u03c3),     some a\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.goto \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 h\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 v : \u03c3 h\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S h\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S TM1.Stmt.halt \u22a2 some val\u271d \u2208 TM1.stmts M S ** case write b q =>\n  cases h\u2081; refine' TM1.stmts_trans _ h\u2082\n  unfold TM1.stmts\u2081\n  exact Finset.mem_insert_of_mem TM1.stmts\u2081_self ** case right.mk.some.mk.some.load \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3),     some a\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.branch \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3),     some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S a_ih\u271d :   \u2200 (v : \u03c3),     some a\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.goto \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 h\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 v : \u03c3 h\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S h\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S TM1.Stmt.halt \u22a2 some val\u271d \u2208 TM1.stmts M S ** case load b q IH =>\n  refine' IH _ (TM1.stmts_trans _ h\u2082) h\u2081 hs\n  unfold TM1.stmts\u2081\n  exact Finset.mem_insert_of_mem TM1.stmts\u2081_self ** case right.mk.some.mk.some.goto \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 h\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d) \u22a2 some val\u271d \u2208 TM1.stmts M S  case right.mk.some.mk.some.halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 v : \u03c3 h\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S h\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S TM1.Stmt.halt \u22a2 some val\u271d \u2208 TM1.stmts M S ** case goto l =>\n  cases h\u2081\n  exact Finset.some_mem_insertNone.2 (Finset.mem_biUnion.2 \u27e8_, hs _ _, TM1.stmts\u2081_self\u27e9) ** case right.mk.some.mk.some.halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 v : \u03c3 h\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S h\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S TM1.Stmt.halt \u22a2 some val\u271d \u2208 TM1.stmts M S ** case halt => cases h\u2081 ** case right.mk.none \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 q' : \u039b'\u2081\u2080 s : Stmt\u2080 v : \u03c3 h\u2081 : (q', s) \u2208 tr M (none, v) a h\u2082 : (none, v) \u2208 \u2191(trStmts M S) \u22a2 q' \u2208 \u2191(trStmts M S) ** cases h\u2081 ** case right.mk.some.mk.none \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v : \u03c3 q : Stmt\u2081 v' : \u03c3 h\u2082 : some q \u2208 TM1.stmts M S h\u2081 : ((none, v'), s) \u2208 tr M (some q, v) a \u22a2 none \u2208 TM1.stmts M S ** exact Multiset.mem_cons_self _ _ ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : Dir d : Stmt\u2081 q :   \u2200 (v : \u03c3),     some d \u2208 TM1.stmts M S \u2192       some (trAux M a d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S d \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.move a\u271d d) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.move a\u271d d) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d d) \u22a2 some val\u271d \u2208 TM1.stmts M S ** cases h\u2081 ** case refl \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : Dir h\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S hs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d) q :   \u2200 (v : \u03c3),     some val\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S \u22a2 some val\u271d \u2208 TM1.stmts M S ** refine' TM1.stmts_trans _ h\u2082 ** case refl \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : Dir h\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S hs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d) q :   \u2200 (v : \u03c3),     some val\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S \u22a2 val\u271d \u2208 TM1.stmts\u2081 (TM1.Stmt.move a\u271d val\u271d) ** unfold TM1.stmts\u2081 ** case refl \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : Dir h\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S hs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d) q :   \u2200 (v : \u03c3),     some val\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S \u22a2 val\u271d \u2208 insert (TM1.Stmt.move a\u271d val\u271d) (TM1.stmts\u2081 val\u271d) ** exact Finset.mem_insert_of_mem TM1.stmts\u2081_self ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393 b : Stmt\u2081 q :   \u2200 (v : \u03c3),     some b \u2208 TM1.stmts M S \u2192       some (trAux M a b v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S b \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.write a\u271d b) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.write a\u271d b) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d b) \u22a2 some val\u271d \u2208 TM1.stmts M S ** cases h\u2081 ** case refl \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393 h\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S hs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d) q :   \u2200 (v : \u03c3),     some val\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192         TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S \u22a2 some val\u271d \u2208 TM1.stmts M S ** refine' TM1.stmts_trans _ h\u2082 ** case refl \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393 h\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S hs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d) q :   \u2200 (v : \u03c3),     some val\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192         TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S \u22a2 val\u271d \u2208 TM1.stmts\u2081 (TM1.Stmt.write a\u271d val\u271d) ** unfold TM1.stmts\u2081 ** case refl \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 v' : \u03c3 val\u271d : Stmt\u2081 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393 h\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S hs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d) q :   \u2200 (v : \u03c3),     some val\u271d \u2208 TM1.stmts M S \u2192       some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192         TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S \u22a2 val\u271d \u2208 insert (TM1.Stmt.write a\u271d val\u271d) (TM1.stmts\u2081 val\u271d) ** exact Finset.mem_insert_of_mem TM1.stmts\u2081_self ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 b : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3),     some q \u2208 TM1.stmts M S \u2192       some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.load b q) \u22a2 some val\u271d \u2208 TM1.stmts M S ** refine' IH _ (TM1.stmts_trans _ h\u2082) h\u2081 hs ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 b : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3),     some q \u2208 TM1.stmts M S \u2192       some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.load b q) \u22a2 q \u2208 TM1.stmts\u2081 (TM1.Stmt.load b q) ** unfold TM1.stmts\u2081 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 b : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3),     some q \u2208 TM1.stmts M S \u2192       some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.load b q) \u22a2 q \u2208 insert (TM1.Stmt.load b q) (TM1.stmts\u2081 q) ** exact Finset.mem_insert_of_mem TM1.stmts\u2081_self ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3),     some q\u2081 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S IH\u2082 :   \u2200 (v : \u03c3),     some q\u2082 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.branch p q\u2081 q\u2082) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082) \u22a2 some val\u271d \u2208 TM1.stmts M S ** cases h : p a v <;> rw [trAux, h] at h\u2081 ** case false \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3),     some q\u2081 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S IH\u2082 :   \u2200 (v : \u03c3),     some q\u2082 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S h\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082) h : p a v = false \u22a2 some val\u271d \u2208 TM1.stmts M S ** refine' IH\u2082 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.2 ** case false \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3),     some q\u2081 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S IH\u2082 :   \u2200 (v : \u03c3),     some q\u2082 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S h\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082) h : p a v = false \u22a2 q\u2082 \u2208 TM1.stmts\u2081 (TM1.Stmt.branch p q\u2081 q\u2082) ** unfold TM1.stmts\u2081 ** case false \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3),     some q\u2081 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S IH\u2082 :   \u2200 (v : \u03c3),     some q\u2082 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S h\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082) h : p a v = false \u22a2 q\u2082 \u2208 insert (TM1.Stmt.branch p q\u2081 q\u2082) (TM1.stmts\u2081 q\u2081 \u222a TM1.stmts\u2081 q\u2082) ** exact Finset.mem_insert_of_mem (Finset.mem_union_right _ TM1.stmts\u2081_self) ** case true \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3),     some q\u2081 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S IH\u2082 :   \u2200 (v : \u03c3),     some q\u2082 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S h\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082) h : p a v = true \u22a2 some val\u271d \u2208 TM1.stmts M S ** refine' IH\u2081 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.1 ** case true \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3),     some q\u2081 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S IH\u2082 :   \u2200 (v : \u03c3),     some q\u2082 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S h\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082) h : p a v = true \u22a2 q\u2081 \u2208 TM1.stmts\u2081 (TM1.Stmt.branch p q\u2081 q\u2082) ** unfold TM1.stmts\u2081 ** case true \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3),     some q\u2081 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S IH\u2082 :   \u2200 (v : \u03c3),     some q\u2082 \u2208 TM1.stmts M S \u2192       some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S v : \u03c3 h\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S h\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082) h : p a v = true \u22a2 q\u2081 \u2208 insert (TM1.Stmt.branch p q\u2081 q\u2082) (TM1.stmts\u2081 q\u2081 \u222a TM1.stmts\u2081 q\u2082) ** exact Finset.mem_insert_of_mem (Finset.mem_union_left _ TM1.stmts\u2081_self) ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 l : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 h\u2082 : some (TM1.Stmt.goto l) \u2208 TM1.stmts M S h\u2081 : some (trAux M a (TM1.Stmt.goto l) v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S (TM1.Stmt.goto l) \u22a2 some val\u271d \u2208 TM1.stmts M S ** cases h\u2081 ** case refl \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 v' : \u03c3 l : \u0393 \u2192 \u03c3 \u2192 \u039b h\u2082 : some (TM1.Stmt.goto l) \u2208 TM1.stmts M S hs : TM1.SupportsStmt S (TM1.Stmt.goto l) \u22a2 some (M (l a v')) \u2208 TM1.stmts M S ** exact Finset.some_mem_insertNone.2 (Finset.mem_biUnion.2 \u27e8_, hs _ _, TM1.stmts\u2081_self\u27e9) ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 inst\u271d : Fintype \u03c3 S : Finset \u039b ss : TM1.Supports M S a : \u0393 s : Stmt\u2080 v' : \u03c3 val\u271d : Stmt\u2081 v : \u03c3 h\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S h\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s) hs : TM1.SupportsStmt S TM1.Stmt.halt \u22a2 some val\u271d \u2208 TM1.stmts M S ** cases h\u2081 ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.sort_univ ** n : \u2115 \u22a2 List.toFinset (Finset.sort (fun x y => x \u2264 y) Finset.univ) = List.toFinset (List.finRange n) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.max'_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 IsGreatest (\u2191(insert a s)) (max (max' s H) a) ** rw [coe_insert, max_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 IsGreatest (insert a \u2191s) (max a (max' s H)) ** exact (isGreatest_max' _ _).insert _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_const ** \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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E \u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c ** cases' (@le_top _ _ _ (\u03bc univ)).lt_or_eq with h\u03bc h\u03bc ** case inl \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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E h\u03bc : \u2191\u2191\u03bc univ < \u22a4 \u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c ** haveI : IsFiniteMeasure \u03bc := \u27e8h\u03bc\u27e9 ** case inl \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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E h\u03bc : \u2191\u2191\u03bc univ < \u22a4 this : IsFiniteMeasure \u03bc \u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c ** simp only [integral, hE, L1.integral] ** case inl \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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E h\u03bc : \u2191\u2191\u03bc univ < \u22a4 this : IsFiniteMeasure \u03bc \u22a2 (if h : True then if hf : Integrable fun x => c then \u2191L1.integralCLM (Integrable.toL1 (fun x => c) hf) else 0     else 0) =     ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c ** exact setToFun_const (dominatedFinMeasAdditive_weightedSMul _) _ ** case inr \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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E h\u03bc : \u2191\u2191\u03bc univ = \u22a4 \u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c ** by_cases hc : c = 0 ** 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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E h\u03bc : \u2191\u2191\u03bc univ = \u22a4 hc : c = 0 \u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c ** simp [hc, integral_zero] ** 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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E h\u03bc : \u2191\u2191\u03bc univ = \u22a4 hc : \u00acc = 0 \u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c ** have : \u00acIntegrable (fun _ : \u03b1 => c) \u03bc := by\n  simp only [integrable_const_iff, not_or]\n  exact \u27e8hc, h\u03bc.not_lt\u27e9 ** 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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E h\u03bc : \u2191\u2191\u03bc univ = \u22a4 hc : \u00acc = 0 this : \u00acIntegrable fun x => c \u22a2 \u222b (x : \u03b1), c \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 c ** simp [integral_undef, *] ** \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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E h\u03bc : \u2191\u2191\u03bc univ = \u22a4 hc : \u00acc = 0 \u22a2 \u00acIntegrable fun x => c ** simp only [integrable_const_iff, not_or] ** \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 g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X c : E h\u03bc : \u2191\u2191\u03bc univ = \u22a4 hc : \u00acc = 0 \u22a2 \u00acc = 0 \u2227 \u00ac\u2191\u2191\u03bc univ < \u22a4 ** exact \u27e8hc, h\u03bc.not_lt\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.degreeOf_rename_of_injective ** 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 p : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 h : Injective f i : \u03c3 \u22a2 degreeOf (f i) (\u2191(rename f) p) = degreeOf i p ** classical\nsimp only [degreeOf, degrees_rename_of_injective h, Multiset.count_map_eq_count' f p.degrees 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\u271d q p : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 h : Injective f i : \u03c3 \u22a2 degreeOf (f i) (\u2191(rename f) p) = degreeOf i p ** simp only [degreeOf, degrees_rename_of_injective h, Multiset.count_map_eq_count' f p.degrees h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurePreserving_piEquivPiSubtypeProd ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : Fintype \u03b9' p : \u03b9 \u2192 Prop inst\u271d : DecidablePred p \u22a2 MeasurePreserving \u2191(MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p) ** set e := (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p).symm ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : Fintype \u03b9' p : \u03b9 \u2192 Prop inst\u271d : DecidablePred p e : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=   MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p) \u22a2 MeasurePreserving \u2191(MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p) ** refine' MeasurePreserving.symm e _ ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : Fintype \u03b9' p : \u03b9 \u2192 Prop inst\u271d : DecidablePred p e : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=   MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p) \u22a2 MeasurePreserving \u2191e ** refine' \u27e8e.measurable, (pi_eq fun s _ => _).symm\u27e9 ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : Fintype \u03b9' p : \u03b9 \u2192 Prop inst\u271d : DecidablePred p e : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=   MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p) s : (i : \u03b9) \u2192 Set (\u03b1 i) x\u271d : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 \u2191\u2191(Measure.map (\u2191e) (Measure.prod (Measure.pi fun i => \u03bc \u2191i) (Measure.pi fun i => \u03bc \u2191i))) (Set.pi univ s) =     \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** have : e \u207b\u00b9' pi univ s =\n    (pi univ fun i : { i // p i } => s i) \u00d7\u02e2 pi univ fun i : { i // \u00acp i } => s i :=\n  Equiv.preimage_piEquivPiSubtypeProd_symm_pi p s ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : Fintype \u03b9' p : \u03b9 \u2192 Prop inst\u271d : DecidablePred p e : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=   MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p) s : (i : \u03b9) \u2192 Set (\u03b1 i) x\u271d : \u2200 (i : \u03b9), MeasurableSet (s i) this : \u2191e \u207b\u00b9' Set.pi univ s = (Set.pi univ fun i => s \u2191i) \u00d7\u02e2 Set.pi univ fun i => s \u2191i \u22a2 \u2191\u2191(Measure.map (\u2191e) (Measure.prod (Measure.pi fun i => \u03bc \u2191i) (Measure.pi fun i => \u03bc \u2191i))) (Set.pi univ s) =     \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** rw [e.map_apply, this, prod_prod, pi_pi, pi_pi] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : 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\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : Fintype \u03b9' p : \u03b9 \u2192 Prop inst\u271d : DecidablePred p e : ((i : Subtype p) \u2192 \u03b1 \u2191i) \u00d7 ((i : { i // \u00acp i }) \u2192 \u03b1 \u2191i) \u2243\u1d50 ((i : \u03b9) \u2192 \u03b1 i) :=   MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd \u03b1 p) s : (i : \u03b9) \u2192 Set (\u03b1 i) x\u271d : \u2200 (i : \u03b9), MeasurableSet (s i) this : \u2191e \u207b\u00b9' Set.pi univ s = (Set.pi univ fun i => s \u2191i) \u00d7\u02e2 Set.pi univ fun i => s \u2191i \u22a2 (\u220f i : Subtype p, \u2191\u2191(\u03bc \u2191i) (s \u2191i)) * \u220f i : { i // \u00acp i }, \u2191\u2191(\u03bc \u2191i) (s \u2191i) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** exact Fintype.prod_subtype_mul_prod_subtype p fun i => \u03bc i (s i) ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.encode_lt_pair ** cf cg : Code \u22a2 encode cf < encode (pair cf cg) \u2227 encode cg < encode (pair cf cg) ** simp only [encodeCode_eq, encodeCode] ** cf cg : Code \u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227     encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ** have := Nat.mul_le_mul_right (Nat.pair cf.encodeCode cg.encodeCode) (by decide : 1 \u2264 2 * 2) ** cf cg : Code this : 1 * Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * 2 * Nat.pair (encodeCode cf) (encodeCode cg) \u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227     encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ** rw [one_mul, mul_assoc] at this ** cf cg : Code this : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) \u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227     encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ** have := lt_of_le_of_lt this (lt_add_of_pos_right _ (by decide : 0 < 4)) ** cf cg : Code this\u271d : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) this : Nat.pair (encodeCode cf) (encodeCode cg) < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227     encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 ** exact \u27e8lt_of_le_of_lt (Nat.left_le_pair _ _) this, lt_of_le_of_lt (Nat.right_le_pair _ _) this\u27e9 ** cf cg : Code \u22a2 1 \u2264 2 * 2 ** decide ** cf cg : Code this : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) \u22a2 0 < 4 ** decide ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.list_getD ** \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 d : \u03b1 \u22a2 Primrec\u2082 fun l n => List.getD l n d ** simp only [List.getD_eq_getD_get?] ** \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 d : \u03b1 \u22a2 Primrec\u2082 fun l n => Option.getD (List.get? l n) d ** exact option_getD.comp\u2082 list_get? (const _) ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_coe_nnreal_real_iff ** \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 f : \u03b1 \u2192 \u211d\u22650 h : Measurable fun x => \u2191(f x) \u22a2 Measurable f ** simpa only [Real.toNNReal_coe] using h.real_toNNReal ** Qed",
        "informal": ""
    },
    {
        "formal": "List.pairwise_filterMap ** \u03b2 : Type u_1 \u03b1 : Type u_2 R : \u03b1 \u2192 \u03b1 \u2192 Prop f : \u03b2 \u2192 Option \u03b1 l : List \u03b2 \u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b') l ** let _S (a a' : \u03b2) := \u2200 b \u2208 f a, \u2200 b' \u2208 f a', R b b' ** \u03b2 : Type u_1 \u03b1 : Type u_2 R : \u03b1 \u2192 \u03b1 \u2192 Prop f : \u03b2 \u2192 Option \u03b1 l : List \u03b2 _S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b' \u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b') l ** simp only [Option.mem_def] ** \u03b2 : Type u_1 \u03b1 : Type u_2 R : \u03b1 \u2192 \u03b1 \u2192 Prop f : \u03b2 \u2192 Option \u03b1 l : List \u03b2 _S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b' \u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l ** induction l with\n| nil => simp only [filterMap, Pairwise.nil]\n| cons a l IH => ?_ ** case cons \u03b2 : Type u_1 \u03b1 : Type u_2 R : \u03b1 \u2192 \u03b1 \u2192 Prop f : \u03b2 \u2192 Option \u03b1 _S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b' a : \u03b2 l : List \u03b2 IH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l \u22a2 Pairwise R (filterMap f (a :: l)) \u2194     Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l) ** match e : f a with\n| none =>\n  rw [filterMap_cons_none _ _ e, pairwise_cons]\n  simp only [e, false_implies, implies_true, true_and, IH]\n| some b =>\n  rw [filterMap_cons_some _ _ _ e]\n  simpa [IH, e] using fun _ =>\n    \u27e8fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab\u27e9 ** case nil \u03b2 : Type u_1 \u03b1 : Type u_2 R : \u03b1 \u2192 \u03b1 \u2192 Prop f : \u03b2 \u2192 Option \u03b1 _S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b' \u22a2 Pairwise R (filterMap f []) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') [] ** simp only [filterMap, Pairwise.nil] ** \u03b2 : Type u_1 \u03b1 : Type u_2 R : \u03b1 \u2192 \u03b1 \u2192 Prop f : \u03b2 \u2192 Option \u03b1 _S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b' a : \u03b2 l : List \u03b2 IH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l e : f a = none \u22a2 Pairwise R (filterMap f (a :: l)) \u2194     Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l) ** rw [filterMap_cons_none _ _ e, pairwise_cons] ** \u03b2 : Type u_1 \u03b1 : Type u_2 R : \u03b1 \u2192 \u03b1 \u2192 Prop f : \u03b2 \u2192 Option \u03b1 _S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b' a : \u03b2 l : List \u03b2 IH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l e : f a = none \u22a2 Pairwise R (filterMap f l) \u2194     (\u2200 (a' : \u03b2), a' \u2208 l \u2192 \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') \u2227       Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l ** simp only [e, false_implies, implies_true, true_and, IH] ** \u03b2 : Type u_1 \u03b1 : Type u_2 R : \u03b1 \u2192 \u03b1 \u2192 Prop f : \u03b2 \u2192 Option \u03b1 _S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b' a : \u03b2 l : List \u03b2 IH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l b : \u03b1 e : f a = some b \u22a2 Pairwise R (filterMap f (a :: l)) \u2194     Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l) ** rw [filterMap_cons_some _ _ _ e] ** \u03b2 : Type u_1 \u03b1 : Type u_2 R : \u03b1 \u2192 \u03b1 \u2192 Prop f : \u03b2 \u2192 Option \u03b1 _S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b' a : \u03b2 l : List \u03b2 IH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l b : \u03b1 e : f a = some b \u22a2 Pairwise R (b :: filterMap f l) \u2194     Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l) ** simpa [IH, e] using fun _ =>\n  \u27e8fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_integral_condexpL1Clm_of_measure_ne_top ** \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 h\u03bcs : \u2191\u2191\u03bc s \u2260 \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 ** refine' @Lp.induction _ _ _ _ _ _ _ ENNReal.one_ne_top\n  (fun f : \u03b1 \u2192\u2081[\u03bc] F' => \u222b x in s, condexpL1Clm F' hm \u03bc f x \u2202\u03bc = \u222b x in s, f x \u2202\u03bc) _ _\n  (isClosed_eq _ _) 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200 (c : F') {s_1 : Set \u03b1} (hs : MeasurableSet s_1) (h\u03bcs : \u2191\u2191\u03bc s_1 < \u22a4),     (fun f => \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)       \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191\u03bc s_1 \u2260 \u22a4) c) ** intro x t ht h\u03bct ** 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : F' t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u222b (x_1 : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) \u2191(simpleFunc.indicatorConst 1 ht (_ : \u2191\u2191\u03bc t \u2260 \u22a4) x)) x_1 \u2202\u03bc =     \u222b (x_1 : \u03b1) in s, \u2191\u2191\u2191(simpleFunc.indicatorConst 1 ht (_ : \u2191\u2191\u03bc t \u2260 \u22a4) x) x_1 \u2202\u03bc ** simp_rw [condexpL1Clm_indicatorConst ht h\u03bct.ne x] ** 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : F' t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u222b (x_1 : \u03b1) in s, \u2191\u2191(\u2191(condexpInd F' hm \u03bc t) x) x_1 \u2202\u03bc =     \u222b (x_1 : \u03b1) in s, \u2191\u2191\u2191(simpleFunc.indicatorConst 1 ht (_ : \u2191\u2191\u03bc t \u2260 \u22a4) x) x_1 \u2202\u03bc ** rw [Lp.simpleFunc.coe_indicatorConst, set_integral_indicatorConstLp (hm _ hs)] ** 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : F' t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u222b (x_1 : \u03b1) in s, \u2191\u2191(\u2191(condexpInd F' hm \u03bc t) x) x_1 \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc (t \u2229 s)) \u2022 x ** exact set_integral_condexpInd hs ht h\u03bcs h\u03bct.ne x ** 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \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 => \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) (Mem\u2112p.toLp f hf) \u2192         (fun f => \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) (Mem\u2112p.toLp g hg) \u2192           (fun f => \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)             (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** intro f g hf_Lp hg_Lp _ hf hg ** 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f g : \u03b1 \u2192 F' hf_Lp : Mem\u2112p f 1 hg_Lp : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc hg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc =     \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc ** simp_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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f g : \u03b1 \u2192 F' hf_Lp : Mem\u2112p f 1 hg_Lp : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc hg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp) + \u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc =     \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc ** rw [set_integral_congr_ae (hm s hs) ((Lp.coeFn_add (condexpL1Clm F' hm \u03bc (hf_Lp.toLp f))\n  (condexpL1Clm F' hm \u03bc (hg_Lp.toLp g))).mono fun x hx _ => hx)] ** 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f g : \u03b1 \u2192 F' hf_Lp : Mem\u2112p f 1 hg_Lp : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc hg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s,       (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp))) x \u2202\u03bc =     \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp + Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc ** rw [set_integral_congr_ae (hm s hs)\n  ((Lp.coeFn_add (hf_Lp.toLp f) (hg_Lp.toLp g)).mono fun x hx _ => hx)] ** 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f g : \u03b1 \u2192 F' hf_Lp : Mem\u2112p f 1 hg_Lp : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc hg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s,       (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp))) x \u2202\u03bc =     \u222b (x : \u03b1) in s, (\u2191\u2191(Mem\u2112p.toLp f hf_Lp) + \u2191\u2191(Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc ** simp_rw [Pi.add_apply] ** 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 f g : \u03b1 \u2192 F' hf_Lp : Mem\u2112p f 1 hg_Lp : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hf : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x \u2202\u03bc hg : \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s,       \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf_Lp)) x + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg_Lp)) x \u2202\u03bc =     \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp f hf_Lp) x + \u2191\u2191(Mem\u2112p.toLp g hg_Lp) x \u2202\u03bc ** rw [integral_add (L1.integrable_coeFn _).integrableOn (L1.integrable_coeFn _).integrableOn,\n  integral_add (L1.integrable_coeFn _).integrableOn (L1.integrable_coeFn _).integrableOn, hf,\n  hg] ** 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 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc ** exact (continuous_set_integral s).comp (condexpL1Clm F' hm \u03bc).continuous ** case refine'_4 \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 h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** exact continuous_set_integral s ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_norm_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 (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, \u2016f x\u2016 \u2202\u03bc ** rw [integral_norm_eq_pos_sub_neg hgi, integral_norm_eq_pos_sub_neg hfi] ** \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 {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264     \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s ** have h_meas_nonneg_g : MeasurableSet[m] {x | 0 \u2264 g x} :=\n  (@stronglyMeasurable_const _ _ m _ _).measurableSet_le hg ** \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 h_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264     \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s ** have h_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x} :=\n  stronglyMeasurable_const.measurableSet_le hf ** \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 h_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x} h_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264     \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s ** have h_meas_nonpos_g : MeasurableSet[m] {x | g x \u2264 0} :=\n  hg.measurableSet_le (@stronglyMeasurable_const _ _ m _ _) ** \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 h_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x} h_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x} h_meas_nonpos_g : MeasurableSet {x | g x \u2264 0} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264     \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s ** have h_meas_nonpos_f : MeasurableSet {x | f x \u2264 0} :=\n  hf.measurableSet_le stronglyMeasurable_const ** \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 h_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x} h_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x} h_meas_nonpos_g : MeasurableSet {x | g x \u2264 0} h_meas_nonpos_f : MeasurableSet {x | f x \u2264 0} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s \u2264     \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s ** refine' sub_le_sub _ _ ** 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' 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 h_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x} h_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x} h_meas_nonpos_g : MeasurableSet {x | g x \u2264 0} h_meas_nonpos_f : MeasurableSet {x | f x \u2264 0} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, g x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s ** rw [Measure.restrict_restrict (hm _ h_meas_nonneg_g), Measure.restrict_restrict h_meas_nonneg_f,\n  hgf _ (@MeasurableSet.inter \u03b1 m _ _ h_meas_nonneg_g hs)\n    ((measure_mono (Set.inter_subset_right _ _)).trans_lt (lt_top_iff_ne_top.mpr h\u03bcs)),\n  \u2190 Measure.restrict_restrict (hm _ h_meas_nonneg_g), \u2190\n  Measure.restrict_restrict h_meas_nonneg_f] ** 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' 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 h_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x} h_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x} h_meas_nonpos_g : MeasurableSet {x | g x \u2264 0} h_meas_nonpos_f : MeasurableSet {x | f x \u2264 0} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 g x}, f x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202Measure.restrict \u03bc s ** exact set_integral_le_nonneg (hm _ h_meas_nonneg_g) hf hfi ** 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' 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 h_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x} h_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x} h_meas_nonpos_g : MeasurableSet {x | g x \u2264 0} h_meas_nonpos_f : MeasurableSet {x | f x \u2264 0} \u22a2 \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | g x \u2264 0}, g x \u2202Measure.restrict \u03bc s ** rw [Measure.restrict_restrict (hm _ h_meas_nonpos_g), Measure.restrict_restrict h_meas_nonpos_f,\n  hgf _ (@MeasurableSet.inter \u03b1 m _ _ h_meas_nonpos_g hs)\n    ((measure_mono (Set.inter_subset_right _ _)).trans_lt (lt_top_iff_ne_top.mpr h\u03bcs)),\n  \u2190 Measure.restrict_restrict (hm _ h_meas_nonpos_g), \u2190\n  Measure.restrict_restrict h_meas_nonpos_f] ** 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' 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 h_meas_nonneg_g : MeasurableSet {x | 0 \u2264 g x} h_meas_nonneg_f : MeasurableSet {x | 0 \u2264 f x} h_meas_nonpos_g : MeasurableSet {x | g x \u2264 0} h_meas_nonpos_f : MeasurableSet {x | f x \u2264 0} \u22a2 \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202Measure.restrict \u03bc s \u2264 \u222b (x : \u03b1) in {x | g x \u2264 0}, f x \u2202Measure.restrict \u03bc s ** exact set_integral_nonpos_le (hm _ h_meas_nonpos_g) hf hfi ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FinStronglyMeasurable.const_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2076 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b2 \ud835\udd5c : Type u_5 inst\u271d\u2074 : TopologicalSpace \ud835\udd5c inst\u271d\u00b3 : AddMonoid \u03b2 inst\u271d\u00b2 : Monoid \ud835\udd5c inst\u271d\u00b9 : DistribMulAction \ud835\udd5c \u03b2 inst\u271d : ContinuousSMul \ud835\udd5c \u03b2 hf : FinStronglyMeasurable f \u03bc c : \ud835\udd5c \u22a2 FinStronglyMeasurable (c \u2022 f) \u03bc ** refine' \u27e8fun n => c \u2022 hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).const_smul c\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2076 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b2 \ud835\udd5c : Type u_5 inst\u271d\u2074 : TopologicalSpace \ud835\udd5c inst\u271d\u00b3 : AddMonoid \u03b2 inst\u271d\u00b2 : Monoid \ud835\udd5c inst\u271d\u00b9 : DistribMulAction \ud835\udd5c \u03b2 inst\u271d : ContinuousSMul \ud835\udd5c \u03b2 hf : FinStronglyMeasurable f \u03bc c : \ud835\udd5c n : \u2115 \u22a2 \u2191\u2191\u03bc (support \u2191((fun n => c \u2022 FinStronglyMeasurable.approx hf n) n)) < \u22a4 ** rw [SimpleFunc.coe_smul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2076 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b2 \ud835\udd5c : Type u_5 inst\u271d\u2074 : TopologicalSpace \ud835\udd5c inst\u271d\u00b3 : AddMonoid \u03b2 inst\u271d\u00b2 : Monoid \ud835\udd5c inst\u271d\u00b9 : DistribMulAction \ud835\udd5c \u03b2 inst\u271d : ContinuousSMul \ud835\udd5c \u03b2 hf : FinStronglyMeasurable f \u03bc c : \ud835\udd5c n : \u2115 \u22a2 \u2191\u2191\u03bc (support (c \u2022 \u2191(FinStronglyMeasurable.approx hf n))) < \u22a4 ** refine' (measure_mono (support_smul_subset_right c _)).trans_lt (hf.fin_support_approx n) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.compProd_eq_sum_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 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 \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m ** 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 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 \u00d7 \u03b3) hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = \u2191\u2191(\u2191(kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) a) s ** simp_rw [kernel.sum_apply' _ 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 \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 \u00d7 \u03b3) hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = \u2211' (n : \u2115) (n_1 : \u2115), \u2191\u2191(\u2191(seq \u03ba n \u2297\u2096 seq \u03b7 n_1) a) s ** rw [compProd_eq_tsum_compProd \u03ba \u03b7 a hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.indepSets_piiUnionInter_of_disjoint ** \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T \u22a2 IndepSets (piiUnionInter s S) (piiUnionInter s T) \u03ba ** rintro t1 t2 \u27e8p1, hp1, f1, ht1_m, ht1_eq\u27e9 \u27e8p2, hp2, f2, ht2_m, ht2_eq\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x \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 ** let g i := ite (i \u2208 p1) (f1 i) Set.univ \u2229 ite (i \u2208 p2) (f2 i) Set.univ ** case intro.intro.intro.intro.intro.intro.intro.intro \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) \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 ** filter_upwards [h_P_inter, h_indep p1 ht1_m, h_indep p2 ht2_m] with a h_P_inter ha1 ha2 ** 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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) a : \u03b1 h_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i) ha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i) h_\u03bcg : \u2200 (n : \u03b9), \u2191\u2191(\u2191\u03ba a) (g n) = (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1 \u22a2 \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t1 * \u2191\u2191(\u2191\u03ba a) t2 ** simp_rw [h_P_inter, h_\u03bcg, Finset.prod_mul_distrib,\n  Finset.prod_ite_mem (p1 \u222a p2) p1 (fun x \u21a6 \u03ba a (f1 x)), Finset.union_inter_cancel_left,\n  Finset.prod_ite_mem (p1 \u222a p2) p2 (fun x => \u03ba a (f2 x)), Finset.union_inter_cancel_right, ht1_eq,\n    \u2190 ha1, ht2_eq, \u2190 ha2] ** \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ hgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ** have h_p1_inter_p2 :\n  ((\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x) =\n    \u22c2 i \u2208 p1 \u222a p2, ite (i \u2208 p1) (f1 i) Set.univ \u2229 ite (i \u2208 p2) (f2 i) Set.univ := by\n  ext1 x\n  simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]\n  exact\n    \u27e8fun h i _ => \u27e8h.1 i, h.2 i\u27e9, fun h =>\n      \u27e8fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi\u27e9\u27e9 ** \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ hgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i h_p1_inter_p2 :   (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x =     \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ** filter_upwards [h_indep _ hgm] with a ha ** 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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ hgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i h_p1_inter_p2 :   (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x =     \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ a : \u03b1 ha : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1 \u222a p2, g i) = \u220f i in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g i) \u22a2 \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ** rw [ht1_eq, ht2_eq, h_p1_inter_p2, \u2190 ha] ** \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ \u22a2 \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i ** intro i hi_mem_union ** \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ i : \u03b9 hi_mem_union : i \u2208 p1 \u222a p2 \u22a2 g i \u2208 s i ** rw [Finset.mem_union] at hi_mem_union ** \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ i : \u03b9 hi_mem_union : i \u2208 p1 \u2228 i \u2208 p2 \u22a2 g i \u2208 s i ** cases' hi_mem_union with hi1 hi2 ** case inl \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ i : \u03b9 hi1 : i \u2208 p1 \u22a2 g i \u2208 s i ** have hi2 : i \u2209 p2 := fun hip2 => Set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2) ** case inl \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ i : \u03b9 hi1 : i \u2208 p1 hi2 : \u00aci \u2208 p2 \u22a2 g i \u2208 s i ** simp_rw [if_pos hi1, if_neg hi2, Set.inter_univ] ** case inl \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ i : \u03b9 hi1 : i \u2208 p1 hi2 : \u00aci \u2208 p2 \u22a2 f1 i \u2208 s i ** exact ht1_m i hi1 ** case inr \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ i : \u03b9 hi2 : i \u2208 p2 \u22a2 g i \u2208 s i ** have hi1 : i \u2209 p1 := fun hip1 => Set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1) ** case inr \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ i : \u03b9 hi2 : i \u2208 p2 hi1 : \u00aci \u2208 p1 \u22a2 g i \u2208 s i ** simp_rw [if_neg hi1, if_pos hi2, Set.univ_inter] ** case inr \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ i : \u03b9 hi2 : i \u2208 p2 hi1 : \u00aci \u2208 p1 \u22a2 f2 i \u2208 s i ** exact ht2_m i hi2 ** \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ hgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i \u22a2 (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x =     \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ ** ext1 x ** 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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ hgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i x : \u03a9 \u22a2 x \u2208 (\u22c2 x \u2208 p1, f1 x) \u2229 \u22c2 x \u2208 p2, f2 x \u2194     x \u2208 \u22c2 i \u2208 p1 \u222a p2, (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ ** simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union] ** 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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ hgm : \u2200 (i : \u03b9), i \u2208 p1 \u222a p2 \u2192 g i \u2208 s i x : \u03a9 \u22a2 ((\u2200 (i : \u03b9), i \u2208 p1 \u2192 x \u2208 f1 i) \u2227 \u2200 (i : \u03b9), i \u2208 p2 \u2192 x \u2208 f2 i) \u2194     \u2200 (i : \u03b9), i \u2208 p1 \u2228 i \u2208 p2 \u2192 (i \u2208 p1 \u2192 x \u2208 f1 i) \u2227 (i \u2208 p2 \u2192 x \u2208 f2 i) ** exact\n  \u27e8fun h i _ => \u27e8h.1 i, h.2 i\u27e9, fun h =>\n    \u27e8fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi\u27e9\u27e9 ** \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) a : \u03b1 h_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i) ha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i) \u22a2 \u2200 (n : \u03b9), \u2191\u2191(\u2191\u03ba a) (g n) = (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1 ** intro 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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) a : \u03b1 h_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i) ha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i) n : \u03b9 \u22a2 \u2191\u2191(\u2191\u03ba a) (g n) = (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1 ** dsimp only ** \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) a : \u03b1 h_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i) ha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i) n : \u03b9 \u22a2 \u2191\u2191(\u2191\u03ba a) ((if n \u2208 p1 then f1 n else Set.univ) \u2229 if n \u2208 p2 then f2 n else Set.univ) =     (if n \u2208 p1 then \u2191\u2191(\u2191\u03ba a) (f1 n) else 1) * if n \u2208 p2 then \u2191\u2191(\u2191\u03ba a) (f2 n) else 1 ** split_ifs with h1 h2 ** case neg \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) a : \u03b1 h_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i) ha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i) n : \u03b9 h1 : n \u2208 p1 h2 : \u00acn \u2208 p2 \u22a2 \u2191\u2191(\u2191\u03ba a) (f1 n \u2229 Set.univ) = \u2191\u2191(\u2191\u03ba a) (f1 n) * 1  case pos \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) a : \u03b1 h_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i) ha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i) n : \u03b9 h1 : \u00acn \u2208 p1 h\u271d : n \u2208 p2 \u22a2 \u2191\u2191(\u2191\u03ba a) (Set.univ \u2229 f2 n) = 1 * \u2191\u2191(\u2191\u03ba a) (f2 n)  case neg \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) a : \u03b1 h_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i) ha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i) n : \u03b9 h1 : \u00acn \u2208 p1 h\u271d : \u00acn \u2208 p2 \u22a2 \u2191\u2191(\u2191\u03ba a) (Set.univ \u2229 Set.univ) = 1 * 1 ** all_goals simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter] ** case pos \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) a : \u03b1 h_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i) ha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i) n : \u03b9 h1 : n \u2208 p1 h2 : n \u2208 p2 \u22a2 \u2191\u2191(\u2191\u03ba a) (f1 n \u2229 f2 n) = \u2191\u2191(\u2191\u03ba a) (f1 n) * \u2191\u2191(\u2191\u03ba a) (f2 n) ** exact absurd rfl (Set.disjoint_iff_forall_ne.mp hST (hp1 h1) (hp2 h2)) ** case neg \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 s : \u03b9 \u2192 Set (Set \u03a9) S T : Set \u03b9 h_indep : iIndepSets s \u03ba hST : Disjoint S T t1 t2 : Set \u03a9 p1 : Finset \u03b9 hp1 : \u2191p1 \u2286 S f1 : \u03b9 \u2192 Set \u03a9 ht1_m : \u2200 (x : \u03b9), x \u2208 p1 \u2192 f1 x \u2208 s x ht1_eq : t1 = \u22c2 x \u2208 p1, f1 x p2 : Finset \u03b9 hp2 : \u2191p2 \u2286 T f2 : \u03b9 \u2192 Set \u03a9 ht2_m : \u2200 (x : \u03b9), x \u2208 p2 \u2192 f2 x \u2208 s x ht2_eq : t2 = \u22c2 x \u2208 p2, f2 x g : \u03b9 \u2192 Set \u03a9 := fun i => (if i \u2208 p1 then f1 i else Set.univ) \u2229 if i \u2208 p2 then f2 i else Set.univ h_P_inter\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) a : \u03b1 h_P_inter : \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u220f n in p1 \u222a p2, \u2191\u2191(\u2191\u03ba a) (g n) ha1 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p1, f1 i) = \u220f i in p1, \u2191\u2191(\u2191\u03ba a) (f1 i) ha2 : \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 p2, f2 i) = \u220f i in p2, \u2191\u2191(\u2191\u03ba a) (f2 i) n : \u03b9 h1 : \u00acn \u2208 p1 h\u271d : \u00acn \u2208 p2 \u22a2 \u2191\u2191(\u2191\u03ba a) (Set.univ \u2229 Set.univ) = 1 * 1 ** simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpInd_empty ** \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) \u22a2 condexpInd G hm \u03bc \u2205 = 0 ** ext1 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) x : G \u22a2 \u2191(condexpInd G hm \u03bc \u2205) x = \u21910 x ** ext1 ** case h.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) x : G \u22a2 \u2191\u2191(\u2191(condexpInd G hm \u03bc \u2205) x) =\u1d50[\u03bc] \u2191\u2191(\u21910 x) ** refine' (condexpInd_ae_eq_condexpIndSMul hm MeasurableSet.empty (by simp) x).trans _ ** case h.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) x : G \u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet \u2205) (_ : \u2191\u2191\u03bc \u2205 \u2260 \u22a4) x) =\u1d50[\u03bc] \u2191\u2191(\u21910 x) ** rw [condexpIndSMul_empty] ** case h.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) x : G \u22a2 \u2191\u21910 =\u1d50[\u03bc] \u2191\u2191(\u21910 x) ** refine' (Lp.coeFn_zero G 2 \u03bc).trans _ ** case h.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) x : G \u22a2 0 =\u1d50[\u03bc] \u2191\u2191(\u21910 x) ** refine' EventuallyEq.trans _ (Lp.coeFn_zero G 1 \u03bc).symm ** case h.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) x : G \u22a2 0 =\u1d50[\u03bc] 0 ** rfl ** \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 \u2191\u2191\u03bc \u2205 \u2260 \u22a4 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.dirac_prod_dirac ** \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 x : \u03b1 y : \u03b2 \u22a2 Measure.prod (dirac x) (dirac y) = dirac (x, y) ** rw [prod_dirac, map_dirac measurable_prod_mk_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.gcd_eq_gcd_ab ** x y : \u2115 \u22a2 \u2191(gcd x y) = \u2191x * gcdA x y + \u2191y * gcdB x y ** have := @xgcdAux_P x y x y 1 0 0 1 (by simp [P]) (by simp [P]) ** x y : \u2115 this : Nat.P x y (xgcdAux x 1 0 y 0 1) \u22a2 \u2191(gcd x y) = \u2191x * gcdA x y + \u2191y * gcdB x y ** rwa [xgcdAux_val, xgcd_val] at this ** x y : \u2115 \u22a2 Nat.P x y (x, 1, 0) ** simp [P] ** x y : \u2115 \u22a2 Nat.P x y (y, 0, 1) ** simp [P] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_smul ** \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 c : \ud835\udd5c f : \u03b1 \u2192 F' \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : \u00acm \u2264 m0 \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \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\u271d g : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \u03bc[f|m] ** refine' (condexp_ae_eq_condexpL1 hm _).trans _ ** 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 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u2191\u2191(condexpL1 hm \u03bc (c \u2022 f)) =\u1d50[\u03bc] c \u2022 \u03bc[f|m] ** rw [condexpL1_smul c 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 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) =\u1d50[\u03bc] c \u2022 \u03bc[f|m] ** refine' (@condexp_ae_eq_condexpL1 _ _ _ _ _ m _ _ hm _ f).mp _ ** 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 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u03bc[f|m]) x = \u2191\u2191(condexpL1 hm \u03bc f) x \u2192 \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) x = (c \u2022 \u03bc[f|m]) x ** refine' (coeFn_smul c (condexpL1 hm \u03bc f)).mono fun x hx1 hx2 => _ ** 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 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) x : \u03b1 hx1 : \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) x = (c \u2022 \u2191\u2191(condexpL1 hm \u03bc f)) x hx2 : (\u03bc[f|m]) x = \u2191\u2191(condexpL1 hm \u03bc f) x \u22a2 \u2191\u2191(c \u2022 condexpL1 hm \u03bc f) x = (c \u2022 \u03bc[f|m]) x ** rw [hx1, Pi.smul_apply, Pi.smul_apply, hx2] ** 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 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : \u00acm \u2264 m0 \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \u03bc[f|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 : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : \u00acm \u2264 m0 \u22a2 0 =\u1d50[\u03bc] c \u2022 0 ** simp ** 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 c : \ud835\udd5c f : \u03b1 \u2192 F' 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 : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[c \u2022 f|m] =\u1d50[\u03bc] c \u2022 \u03bc[f|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 : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 0 =\u1d50[\u03bc] c \u2022 0 ** simp ** 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 c : \ud835\udd5c f : \u03b1 \u2192 F' hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 0 =\u1d50[\u03bc] 0 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.card_insert_eq_ite ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 \u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1 ** by_cases h : a \u2208 s ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 h : a \u2208 s \u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1 ** rw [card_insert_of_mem h, if_pos h] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 h : \u00aca \u2208 s \u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1 ** rw [card_insert_of_not_mem h, if_neg h] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.erase_cons ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a b : \u03b1 l : List \u03b1 h : b = a \u22a2 List.erase (b :: l) a = if b = a then l else b :: List.erase l a ** simp [List.erase, h] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a b : \u03b1 l : List \u03b1 h : \u00acb = a \u22a2 List.erase (b :: l) a = if b = a then l else b :: List.erase l a ** simp [List.erase, h, (beq_eq_false_iff_ne _ _).2 h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.NullMeasurable.congr ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b1 g : \u03b1 \u2192 \u03b2 hf : NullMeasurable f hg : f =\u1d50[\u03bc] g s : Set \u03b2 hs : MeasurableSet s x : \u03b1 hx : f x = g x \u22a2 x \u2208 f \u207b\u00b9' s \u2194 x \u2208 g \u207b\u00b9' s ** rw [mem_preimage, mem_preimage, hx] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.neg_smul ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : Ring \u03b1 inst\u271d\u00b9 : AddCommGroup \u03b2 inst\u271d : Module \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 : Ring \u03b1 inst\u271d\u00b9 : AddCommGroup \u03b2 inst\u271d : Module \u03b1 \u03b2 a : \u03b1 s : Set \u03b1 t : Set \u03b2 \u22a2 Neg.neg '' s \u2022 t = Neg.neg '' (s \u2022 t) ** exact image2_image_left_comm neg_smul ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.le_invariants_iterate ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 n : \u2115 \u22a2 invariants f \u2264 invariants f^[n] ** induction n with\n| zero => simp [invariants_le]\n| succ n ihn => exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _) ** case zero \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 \u22a2 invariants f \u2264 invariants f^[Nat.zero] ** simp [invariants_le] ** case succ \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u03b1 n : \u2115 ihn : invariants f \u2264 invariants f^[n] \u22a2 invariants f \u2264 invariants f^[Nat.succ n] ** exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.regular_of_isMulLeftInvariant ** 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 : SigmaFinite \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K h2K : Set.Nonempty (interior K) h\u03bcK : \u2191\u2191\u03bc K \u2260 \u22a4 \u22a2 Regular \u03bc ** rw [haarMeasure_unique \u03bc \u27e8\u27e8K, hK\u27e9, h2K\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 : SigmaFinite \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K h2K : Set.Nonempty (interior K) h\u03bcK : \u2191\u2191\u03bc K \u2260 \u22a4 \u22a2 Regular     (\u2191\u2191\u03bc \u2191{ toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K } \u2022       haarMeasure { toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K }) ** exact Regular.smul h\u03bcK ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.orderEmbOfFin_last ** \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 := k - 1, isLt := (_ : k - 1 < k) } = max' s (_ : Finset.Nonempty s) ** simp [orderEmbOfFin_apply, max'_eq_sorted_last, h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.integral_comp_inv_mul_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 : Set E g : \u211d \u2192 F a : \u211d \u22a2 \u222b (x : \u211d), g (x * a\u207b\u00b9) = |a| \u2022 \u222b (y : \u211d), g y ** simpa only [mul_comm] using integral_comp_inv_mul_left g a ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_closedBall_center ** E\u271d : Type u_1 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d inst\u271d\u00b9\u2070 : MeasurableSpace E\u271d inst\u271d\u2079 : BorelSpace E\u271d inst\u271d\u2078 : FiniteDimensional \u211d E\u271d \u03bc\u271d : Measure E\u271d inst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d F : Type u_2 inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F s : Set E\u271d E : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc x : E r : \u211d this : closedBall 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' closedBall x r \u22a2 \u2191\u2191\u03bc (closedBall x r) = \u2191\u2191\u03bc (closedBall 0 r) ** rw [this, measure_preimage_add] ** E\u271d : Type u_1 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d inst\u271d\u00b9\u2070 : MeasurableSpace E\u271d inst\u271d\u2079 : BorelSpace E\u271d inst\u271d\u2078 : FiniteDimensional \u211d E\u271d \u03bc\u271d : Measure E\u271d inst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d F : Type u_2 inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F s : Set E\u271d E : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc x : E r : \u211d \u22a2 closedBall 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' closedBall x r ** simp [preimage_add_closedBall] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_measurableEmbedding_real ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 \u22a2 \u2203 f, MeasurableEmbedding f ** obtain \u27e8s, hs, \u27e8e\u27e9\u27e9 := exists_subset_real_measurableEquiv \u03b1 ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 s : Set \u211d hs : MeasurableSet s e : \u03b1 \u2243\u1d50 \u2191s \u22a2 \u2203 f, MeasurableEmbedding f ** exact \u27e8(\u2191) \u2218 e, (MeasurableEmbedding.subtype_coe hs).comp e.measurableEmbedding\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.integral_truncation_le_integral_of_nonneg ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : Integrable f h'f : 0 \u2264 f A : \u211d \u22a2 \u222b (x : \u03b1), truncation f A x \u2202\u03bc \u2264 \u222b (x : \u03b1), f x \u2202\u03bc ** apply integral_mono_of_nonneg\n  (eventually_of_forall fun x => ?_) hf (eventually_of_forall fun x => ?_) ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : Integrable f h'f : 0 \u2264 f A : \u211d x : \u03b1 \u22a2 OfNat.ofNat 0 x \u2264 truncation f A x ** exact truncation_nonneg _ (h'f x) ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : Integrable f h'f : 0 \u2264 f A : \u211d x : \u03b1 \u22a2 truncation f A x \u2264 f x ** calc\n  truncation f A x \u2264 |truncation f A x| := le_abs_self _\n  _ \u2264 |f x| := (abs_truncation_le_abs_self _ _ _)\n  _ = f x := abs_of_nonneg (h'f x) ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.ediv_eq_iff_eq_mul_left ** a b c : Int H : b \u2260 0 H' : b \u2223 a \u22a2 a / b = c \u2194 a = c * b ** rw [Int.mul_comm] ** a b c : Int H : b \u2260 0 H' : b \u2223 a \u22a2 a / b = c \u2194 a = b * c ** exact Int.ediv_eq_iff_eq_mul_right H H' ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.zero_mem_smul_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u2074 : Zero \u03b1 inst\u271d\u00b3 : Zero \u03b2 inst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2 inst\u271d\u00b9 : DecidableEq \u03b2 s : Finset \u03b1 t : Finset \u03b2 inst\u271d : NoZeroSMulDivisors \u03b1 \u03b2 a : \u03b1 \u22a2 0 \u2208 s \u2022 t \u2194 0 \u2208 s \u2227 Finset.Nonempty t \u2228 0 \u2208 t \u2227 Finset.Nonempty s ** rw [\u2190 mem_coe, coe_smul, Set.zero_mem_smul_iff] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u2074 : Zero \u03b1 inst\u271d\u00b3 : Zero \u03b2 inst\u271d\u00b2 : SMulWithZero \u03b1 \u03b2 inst\u271d\u00b9 : DecidableEq \u03b2 s : Finset \u03b1 t : Finset \u03b2 inst\u271d : NoZeroSMulDivisors \u03b1 \u03b2 a : \u03b1 \u22a2 0 \u2208 \u2191s \u2227 Set.Nonempty \u2191t \u2228 0 \u2208 \u2191t \u2227 Set.Nonempty \u2191s \u2194 0 \u2208 s \u2227 Finset.Nonempty t \u2228 0 \u2208 t \u2227 Finset.Nonempty s ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.self_eq_mass_mul_normalize ** \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 s : Set \u03a9 \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s ** obtain rfl | h := eq_or_ne \u03bc 0 ** case inr \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 s : Set \u03a9 h : \u03bc \u2260 0 \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s ** have mass_nonzero : \u03bc.mass \u2260 0 := by rwa [\u03bc.mass_nonzero_iff] ** case inr \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 s : Set \u03a9 h : \u03bc \u2260 0 mass_nonzero : mass \u03bc \u2260 0 \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s ** simp only [normalize, dif_neg mass_nonzero] ** case inr \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 s : Set \u03a9 h : \u03bc \u2260 0 mass_nonzero : mass \u03bc \u2260 0 \u22a2 ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s) =     mass \u03bc *       ENNReal.toNNReal (\u2191\u2191\u2191{ val := \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc), property := (_ : IsProbabilityMeasure \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc)) } s) ** change \u03bc s = mass \u03bc * ((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) s ** case inr \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 s : Set \u03a9 h : \u03bc \u2260 0 mass_nonzero : mass \u03bc \u2260 0 \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) s)) s ** simp only [toMeasure_smul, Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n  Measure.nnreal_smul_coe_apply, ne_eq, mass_zero_iff, ENNReal.toNNReal_mul, ENNReal.toNNReal_coe,\n  mul_inv_cancel_left\u2080 mass_nonzero] ** case inl \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 s : Set \u03a9 \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u21910 s)) s = mass 0 * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize 0) s)) s ** simp only [zero_mass, coeFn_zero, Pi.zero_apply, zero_mul] ** case inl \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 s : Set \u03a9 \u22a2 ENNReal.toNNReal (\u2191\u2191\u21910 s) = 0 ** rfl ** \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 s : Set \u03a9 h : \u03bc \u2260 0 \u22a2 mass \u03bc \u2260 0 ** rwa [\u03bc.mass_nonzero_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_comp_mul_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 a b c d : \u211d f : \u211d \u2192 E hc : c \u2260 0 \u22a2 \u222b (x : \u211d) in a..b, f (x * c) = c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a * c..b * c, f x ** have A : MeasurableEmbedding fun x => x * c :=\n  (Homeomorph.mulRight\u2080 c hc).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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f : \u211d \u2192 E hc : c \u2260 0 A : MeasurableEmbedding fun x => x * c \u22a2 \u222b (x : \u211d) in a..b, f (x * c) = c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a * c..b * c, f x ** conv_rhs => rw [\u2190 Real.smul_map_volume_mul_right hc] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f : \u211d \u2192 E hc : c \u2260 0 A : MeasurableEmbedding fun x => x * c \u22a2 \u222b (x : \u211d) in a..b, f (x * c) =     c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a * c..b * c, f x \u2202ENNReal.ofReal |c| \u2022 Measure.map (fun x => x * c) volume ** simp_rw [integral_smul_measure, intervalIntegral, A.set_integral_map,\n  ENNReal.toReal_ofReal (abs_nonneg c)] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f : \u211d \u2192 E hc : c \u2260 0 A : MeasurableEmbedding fun x => x * c \u22a2 (\u222b (x : \u211d) in Ioc a b, f (x * c)) - \u222b (x : \u211d) in Ioc b a, f (x * c) =     c\u207b\u00b9 \u2022       |c| \u2022         ((\u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (a * c) (b * c), f (x * c)) -           \u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (b * c) (a * c), f (x * c)) ** cases' hc.lt_or_lt with h h ** case inl \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f : \u211d \u2192 E hc : c \u2260 0 A : MeasurableEmbedding fun x => x * c h : c < 0 \u22a2 (\u222b (x : \u211d) in Ioc a b, f (x * c)) - \u222b (x : \u211d) in Ioc b a, f (x * c) =     c\u207b\u00b9 \u2022       |c| \u2022         ((\u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (a * c) (b * c), f (x * c)) -           \u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (b * c) (a * c), f (x * c)) ** simp [h, mul_div_cancel, hc, abs_of_neg,\n  Measure.restrict_congr_set (\u03b1 := \u211d) (\u03bc := volume) Ico_ae_eq_Ioc] ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f : \u211d \u2192 E hc : c \u2260 0 A : MeasurableEmbedding fun x => x * c h : 0 < c \u22a2 (\u222b (x : \u211d) in Ioc a b, f (x * c)) - \u222b (x : \u211d) in Ioc b a, f (x * c) =     c\u207b\u00b9 \u2022       |c| \u2022         ((\u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (a * c) (b * c), f (x * c)) -           \u222b (x : \u211d) in (fun x => x * c) \u207b\u00b9' Ioc (b * c) (a * c), f (x * c)) ** simp [h, mul_div_cancel, hc, abs_of_pos] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Adapted.measurable_upcrossingsBefore ** \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 : Adapted \u2131 f hab : a < b \u22a2 Measurable (upcrossingsBefore a b f N) ** have : upcrossingsBefore a b f N = fun \u03c9 =>\n    \u2211 i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n \u03c9 < N}.indicator 1 i := by\n  ext \u03c9\n  exact upcrossingsBefore_eq_sum 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 : Adapted \u2131 f hab : a < b this :   upcrossingsBefore a b f N = fun \u03c9 =>     \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i \u22a2 Measurable (upcrossingsBefore a b f N) ** rw [this] ** \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 : Adapted \u2131 f hab : a < b this :   upcrossingsBefore a b f N = fun \u03c9 =>     \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i \u22a2 Measurable fun \u03c9 => \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i ** exact Finset.measurable_sum _ fun i _ => Measurable.indicator measurable_const <|\n  \u2131.le N _ (hf.isStoppingTime_upperCrossingTime.measurableSet_lt_of_pred N) ** \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 : Adapted \u2131 f hab : a < b \u22a2 upcrossingsBefore a b f N = fun \u03c9 =>     \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i ** 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 hf : Adapted \u2131 f hab : a < b \u03c9 : \u03a9 \u22a2 upcrossingsBefore a b f N \u03c9 = \u2211 i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n \u03c9 < N} 1 i ** exact upcrossingsBefore_eq_sum hab ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hausdorffMeasure_segment ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2079 : EMetricSpace X inst\u271d\u2078 : EMetricSpace Y inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : BorelSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : BorelSpace Y \ud835\udd5c : Type u_4 E\u271d : Type u_5 P : Type u_6 E : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E x y : E \u22a2 \u2191\u2191\u03bcH[1] (segment \u211d x y) = edist x y ** rw [\u2190 affineSegment_eq_segment, hausdorffMeasure_affineSegment] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.lcm_dvd ** m n k : Nat H1 : m \u2223 k H2 : n \u2223 k \u22a2 lcm m n \u2223 k ** match eq_zero_or_pos k with\n| .inl h => rw [h]; exact Nat.dvd_zero _\n| .inr kpos =>\n  apply Nat.dvd_of_mul_dvd_mul_left (gcd_pos_of_pos_left n (pos_of_dvd_of_pos H1 kpos))\n  rw [gcd_mul_lcm, \u2190 gcd_mul_right, Nat.mul_comm n k]\n  exact dvd_gcd (Nat.mul_dvd_mul_left _ H2) (Nat.mul_dvd_mul_right H1 _) ** m n k : Nat H1 : m \u2223 k H2 : n \u2223 k h : k = 0 \u22a2 lcm m n \u2223 k ** rw [h] ** m n k : Nat H1 : m \u2223 k H2 : n \u2223 k h : k = 0 \u22a2 lcm m n \u2223 0 ** exact Nat.dvd_zero _ ** m n k : Nat H1 : m \u2223 k H2 : n \u2223 k kpos : k > 0 \u22a2 lcm m n \u2223 k ** apply Nat.dvd_of_mul_dvd_mul_left (gcd_pos_of_pos_left n (pos_of_dvd_of_pos H1 kpos)) ** m n k : Nat H1 : m \u2223 k H2 : n \u2223 k kpos : k > 0 \u22a2 gcd m n * lcm m n \u2223 gcd m n * k ** rw [gcd_mul_lcm, \u2190 gcd_mul_right, Nat.mul_comm n k] ** m n k : Nat H1 : m \u2223 k H2 : n \u2223 k kpos : k > 0 \u22a2 m * n \u2223 gcd (m * k) (k * n) ** exact dvd_gcd (Nat.mul_dvd_mul_left _ H2) (Nat.mul_dvd_mul_right H1 _) ** Qed",
        "informal": ""
    },
    {
        "formal": "integrableOn_Ico_iff_integrableOn_Ioo ** \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 {a} \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": "surjOn_Ioc_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 b : \u03b1 \u22a2 SurjOn f (Ioc a b) (Ioc (f a) (f b)) ** simpa using surjOn_Ico_of_monotone_surjective h_mono.dual h_surj (toDual b) (toDual a) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.continuous_L1_toL1 ** \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc \u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f) ** by_cases hc'0 : 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 g : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 \u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f) ** rw [Metric.continuous_iff] ** 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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 \u22a2 \u2200 (b : { x // x \u2208 Lp G 1 }) (\u03b5 : \u211d),     \u03b5 > 0 \u2192       \u2203 \u03b4,         \u03b4 > 0 \u2227           \u2200 (a : { x // x \u2208 Lp G 1 }),             dist a b < \u03b4 \u2192               dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191b (_ : Integrable \u2191\u2191b)) < \u03b5 ** intro f \u03b5 h\u03b5_pos ** 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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 \u22a2 \u2203 \u03b4,     \u03b4 > 0 \u2227       \u2200 (a : { x // x \u2208 Lp G 1 }),         dist a f < \u03b4 \u2192 dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5 ** use \u03b5 / 2 / c'.toReal ** 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\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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 \u22a2 \u03b5 / 2 / ENNReal.toReal c' > 0 \u2227     \u2200 (a : { x // x \u2208 Lp G 1 }),       dist a f < \u03b5 / 2 / ENNReal.toReal c' \u2192         dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5 ** refine' \u27e8div_pos (half_pos h\u03b5_pos) (toReal_pos hc'0 hc'), _\u27e9 ** 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\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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 \u22a2 \u2200 (a : { x // x \u2208 Lp G 1 }),     dist a f < \u03b5 / 2 / ENNReal.toReal c' \u2192       dist (Integrable.toL1 \u2191\u2191a (_ : Integrable \u2191\u2191a)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5 ** intro g hfg ** 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\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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : dist g f < \u03b5 / 2 / ENNReal.toReal c' \u22a2 dist (Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) (Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) < \u03b5 ** rw [Lp.dist_def] at hfg \u22a2 ** 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\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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' \u22a2 ENNReal.toReal       (snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc') <     \u03b5 ** let h_int := fun f' : \u03b1 \u2192\u2081[\u03bc] G => (L1.integrable_coeFn f').of_measure_le_smul c' hc' h\u03bc'_le ** 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\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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') \u22a2 ENNReal.toReal       (snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc') <     \u03b5 ** have :\n  snorm (\u21d1(Integrable.toL1 g (h_int g)) - \u21d1(Integrable.toL1 f (h_int f))) 1 \u03bc' =\n    snorm (\u21d1g - \u21d1f) 1 \u03bc' :=\n  snorm_congr_ae ((Integrable.coeFn_toL1 _).sub (Integrable.coeFn_toL1 _)) ** 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\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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u22a2 ENNReal.toReal       (snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc') <     \u03b5 ** rw [this] ** 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\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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') < \u03b5 ** have h_snorm_ne_top : snorm (\u21d1g - \u21d1f) 1 \u03bc \u2260 \u221e := by\n  rw [\u2190 snorm_congr_ae (Lp.coeFn_sub _ _)]; exact Lp.snorm_ne_top _ ** 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\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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 \u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') < \u03b5 ** have h_snorm_ne_top' : snorm (\u21d1g - \u21d1f) 1 \u03bc' \u2260 \u221e := by\n  refine' ((snorm_mono_measure _ h\u03bc'_le).trans_lt _).ne\n  rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]\n  refine' ENNReal.mul_lt_top _ h_snorm_ne_top\n  simp [hc', hc'0] ** 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\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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 h_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4 \u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') < \u03b5 ** calc\n  (snorm (\u21d1g - \u21d1f) 1 \u03bc').toReal \u2264 (c' * snorm (\u21d1g - \u21d1f) 1 \u03bc).toReal := by\n    rw [toReal_le_toReal h_snorm_ne_top' (ENNReal.mul_ne_top hc' h_snorm_ne_top)]\n    refine' (snorm_mono_measure (\u21d1g - \u21d1f) h\u03bc'_le).trans _\n    rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]\n    simp\n  _ = c'.toReal * (snorm (\u21d1g - \u21d1f) 1 \u03bc).toReal := toReal_mul\n  _ \u2264 c'.toReal * (\u03b5 / 2 / c'.toReal) :=\n    (mul_le_mul le_rfl hfg.le toReal_nonneg toReal_nonneg)\n  _ = \u03b5 / 2 := by\n    refine' mul_div_cancel' (\u03b5 / 2) _; rw [Ne.def, toReal_eq_zero_iff]; simp [hc', hc'0]\n  _ < \u03b5 := half_lt_self h\u03b5_pos ** 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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 \u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f) ** have h\u03bc'0 : \u03bc' = 0 := by rw [\u2190 Measure.nonpos_iff_eq_zero']; refine' h\u03bc'_le.trans _; simp [hc'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\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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 h\u03bc'0 : \u03bc' = 0 \u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f) ** have h_im_zero :\n  (fun f : \u03b1 \u2192\u2081[\u03bc] G =>\n      (Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f)).toL1 f) =\n    0 := by\n  ext1 f; ext1; simp_rw [h\u03bc'0]; simp only [ae_zero, EventuallyEq, eventually_bot] ** 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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 h\u03bc'0 : \u03bc' = 0 h_im_zero : (fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) = 0 \u22a2 Continuous fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f) ** rw [h_im_zero] ** 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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 h\u03bc'0 : \u03bc' = 0 h_im_zero : (fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) = 0 \u22a2 Continuous 0 ** exact continuous_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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 \u22a2 \u03bc' = 0 ** rw [\u2190 Measure.nonpos_iff_eq_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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 \u22a2 \u03bc' \u2264 0 ** refine' h\u03bc'_le.trans _ ** \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 \u22a2 c' \u2022 \u03bc \u2264 0 ** simp [hc'0] ** \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 h\u03bc'0 : \u03bc' = 0 \u22a2 (fun f => Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) = 0 ** 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\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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 h\u03bc'0 : \u03bc' = 0 f : { x // x \u2208 Lp G 1 } \u22a2 Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f) = OfNat.ofNat 0 f ** ext1 ** case h.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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 h\u03bc'0 : \u03bc' = 0 f : { x // x \u2208 Lp G 1 } \u22a2 \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) =\u1d50[\u03bc'] \u2191\u2191(OfNat.ofNat 0 f) ** simp_rw [h\u03bc'0] ** case h.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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : c' = 0 h\u03bc'0 : \u03bc' = 0 f : { x // x \u2208 Lp G 1 } \u22a2 \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) =\u1d50[0] \u2191\u2191(OfNat.ofNat 0 f) ** simp only [ae_zero, EventuallyEq, eventually_bot] ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 ** rw [\u2190 snorm_congr_ae (Lp.coeFn_sub _ _)] ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u22a2 snorm (\u2191\u2191(g - f)) 1 \u03bc \u2260 \u22a4 ** exact Lp.snorm_ne_top _ ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 \u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4 ** refine' ((snorm_mono_measure _ h\u03bc'_le).trans_lt _).ne ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 \u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 (c' \u2022 \u03bc) < \u22a4 ** rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_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 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 \u22a2 c' ^ ENNReal.toReal (1 / 1) * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc < \u22a4 ** refine' ENNReal.mul_lt_top _ h_snorm_ne_top ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 \u22a2 c' ^ ENNReal.toReal (1 / 1) \u2260 \u22a4 ** simp [hc', hc'0] ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 h_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4 \u22a2 ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc') \u2264 ENNReal.toReal (c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) ** rw [toReal_le_toReal h_snorm_ne_top' (ENNReal.mul_ne_top hc' h_snorm_ne_top)] ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 h_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4 \u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2264 c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc ** refine' (snorm_mono_measure (\u21d1g - \u21d1f) h\u03bc'_le).trans _ ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 h_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4 \u22a2 snorm (\u2191\u2191g - \u2191\u2191f) 1 (c' \u2022 \u03bc) \u2264 c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc ** rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_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 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 h_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4 \u22a2 c' ^ ENNReal.toReal (1 / 1) * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2264 c' * snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc ** simp ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 h_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4 \u22a2 ENNReal.toReal c' * (\u03b5 / 2 / ENNReal.toReal c') = \u03b5 / 2 ** refine' mul_div_cancel' (\u03b5 / 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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 h_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4 \u22a2 ENNReal.toReal c' \u2260 0 ** rw [Ne.def, toReal_eq_zero_iff] ** \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\u271d : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hc'0 : \u00acc' = 0 f : { x // x \u2208 Lp G 1 } \u03b5 : \u211d h\u03b5_pos : \u03b5 > 0 g : { x // x \u2208 Lp G 1 } hfg : ENNReal.toReal (snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc) < \u03b5 / 2 / ENNReal.toReal c' h_int : \u2200 (f' : { x // x \u2208 Lp G 1 }), Integrable \u2191\u2191f' :=   fun f' => Integrable.of_measure_le_smul c' hc' h\u03bc'_le (L1.integrable_coeFn f') this :   snorm (\u2191\u2191(Integrable.toL1 \u2191\u2191g (_ : Integrable \u2191\u2191g)) - \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f))) 1 \u03bc' =     snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' h_snorm_ne_top : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc \u2260 \u22a4 h_snorm_ne_top' : snorm (\u2191\u2191g - \u2191\u2191f) 1 \u03bc' \u2260 \u22a4 \u22a2 \u00ac(c' = 0 \u2228 c' = \u22a4) ** simp [hc', hc'0] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.quasiMeasurePreserving_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 : IsMulLeftInvariant \u03bc \u22a2 QuasiMeasurePreserving Inv.inv ** refine' \u27e8measurable_inv, AbsolutelyContinuous.mk fun s hsm h\u03bcs => _\u27e9 ** 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\u271d : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulLeftInvariant \u03bc s : Set G hsm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191(map Inv.inv \u03bc) s = 0 ** rw [map_apply measurable_inv hsm, inv_preimage] ** 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\u271d : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulLeftInvariant \u03bc s : Set G hsm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0 ** have hf : Measurable fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9) :=\n  (measurable_snd.mul measurable_fst).prod_mk measurable_fst.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\u271d : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulLeftInvariant \u03bc s : Set G hsm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9) \u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0 ** suffices map (fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9)) (\u03bc.prod \u03bc) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0 by\n  simpa only [(measurePreserving_mul_prod_inv \u03bc \u03bc).map_eq, prod_prod, mul_eq_zero (M\u2080 := \u211d\u22650\u221e),\n    or_self_iff] using 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\u271d : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulLeftInvariant \u03bc s : Set G hsm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9) \u22a2 \u2191\u2191(map (fun z => (z.2 * z.1, z.1\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0 ** have hsm' : MeasurableSet (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) := hsm.inv.prod hsm.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\u271d : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulLeftInvariant \u03bc s : Set G hsm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9) hsm' : MeasurableSet (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) \u22a2 \u2191\u2191(map (fun z => (z.2 * z.1, z.1\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0 ** simp_rw [map_apply hf hsm', prod_apply_symm (\u03bc := \u03bc) (\u03bd := \u03bc) (hf hsm'), preimage_preimage,\n  mk_preimage_prod, inv_preimage, inv_inv, measure_mono_null (inter_subset_right _ _) h\u03bcs,\n  lintegral_zero] ** 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\u271d : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulLeftInvariant \u03bc s : Set G hsm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = 0 hf : Measurable fun z => (z.2 * z.1, z.1\u207b\u00b9) this : \u2191\u2191(map (fun z => (z.2 * z.1, z.1\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0 \u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0 ** simpa only [(measurePreserving_mul_prod_inv \u03bc \u03bc).map_eq, prod_prod, mul_eq_zero (M\u2080 := \u211d\u22650\u221e),\n  or_self_iff] using this ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.mem_ite_empty_left ** \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 \u00acp \u2227 x \u2208 t ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image\u2082_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 h : Finset.Nonempty s \u22a2 \u2191(image\u2082 (fun x y => y) s t) = \u2191t ** 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 h : Finset.Nonempty s \u22a2 image2 (fun x y => y) \u2191s \u2191t = \u2191t ** exact image2_right h ** Qed",
        "informal": ""
    },
    {
        "formal": "AddCircle.integral_preimage ** T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E \u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b ** have m : MeasurableSet (Ioc t (t + T)) := measurableSet_Ioc ** T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E m : MeasurableSet (Ioc t (t + T)) \u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b ** have := integral_map_equiv (\u03bc := volume) (measurableEquivIoc T t).symm f ** T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E m : MeasurableSet (Ioc t (t + T)) this :   \u222b (y : AddCircle T), f y \u2202Measure.map (\u2191(MeasurableEquiv.symm (measurableEquivIoc T t))) volume =     \u222b (x : \u2191(Ioc t (t + T))), f (\u2191(MeasurableEquiv.symm (measurableEquivIoc T t)) x) \u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b ** simp only [measurableEquivIoc, equivIoc, QuotientAddGroup.equivIocMod, MeasurableEquiv.symm_mk,\n  MeasurableEquiv.coe_mk, Equiv.coe_fn_symm_mk] at this ** T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E m : MeasurableSet (Ioc t (t + T)) this : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x \u22a2 \u222b (a : \u211d) in Ioc t (t + T), f \u2191a = \u222b (b : AddCircle T), f b ** rw [\u2190 (AddCircle.measurePreserving_mk T t).map_eq, set_integral_eq_subtype m, \u2190 this] ** T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E m : MeasurableSet (Ioc t (t + T)) this : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x \u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume =     \u222b (b : AddCircle T), f b \u2202Measure.map QuotientAddGroup.mk (Measure.restrict volume (Ioc t (t + T))) ** have : ((\u2191) : Ioc t (t + T) \u2192 AddCircle T) = ((\u2191) : \u211d \u2192 AddCircle T) \u2218 ((\u2191) : _ \u2192 \u211d) := by\n  ext1 x; rfl ** T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E m : MeasurableSet (Ioc t (t + T)) this\u271d : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x this : (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val \u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume =     \u222b (b : AddCircle T), f b \u2202Measure.map QuotientAddGroup.mk (Measure.restrict volume (Ioc t (t + T))) ** simp_rw [this] ** T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E m : MeasurableSet (Ioc t (t + T)) this\u271d : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x this : (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val \u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map (QuotientAddGroup.mk \u2218 Subtype.val) volume =     \u222b (y : AddCircle T), f y \u2202Measure.map QuotientAddGroup.mk (Measure.restrict volume (Ioc t (t + T))) ** rw [\u2190 map_map AddCircle.measurable_mk' measurable_subtype_coe, \u2190 map_comap_subtype_coe m] ** T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E m : MeasurableSet (Ioc t (t + T)) this\u271d : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x this : (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val \u22a2 \u222b (y : AddCircle T), f y \u2202Measure.map QuotientAddGroup.mk (Measure.map Subtype.val volume) =     \u222b (y : AddCircle T),       f y \u2202Measure.map QuotientAddGroup.mk (Measure.map Subtype.val (Measure.comap Subtype.val volume)) ** rfl ** T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E m : MeasurableSet (Ioc t (t + T)) this : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x \u22a2 (fun x => \u2191\u2191x) = QuotientAddGroup.mk \u2218 Subtype.val ** ext1 x ** case h T : \u211d hT : Fact (0 < T) E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E t : \u211d f : AddCircle T \u2192 E m : MeasurableSet (Ioc t (t + T)) this : \u222b (y : AddCircle T), f y \u2202Measure.map (fun x => \u2191\u2191x) volume = \u222b (x : \u2191(Ioc t (t + T))), f \u2191\u2191x x : \u2191(Ioc t (t + T)) \u22a2 \u2191\u2191x = (QuotientAddGroup.mk \u2218 Subtype.val) x ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.sum_zero ** \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 \u22a2 (kernel.sum fun x => 0) = 0 ** 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 : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(kernel.sum fun x => 0) a) s = \u2191\u2191(\u21910 a) s ** rw [sum_apply' _ 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 : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 \u2211' (n : \u03b9), \u2191\u2191(\u21910 a) s = \u2191\u2191(\u21910 a) s ** simp only [zero_apply, Measure.coe_zero, Pi.zero_apply, tsum_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.MeasurableSet.nullMeasurableSet_subtype_coe ** \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 : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t \u22a2 NullMeasurableSet (Subtype.val '' t) ** refine'\n  generateFrom_induction (p := fun t : Set s => NullMeasurableSet ((\u2191) '' t) \u03bc)\n    { t : Set s | \u2203 s' : Set \u03b1, MeasurableSet s' \u2227 (\u2191) \u207b\u00b9' s' = t } _ _ _ _ ht ** case 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 t : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t \u22a2 \u2200 (t : Set \u2191s),     t \u2208 {t | \u2203 s', MeasurableSet s' \u2227 Subtype.val \u207b\u00b9' s' = t} \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t ** rintro t' \u27e8s', hs', rfl\u27e9 ** case refine'_1.intro.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 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'\u271d t\u271d : Set \u03b1 t : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t s' : Set \u03b1 hs' : MeasurableSet s' \u22a2 NullMeasurableSet (Subtype.val '' (Subtype.val \u207b\u00b9' s')) ** rw [Subtype.image_preimage_coe] ** case refine'_1.intro.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 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'\u271d t\u271d : Set \u03b1 t : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t s' : Set \u03b1 hs' : MeasurableSet s' \u22a2 NullMeasurableSet (s' \u2229 s) ** exact hs'.nullMeasurableSet.inter hs ** case 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 t : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t \u22a2 (fun t => NullMeasurableSet (Subtype.val '' t)) \u2205 ** simp only [image_empty, nullMeasurableSet_empty] ** case 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 t : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t \u22a2 \u2200 (t : Set \u2191s), (fun t => NullMeasurableSet (Subtype.val '' t)) t \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t\u1d9c ** intro t' ** case 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 t : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t t' : Set \u2191s \u22a2 (fun t => NullMeasurableSet (Subtype.val '' t)) t' \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t'\u1d9c ** simp only [\u2190 range_diff_image Subtype.coe_injective, Subtype.range_coe_subtype, setOf_mem_eq] ** case 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 t : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t t' : Set \u2191s \u22a2 NullMeasurableSet (Subtype.val '' t') \u2192 NullMeasurableSet (s \\ (fun a => \u2191a) '' t') ** exact hs.diff ** case refine'_4 \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 : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t \u22a2 \u2200 (f : \u2115 \u2192 Set \u2191s),     (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) \u2192       (fun t => NullMeasurableSet (Subtype.val '' t)) (\u22c3 i, f i) ** intro f ** case refine'_4 \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 : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t f : \u2115 \u2192 Set \u2191s \u22a2 (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) \u2192     (fun t => NullMeasurableSet (Subtype.val '' t)) (\u22c3 i, f i) ** dsimp only [] ** case refine'_4 \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 : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t f : \u2115 \u2192 Set \u2191s \u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n)) \u2192 NullMeasurableSet (Subtype.val '' \u22c3 i, f i) ** rw [image_iUnion] ** case refine'_4 \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 : Set \u2191s hs : NullMeasurableSet s ht : MeasurableSet t f : \u2115 \u2192 Set \u2191s \u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n)) \u2192 NullMeasurableSet (\u22c3 i, Subtype.val '' f i) ** exact NullMeasurableSet.iUnion ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.sub_to_nat ** \u03b1 : Type u_1 m n : Num \u22a2 \u2191(ofZNum (sub' m n)) = \u2191m - \u2191n ** rw [ofZNum_toNat, cast_sub', \u2190 to_nat_to_int, \u2190 to_nat_to_int, Int.toNat_sub] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.expand_WF ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H : Buckets.WF buckets x\u271d\u00b3 : AssocList \u03b1 \u03b2 x\u271d\u00b2 : x\u271d\u00b3 \u2208 (Buckets.mk (Array.size buckets.val * 2)).val.data x\u271d\u00b9 : \u03b1 \u00d7 \u03b2 x\u271d : x\u271d\u00b9 \u2208 AssocList.toList x\u271d\u00b3 \u22a2 (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) < 0) x\u271d\u00b9.fst x\u271d\u00b9.snd ** simp_all [Buckets.mk, List.mem_replicate] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket \u22a2 Buckets.WF (expand.go i source target) ** unfold expand.go ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket \u22a2 Buckets.WF     (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) ** split ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source \u22a2 Buckets.WF     (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) ** refine go (i+1) (fun _ hl => ?_) (fun i h => ?_) ?_ ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source inst\u271d\u00b9 : LawfulHashable \u03b1 inst\u271d : PartialEquivBEq \u03b1 x\u271d : AssocList \u03b1 \u03b2 hl : x\u271d \u2208 (Array.set source { val := i, isLt := H } AssocList.nil).data \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d) ** match List.mem_or_eq_of_mem_set hl with\n| .inl hl => exact hs\u2081 _ hl\n| .inr e => exact e \u25b8 .nil ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source inst\u271d\u00b9 : LawfulHashable \u03b1 inst\u271d : PartialEquivBEq \u03b1 x\u271d : AssocList \u03b1 \u03b2 hl\u271d : x\u271d \u2208 (Array.set source { val := i, isLt := H } AssocList.nil).data hl : x\u271d \u2208 source.data \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d) ** exact hs\u2081 _ hl ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source inst\u271d\u00b9 : LawfulHashable \u03b1 inst\u271d : PartialEquivBEq \u03b1 x\u271d : AssocList \u03b1 \u03b2 hl : x\u271d \u2208 (Array.set source { val := i, isLt := H } AssocList.nil).data e : x\u271d = AssocList.nil \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d) ** exact e \u25b8 .nil ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i\u271d : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket H : i\u271d < Array.size source i : Nat h : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil) \u22a2 AssocList.All     (fun k x =>       USize.toNat (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil)) = i)     (Array.set source { val := i\u271d, isLt := H } AssocList.nil)[i] ** simp [Array.getElem_eq_data_get, List.get_set] ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i\u271d : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket H : i\u271d < Array.size source i : Nat h : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil) \u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = i)     (if i\u271d = i then AssocList.nil else List.get source.data { val := i, isLt := (_ : i < List.length source.data) }) ** split ** case refine_2.inl \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i\u271d : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket H : i\u271d < Array.size source i : Nat h : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil) h\u271d : i\u271d = i \u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = i) AssocList.nil ** intro. ** case refine_2.inr \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i\u271d : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket H : i\u271d < Array.size source i : Nat h : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil) h\u271d : \u00aci\u271d = i \u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = i)     (List.get source.data { val := i, isLt := (_ : i < List.length source.data) }) ** exact hs\u2082 _ (by simp_all) ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i\u271d : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i\u271d) bucket H : i\u271d < Array.size source i : Nat h : i < Array.size (Array.set source { val := i\u271d, isLt := H } AssocList.nil) h\u271d : \u00aci\u271d = i \u22a2 i < Array.size source ** simp_all ** case refine_3 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source \u22a2 Buckets.WF (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })).val.data \u2192         AssocList.All           (fun k x =>             USize.toNat                 (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i, isLt := H } AssocList.nil)) <               i + 1)           bucket ** let rank (k : \u03b1) := ((hash k).toUSize % source.size).toNat ** case refine_3 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source rank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source) \u22a2 Buckets.WF (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })).val.data \u2192         AssocList.All           (fun k x =>             USize.toNat                 (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i, isLt := H } AssocList.nil)) <               i + 1)           bucket ** have := expand_WF.foldl rank ?_ (hs\u2082 _ H) ht.1 (fun _ h\u2081 _ h\u2082 => ?_) ** case refine_3.refine_3 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source rank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source) this :   Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data \u2192         AssocList.All (fun k x => rank k \u2264 i) bucket \u22a2 Buckets.WF (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (AssocList.foldl reinsertAux target (Array.get source { val := i, isLt := H })).val.data \u2192         AssocList.All           (fun k x =>             USize.toNat                 (UInt64.toUSize (hash k) % Array.size (Array.set source { val := i, isLt := H } AssocList.nil)) <               i + 1)           bucket ** simp ** case refine_3.refine_3 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source rank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source) this :   Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data \u2192         AssocList.All (fun k x => rank k \u2264 i) bucket \u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i + 1) bucket ** exact \u27e8this.1, fun _ h\u2081 _ h\u2082 => Nat.lt_succ_of_le (this.2 _ h\u2081 _ h\u2082)\u27e9 ** case refine_3.refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source rank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source) \u22a2 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList source[i]) ** exact hs\u2081 _ (Array.getElem_mem_data ..) ** case refine_3.refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source rank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source) x\u271d\u00b9 : AssocList \u03b1 \u03b2 h\u2081 : x\u271d\u00b9 \u2208 target.val.data x\u271d : \u03b1 \u00d7 \u03b2 h\u2082 : x\u271d \u2208 AssocList.toList x\u271d\u00b9 \u22a2 (fun k x =>       rank k \u2264 i \u2227         \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),           x \u2208 AssocList.toList source[i] \u2192 \u00ac(x.fst == k) = true)     x\u271d.fst x\u271d.snd ** have := ht.2 _ h\u2081 _ h\u2082 ** case refine_3.refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source rank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source) x\u271d\u00b9 : AssocList \u03b1 \u03b2 h\u2081 : x\u271d\u00b9 \u2208 target.val.data x\u271d : \u03b1 \u00d7 \u03b2 h\u2082 : x\u271d \u2208 AssocList.toList x\u271d\u00b9 this : (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) x\u271d.fst x\u271d.snd \u22a2 (fun k x =>       rank k \u2264 i \u2227         \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),           x \u2208 AssocList.toList source[i] \u2192 \u00ac(x.fst == k) = true)     x\u271d.fst x\u271d.snd ** refine \u27e8Nat.le_of_lt this, fun _ h h' => Nat.ne_of_lt this ?_\u27e9 ** case refine_3.refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H\u271d : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket H : i < Array.size source rank : \u03b1 \u2192 Nat := fun k => USize.toNat (UInt64.toUSize (hash k) % Array.size source) x\u271d\u00b2 : AssocList \u03b1 \u03b2 h\u2081 : x\u271d\u00b2 \u2208 target.val.data x\u271d\u00b9 : \u03b1 \u00d7 \u03b2 h\u2082 : x\u271d\u00b9 \u2208 AssocList.toList x\u271d\u00b2 this : (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) x\u271d\u00b9.fst x\u271d\u00b9.snd inst\u271d\u00b9 : PartialEquivBEq \u03b1 inst\u271d : LawfulHashable \u03b1 x\u271d : \u03b1 \u00d7 \u03b2 h : x\u271d \u2208 AssocList.toList source[i] h' : (x\u271d.fst == x\u271d\u00b9.fst) = true \u22a2 USize.toNat (UInt64.toUSize (hash x\u271d\u00b9.fst) % Array.size source) = i ** exact LawfulHashable.hash_eq h' \u25b8 hs\u2082 _ H _ h ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 H : Buckets.WF buckets i : Nat source : Array (AssocList \u03b1 \u03b2) hs\u2081 :   \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 source.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) hs\u2082 :   \u2200 (j : Nat) (h : j < Array.size source),     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) = j) source[j] target : Buckets \u03b1 \u03b2 ht :   Buckets.WF target \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 target.val.data \u2192         AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size source) < i) bucket h\u271d : \u00aci < Array.size source \u22a2 Buckets.WF target ** exact ht.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "aemeasurable_add_measure_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\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f g : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 \u22a2 AEMeasurable f \u2194 AEMeasurable f \u2227 AEMeasurable f ** rw [\u2190 sum_cond, aemeasurable_sum_measure_iff, Bool.forall_bool, and_comm] ** \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 g : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 \u22a2 AEMeasurable f \u2227 AEMeasurable f \u2194 AEMeasurable f \u2227 AEMeasurable f ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_prod_smul ** \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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd ** 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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : CompleteSpace E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd  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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : \u00acCompleteSpace E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd ** swap ** 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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : \u00acCompleteSpace E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd  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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : CompleteSpace E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd ** simp [integral, hE] ** 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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : CompleteSpace E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd ** by_cases h : Integrable (fun z : \u03b1 \u00d7 \u03b2 => f z.1 \u2022 g z.2) (\u03bc.prod \u03bd) ** 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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : CompleteSpace E h : \u00acIntegrable fun z => f z.1 \u2022 g z.2 \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd ** have H : \u00acIntegrable f \u03bc \u2228 \u00acIntegrable g \u03bd := by\n  contrapose! h\n  exact h.1.prod_smul h.2 ** 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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : CompleteSpace E h : \u00acIntegrable fun z => f z.1 \u2022 g z.2 H : \u00acIntegrable f \u2228 \u00acIntegrable g \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd ** cases' H with H H <;> simp [integral_undef h, integral_undef H] ** 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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : CompleteSpace E h : Integrable fun z => f z.1 \u2022 g z.2 \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2022 g z.2 \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd ** rw [integral_prod _ h] ** 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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : CompleteSpace E h : Integrable fun z => f z.1 \u2022 g z.2 \u22a2 \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y).1 \u2022 g (x, y).2 \u2202\u03bd \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 \u222b (y : \u03b2), g y \u2202\u03bd ** simp_rw [integral_smul, integral_smul_const] ** \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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : CompleteSpace E h : \u00acIntegrable fun z => f z.1 \u2022 g z.2 \u22a2 \u00acIntegrable f \u2228 \u00acIntegrable g ** contrapose! 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\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1' inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2' inst\u271d\u2078 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : SigmaFinite \u03bd inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup E' inst\u271d\u00b2 : NormedSpace \u211d E' \ud835\udd5c : Type u_8 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hE : CompleteSpace E h : Integrable f \u2227 Integrable g \u22a2 Integrable fun z => f z.1 \u2022 g z.2 ** exact h.1.prod_smul h.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.subsingleton_iff_le_one ** n : Nat \u22a2 Subsingleton (Fin n) \u2194 n \u2264 1 ** (match n with | 0 | 1 | n+2 => ?_) <;> try simp ** n : Nat \u22a2 Subsingleton (Fin n) \u2194 n \u2264 1 ** match n with | 0 | 1 | n+2 => ?_ ** case refine_2 n : Nat \u22a2 Subsingleton (Fin 1) \u2194 1 \u2264 1 ** simp ** case refine_1 n : Nat \u22a2 Subsingleton (Fin 0) ** exact \u27e8fun.\u27e9 ** case refine_2 n : Nat \u22a2 Subsingleton (Fin 1) ** exact \u27e8fun \u27e80, _\u27e9 \u27e80, _\u27e9 => rfl\u27e9 ** case refine_3 n\u271d n : Nat \u22a2 Subsingleton (Fin (n + 2)) \u2194 n + 2 \u2264 1 ** exact iff_of_false (fun h => Fin.ne_of_lt zero_lt_one (h.elim ..)) (of_decide_eq_false rfl) ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.units_eq_one_or ** u : \u2124\u02e3 \u22a2 u = 1 \u2228 u = -1 ** simpa only [Units.ext_iff, units_natAbs] using natAbs_eq u ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.disjoint_union_right ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 x y z : Finmap \u03b2 \u22a2 Disjoint x (y \u222a z) \u2194 Disjoint x y \u2227 Disjoint x z ** rw [Disjoint.symm_iff, disjoint_union_left, Disjoint.symm_iff _ x, Disjoint.symm_iff _ x] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.induction_on_pi_of_choice ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) \u22a2 p f ** cases nonempty_fintype \u03b9 ** case intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 \u22a2 p f ** induction' hs : univ.sigma f using Finset.strongInductionOn with s ihs generalizing f ** case intro.a \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs\u271d : Finset.sigma univ f\u271d = x\u271d s : Finset ((i : \u03b9) \u00d7 \u03b1 i) ihs : \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)), t \u2282 s \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f f : (i : \u03b9) \u2192 Finset (\u03b1 i) hs : Finset.sigma univ f = s \u22a2 p f ** subst s ** case intro.a \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f \u22a2 p f ** cases' eq_empty_or_nonempty (univ.sigma f) with he hne ** case intro.a.inl \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f he : Finset.sigma univ f = \u2205 \u22a2 p f ** convert h0 using 1 ** case h.e'_1 \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f he : Finset.sigma univ f = \u2205 \u22a2 f = fun x => \u2205 ** simpa [funext_iff] using he ** case intro.a.inr \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hne : Finset.Nonempty (Finset.sigma univ f) \u22a2 p f ** rcases sigma_nonempty.1 hne with \u27e8i, -, hi\u27e9 ** case intro.a.inr.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) \u22a2 p f ** rcases H_ex i (f i) hi with \u27e8x, x_mem, hr\u27e9 ** case intro.a.inr.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i hr : r i x (erase (f i) x) \u22a2 p f ** set g := update f i ((f i).erase x) with hg ** case intro.a.inr.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i hr : r i x (erase (f i) x) g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) hg : g = update f i (erase (f i) x) \u22a2 p f ** have hx' : x \u2209 g i := by\n  rw [hg, update_same]\n  apply not_mem_erase ** case intro.a.inr.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i hr : r i x (erase (f i) x) g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) hg : g = update f i (erase (f i) x) hx' : \u00acx \u2208 g i \u22a2 p f ** rw [show f = update g i (insert x (g i)) by\n  rw [hg, update_idem, update_same, insert_erase x_mem, update_eq_self]] at hr ihs \u22a2 ** case intro.a.inr.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hr : r i x (erase (update g i (insert x (g i)) i) x) hg : g = update f i (erase (f i) x) hx' : \u00acx \u2208 g i \u22a2 p (update g i (insert x (g i))) ** clear hg ** case intro.a.inr.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hr : r i x (erase (update g i (insert x (g i)) i) x) hx' : \u00acx \u2208 g i \u22a2 p (update g i (insert x (g i))) ** rw [update_same, erase_insert hx'] at hr ** case intro.a.inr.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hr : r i x (g i) hx' : \u00acx \u2208 g i \u22a2 p (update g i (insert x (g i))) ** refine step _ _ _ hr (ihs (univ.sigma g) ?_ _ rfl) ** case intro.a.inr.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hr : r i x (g i) hx' : \u00acx \u2208 g i \u22a2 Finset.sigma univ g \u2282 Finset.sigma univ (update g i (insert x (g i))) ** rw [ssubset_iff_of_subset (sigma_mono (Subset.refl _) _)] ** case intro.a.inr.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hr : r i x (g i) hx' : \u00acx \u2208 g i \u22a2 \u2203 x_1, x_1 \u2208 Finset.sigma univ (update g i (insert x (g i))) \u2227 \u00acx_1 \u2208 Finset.sigma univ g  \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hr : r i x (g i) hx' : \u00acx \u2208 g i \u22a2 \u2200 (i_1 : \u03b9), g i_1 \u2286 update g i (insert x (g i)) i_1 ** exacts [\u27e8\u27e8i, x\u27e9, mem_sigma.2 \u27e8mem_univ _, by simp\u27e9, by simp [hx']\u27e9,\n  (@le_update_iff _ _ _ _ g g i _).2 \u27e8subset_insert _ _, fun _ _ \u21a6 le_rfl\u27e9] ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i hr : r i x (erase (f i) x) g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) hg : g = update f i (erase (f i) x) \u22a2 \u00acx \u2208 g i ** rw [hg, update_same] ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ f \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i hr : r i x (erase (f i) x) g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) hg : g = update f i (erase (f i) x) \u22a2 \u00acx \u2208 erase (f i) x ** apply not_mem_erase ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hr : r i x (erase (update g i (insert x (g i)) i) x) hg : g = update f i (erase (f i) x) hx' : \u00acx \u2208 g i \u22a2 f = update g i (insert x (g i)) ** rw [hg, update_idem, update_same, insert_erase x_mem, update_eq_self] ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hr : r i x (g i) hx' : \u00acx \u2208 g i \u22a2 { fst := i, snd := x }.snd \u2208 update g i (insert x (g i)) { fst := i, snd := x }.fst ** simp ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : Finite \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) r : (i : \u03b9) \u2192 \u03b1 i \u2192 Finset (\u03b1 i) \u2192 Prop H_ex : \u2200 (i : \u03b9) (s : Finset (\u03b1 i)), Finset.Nonempty s \u2192 \u2203 x, x \u2208 s \u2227 r i x (erase s x) p : ((i : \u03b9) \u2192 Finset (\u03b1 i)) \u2192 Prop f\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i) h0 : p fun x => \u2205 step : \u2200 (g : (i : \u03b9) \u2192 Finset (\u03b1 i)) (i : \u03b9) (x : \u03b1 i), r i x (g i) \u2192 p g \u2192 p (update g i (insert x (g i))) val\u271d : Fintype \u03b9 x\u271d : Finset ((i : \u03b9) \u00d7 \u03b1 i) hs : Finset.sigma univ f\u271d = x\u271d f : (i : \u03b9) \u2192 Finset (\u03b1 i) hne : Finset.Nonempty (Finset.sigma univ f) i : \u03b9 hi : Finset.Nonempty (f i) x : \u03b1 i x_mem : x \u2208 f i g : (a : \u03b9) \u2192 Finset (\u03b1 a) := update f i (erase (f i) x) ihs :   \u2200 (t : Finset ((i : \u03b9) \u00d7 \u03b1 i)),     t \u2282 Finset.sigma univ (update g i (insert x (g i))) \u2192 \u2200 (f : (i : \u03b9) \u2192 Finset (\u03b1 i)), Finset.sigma univ f = t \u2192 p f hr : r i x (g i) hx' : \u00acx \u2208 g i \u22a2 \u00ac{ fst := i, snd := x } \u2208 Finset.sigma univ g ** simp [hx'] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.integrable_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 hf : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a) \u22a2 Integrable f \u2194 (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, Integrable fun y => f (x, y)) \u2227 Integrable fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x) ** simp only [Integrable, hasFiniteIntegral_compProd_iff' hf, hf.norm.integral_kernel_compProd,\n  hf, hf.compProd_mk_left, eventually_and, true_and_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_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 f\u271d : \u03b1 \u2192 \u211d inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d 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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u22a2 0 \u2264\u1d50[\u03bc] f ** apply ae_of_forall_measure_lt_top_ae_restrict ** case h \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 f\u271d : \u03b1 \u2192 \u211d inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d 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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, OfNat.ofNat 0 x \u2264 f x ** intro t t_meas t_lt_top ** case h \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e f\u271d : \u03b1 \u2192 \u211d inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d 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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc t : Set \u03b1 t_meas : MeasurableSet t t_lt_top : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, OfNat.ofNat 0 x \u2264 f x ** apply ae_nonneg_restrict_of_forall_set_integral_nonneg_inter (hf_int_finite t t_meas t_lt_top) ** case h \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e f\u271d : \u03b1 \u2192 \u211d inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d 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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc t : Set \u03b1 t_meas : MeasurableSet t t_lt_top : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (s \u2229 t) < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc ** intro s s_meas _ ** case h \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t\u271d : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e f\u271d : \u03b1 \u2192 \u211d inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d 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 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc t : Set \u03b1 t_meas : MeasurableSet t t_lt_top : \u2191\u2191\u03bc t < \u22a4 s : Set \u03b1 s_meas : MeasurableSet s a\u271d : \u2191\u2191\u03bc (s \u2229 t) < \u22a4 \u22a2 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc ** exact\n  hf_zero _ (s_meas.inter t_meas)\n    (lt_of_le_of_lt (measure_mono (Set.inter_subset_right _ _)) t_lt_top) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.mk_preimage_prod_left_fn_eq_if ** \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 inst\u271d : DecidablePred fun x => x \u2208 t f : \u03b3 \u2192 \u03b1 \u22a2 (fun a => (f a, b)) \u207b\u00b9' s \u00d7\u02e2 t = if b \u2208 t then f \u207b\u00b9' s else \u2205 ** rw [\u2190 mk_preimage_prod_left_eq_if, prod_preimage_left, preimage_preimage] ** 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 c : \u211d f : \u03b1 \u2192 E \u22a2 setToFun \u03bc (fun s => c \u2022 T s) (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) 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 c : \u211d f : \u03b1 \u2192 E hf : Integrable f \u22a2 setToFun \u03bc (fun s => c \u2022 T s) (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) f = c \u2022 setToFun \u03bc T hT f ** simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left hT 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\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 c : \u211d f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 setToFun \u03bc (fun s => c \u2022 T s) (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) f = c \u2022 setToFun \u03bc T hT f ** simp_rw [setToFun_undef _ hf, smul_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.null_of_totalVariation_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 i : Set \u03b1 hs : \u2191\u2191(totalVariation s) i = 0 \u22a2 \u2191s i = 0 ** rw [totalVariation, Measure.coe_add, Pi.add_apply, add_eq_zero_iff] at hs ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 i : Set \u03b1 hs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0 \u22a2 \u2191s i = 0 ** rw [\u2190 toSignedMeasure_toJordanDecomposition s, toSignedMeasure, VectorMeasure.coe_sub,\n  Pi.sub_apply, Measure.toSignedMeasure_apply, Measure.toSignedMeasure_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 i : Set \u03b1 hs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0 \u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -       if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =     0 ** by_cases hi : MeasurableSet i ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 i : Set \u03b1 hs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0 hi : MeasurableSet i \u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -       if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =     0 ** rw [if_pos hi, if_pos hi] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 i : Set \u03b1 hs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0 hi : MeasurableSet i \u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) = 0 ** simp [hs.1, hs.2] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 i : Set \u03b1 hs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0 hi : \u00acMeasurableSet i \u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -       if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =     0 ** simp [if_neg hi] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.testAgainstNN_smul ** \u03a9 : Type u_1 inst\u271d\u00b9\u2070 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2079 : SMul R \u211d\u22650 inst\u271d\u2078 : SMul R \u211d\u22650\u221e inst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u2075 : TopologicalSpace \u03a9 inst\u271d\u2074 : OpensMeasurableSpace \u03a9 inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650 inst\u271d\u00b2 : PseudoMetricSpace R inst\u271d\u00b9 : Zero R inst\u271d : BoundedSMul R \u211d\u22650 \u03bc : FiniteMeasure \u03a9 c : R f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 testAgainstNN \u03bc (c \u2022 f) = c \u2022 testAgainstNN \u03bc f ** simp only [\u2190 ENNReal.coe_eq_coe, BoundedContinuousFunction.coe_smul, testAgainstNN_coe_eq,\n  ENNReal.coe_smul] ** \u03a9 : Type u_1 inst\u271d\u00b9\u2070 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2079 : SMul R \u211d\u22650 inst\u271d\u2078 : SMul R \u211d\u22650\u221e inst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u2075 : TopologicalSpace \u03a9 inst\u271d\u2074 : OpensMeasurableSpace \u03a9 inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650 inst\u271d\u00b2 : PseudoMetricSpace R inst\u271d\u00b9 : Zero R inst\u271d : BoundedSMul R \u211d\u22650 \u03bc : FiniteMeasure \u03a9 c : R f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 \u222b\u207b (\u03c9 : \u03a9), c \u2022 \u2191(\u2191f \u03c9) \u2202\u2191\u03bc = c \u2022 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc ** simp_rw [\u2190 smul_one_smul \u211d\u22650\u221e c (f _ : \u211d\u22650\u221e), \u2190 smul_one_smul \u211d\u22650\u221e c (lintegral _ _ : \u211d\u22650\u221e),\n  smul_eq_mul] ** \u03a9 : Type u_1 inst\u271d\u00b9\u2070 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2079 : SMul R \u211d\u22650 inst\u271d\u2078 : SMul R \u211d\u22650\u221e inst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u2075 : TopologicalSpace \u03a9 inst\u271d\u2074 : OpensMeasurableSpace \u03a9 inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650 inst\u271d\u00b2 : PseudoMetricSpace R inst\u271d\u00b9 : Zero R inst\u271d : BoundedSMul R \u211d\u22650 \u03bc : FiniteMeasure \u03a9 c : R f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 \u222b\u207b (\u03c9 : \u03a9), c \u2022 1 * \u2191(\u2191f \u03c9) \u2202\u2191\u03bc = c \u2022 1 * \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc ** exact\n  @lintegral_const_mul _ _ (\u03bc : Measure \u03a9) (c \u2022 (1 : \u211d\u22650\u221e)) _ f.measurable_coe_ennreal_comp ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.cast_succ ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 n : ZNum \u22a2 \u2191(succ n) = \u2191n + 1 ** rw [\u2190 add_one, cast_add, cast_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.continuous_set_integral ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d : NormedSpace \u211d E s : Set \u03b1 \u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** haveI : Fact ((1 : \u211d\u22650\u221e) \u2264 1) := \u27e8le_rfl\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d : NormedSpace \u211d E s : Set \u03b1 this : Fact (1 \u2264 1) \u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** have h_comp :\n  (fun f : \u03b1 \u2192\u2081[\u03bc] E => \u222b x in s, f x \u2202\u03bc) =\n    integral (\u03bc.restrict s) \u2218 fun f => LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s f := by\n  ext1 f\n  rw [Function.comp_apply, integral_congr_ae (LpToLpRestrictCLM_coeFn \u211d s f)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d : NormedSpace \u211d E s : Set \u03b1 this : Fact (1 \u2264 1) h_comp :   (fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) =     integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f) \u22a2 Continuous fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** rw [h_comp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d : NormedSpace \u211d E s : Set \u03b1 this : Fact (1 \u2264 1) h_comp :   (fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) =     integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f) \u22a2 Continuous (integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f)) ** exact continuous_integral.comp (LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s).continuous ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d : NormedSpace \u211d E s : Set \u03b1 this : Fact (1 \u2264 1) \u22a2 (fun f => \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc) =     integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f) ** ext1 f ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d : NormedSpace \u211d E s : Set \u03b1 this : Fact (1 \u2264 1) f : { x // x \u2208 Lp E 1 } \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = (integral (Measure.restrict \u03bc s) \u2218 fun f => \u2191\u2191(\u2191(LpToLpRestrictCLM \u03b1 E \u211d \u03bc 1 s) f)) f ** rw [Function.comp_apply, integral_congr_ae (LpToLpRestrictCLM_coeFn \u211d s f)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.sym2_eq_empty ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 m : Sym2 \u03b1 \u22a2 Finset.sym2 s = \u2205 \u2194 s = \u2205 ** rw [Finset.sym2, image_eq_empty, product_eq_empty, or_self_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq0_lt : 0 < q hfq : snorm' f q \u03bc < \u22a4 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc < \u22a4 ** rw [lintegral_rpow_nnnorm_eq_rpow_snorm' hq0_lt] ** \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 hq0_lt : 0 < q hfq : snorm' f q \u03bc < \u22a4 \u22a2 snorm' (fun a => f a) q \u03bc ^ q < \u22a4 ** exact ENNReal.rpow_lt_top_of_nonneg (le_of_lt hq0_lt) (ne_of_lt hfq) ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.div_eq_ediv_of_dvd ** a b : Int h : b \u2223 a \u22a2 div a b = a / b ** if b0 : b = 0 then simp [b0]\nelse rw [Int.div_eq_iff_eq_mul_left b0 h, \u2190 Int.ediv_eq_iff_eq_mul_left b0 h] ** a b : Int h : b \u2223 a b0 : b = 0 \u22a2 div a b = a / b ** simp [b0] ** a b : Int h : b \u2223 a b0 : \u00acb = 0 \u22a2 div a b = a / b ** rw [Int.div_eq_iff_eq_mul_left b0 h, \u2190 Int.ediv_eq_iff_eq_mul_left b0 h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.LocallyIntegrableOn.continuousOn_smul ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace X inst\u271d\u2079 : TopologicalSpace X inst\u271d\u2078 : MeasurableSpace Y inst\u271d\u2077 : TopologicalSpace Y inst\u271d\u2076 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s\u271d : Set X inst\u271d\u2075 : OpensMeasurableSpace X A K : Set X inst\u271d\u2074 : LocallyCompactSpace X inst\u271d\u00b3 : T2Space X \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NormedField \ud835\udd5c inst\u271d\u00b9 : SecondCountableTopologyEither X \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : X \u2192 E g : X \u2192 \ud835\udd5c s : Set X hs : IsOpen s hf : LocallyIntegrableOn f s hg : ContinuousOn g s \u22a2 LocallyIntegrableOn (fun x => g x \u2022 f x) s ** rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf \u22a2 ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace X inst\u271d\u2079 : TopologicalSpace X inst\u271d\u2078 : MeasurableSpace Y inst\u271d\u2077 : TopologicalSpace Y inst\u271d\u2076 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s\u271d : Set X inst\u271d\u2075 : OpensMeasurableSpace X A K : Set X inst\u271d\u2074 : LocallyCompactSpace X inst\u271d\u00b3 : T2Space X \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NormedField \ud835\udd5c inst\u271d\u00b9 : SecondCountableTopologyEither X \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : X \u2192 E g : X \u2192 \ud835\udd5c s : Set X hs : IsOpen s hf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k hg : ContinuousOn g s \u22a2 \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn (fun x => g x \u2022 f x) k ** exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_smul (hg.mono hk_sub) hk_c ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.Mem\u2112p.uniformIntegrable_of_identDistrib_aux ** \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u22a2 UniformIntegrable f p \u03bc ** refine' uniformIntegrable_of' hp hp' hfmeas fun \u03b5 h\u03b5 => _ ** \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** by_cases h\u03b9 : Nonempty \u03b9 ** case pos \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5  case neg \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : \u00acNonempty \u03b9 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** swap ** case pos \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8C, hC\u2081, hC\u2082\u27e9 := h\u2112p.snorm_indicator_norm_ge_pos_le \u03bc (hfmeas _) h\u03b5 ** case pos.intro.intro \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e8\u27e8C, hC\u2081.le\u27e9, fun i => le_trans (le_of_eq _) hC\u2082\u27e9 ** case pos.intro.intro \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 \u22a2 snorm (Set.indicator {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc =     snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc ** have : {x | (\u27e8C, hC\u2081.le\u27e9 : \u211d\u22650) \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} := by\n  ext x\n  simp_rw [\u2190 norm_toNNReal]\n  exact Real.le_toNNReal_iff_coe_le (norm_nonneg _) ** case pos.intro.intro \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} \u22a2 snorm (Set.indicator {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc =     snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc ** rw [this, \u2190 snorm_norm, \u2190 snorm_norm (Set.indicator _ _)] ** case pos.intro.intro \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} \u22a2 snorm (fun x => \u2016Set.indicator {x | C \u2264 \u2016f i x\u2016} (f i) x\u2016) p \u03bc =     snorm (fun x => \u2016Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j) x\u2016) p \u03bc ** simp_rw [norm_indicator_eq_indicator_norm, coe_nnnorm] ** case pos.intro.intro \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} \u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =     snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc ** let F : E \u2192 \u211d := (fun x : E => if (\u27e8C, hC\u2081.le\u27e9 : \u211d\u22650) \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0) ** case pos.intro.intro \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} F : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0 \u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =     snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc ** have F_meas : Measurable F := by\n  apply measurable_norm.indicator (measurableSet_le measurable_const measurable_nnnorm) ** case pos.intro.intro \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} F : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0 F_meas : Measurable F \u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =     snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc ** have : \u2200 k, (fun x \u21a6 Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a \u21a6 \u2016f k a\u2016) x) = F \u2218 f k := by\n  intro k\n  ext x\n  simp only [Set.indicator, Set.mem_setOf_eq]; norm_cast ** case pos.intro.intro \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this\u271d : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} F : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0 F_meas : Measurable F this : \u2200 (k : \u03b9), (fun x => Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x) = F \u2218 f k \u22a2 snorm (fun x => Set.indicator {x | C \u2264 \u2016f i x\u2016} (fun a => \u2016f i a\u2016) x) p \u03bc =     snorm (fun x => Set.indicator {x | C \u2264 \u2016f j x\u2016} (fun a => \u2016f j a\u2016) x) p \u03bc ** rw [this, this, \u2190 snorm_map_measure F_meas.aestronglyMeasurable (hf i).aemeasurable_fst,\n  (hf i).map_eq, snorm_map_measure F_meas.aestronglyMeasurable (hf j).aemeasurable_fst] ** case neg \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : \u00acNonempty \u03b9 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** exact \u27e80, fun i => False.elim (h\u03b9 <| Nonempty.intro i)\u27e9 ** \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 \u22a2 {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} ** ext x ** case h \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 x : \u03b1 \u22a2 x \u2208 {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} \u2194 x \u2208 {x | C \u2264 \u2016f i x\u2016} ** simp_rw [\u2190 norm_toNNReal] ** case h \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 x : \u03b1 \u22a2 x \u2208 {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 Real.toNNReal \u2016f i x\u2016} \u2194 x \u2208 {x | C \u2264 \u2016f i x\u2016} ** exact Real.le_toNNReal_iff_coe_le (norm_nonneg _) ** \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} F : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0 \u22a2 Measurable F ** apply measurable_norm.indicator (measurableSet_le measurable_const measurable_nnnorm) ** \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} F : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0 F_meas : Measurable F \u22a2 \u2200 (k : \u03b9), (fun x => Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x) = F \u2218 f k ** intro k ** \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} F : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0 F_meas : Measurable F k : \u03b9 \u22a2 (fun x => Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x) = F \u2218 f k ** ext x ** case h \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} F : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0 F_meas : Measurable F k : \u03b9 x : \u03b1 \u22a2 Set.indicator {x | C \u2264 \u2016f k x\u2016} (fun a => \u2016f k a\u2016) x = (F \u2218 f k) x ** simp only [Set.indicator, Set.mem_setOf_eq] ** case h \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 hfmeas : \u2200 (i : \u03b9), StronglyMeasurable (f i) hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u03b9 : Nonempty \u03b9 C : \u211d hC\u2081 : 0 < C hC\u2082 : snorm (Set.indicator {x | C \u2264 \u2191\u2016f j x\u2016\u208a} (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 this : {x | { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016f i x\u2016\u208a} = {x | C \u2264 \u2016f i x\u2016} F : E \u2192 \u211d := fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0 F_meas : Measurable F k : \u03b9 x : \u03b1 \u22a2 (if C \u2264 \u2016f k x\u2016 then \u2016f k x\u2016 else 0) =     ((fun x => if { val := C, property := (_ : 0 \u2264 C) } \u2264 \u2016x\u2016\u208a then \u2016x\u2016 else 0) \u2218 f k) x ** norm_cast ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.univ_pi_Ioo_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 => Ioo (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 => Ioo (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Icc (f i) (g i) ** exact pi_Ioo_ae_eq_pi_Icc ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.lintegral_rpow_funMulInvSnorm_eq_one ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0_lt : 0 < p f : \u03b1 \u2192 \u211d\u22650\u221e hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 \u22a2 \u222b\u207b (c : \u03b1), funMulInvSnorm f p \u03bc c ^ p \u2202\u03bc = 1 ** simp_rw [funMulInvSnorm_rpow hp0_lt] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0_lt : 0 < p f : \u03b1 \u2192 \u211d\u22650\u221e hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 \u22a2 \u222b\u207b (c : \u03b1), f c ^ p * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2202\u03bc = 1 ** rw [lintegral_mul_const', ENNReal.mul_inv_cancel hf_nonzero hf_top] ** case hr \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0_lt : 0 < p f : \u03b1 \u2192 \u211d\u22650\u221e hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 \u22a2 (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2260 \u22a4 ** rwa [inv_ne_top] ** Qed",
        "informal": ""
    },
    {
        "formal": "QPF.liftpPreservation_iff_uniform ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u22a2 LiftpPreservation \u2194 IsUniform ** rw [\u2190 suppPreservation_iff_liftpPreservation, suppPreservation_iff_uniform] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Iterator.ValidFor.next ** l : List Char c : Char r : List Char it : Iterator h : ValidFor l (c :: r) it \u22a2 ValidFor (c :: l) r (Iterator.next it) ** cases h.out' ** case refl l : List Char c : Char r : List Char h : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } } \u22a2 ValidFor (c :: l) r     (Iterator.next { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }) ** simp only [Iterator.next, next_of_valid l.reverse c r] ** case refl l : List Char c : Char r : List Char h : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } } \u22a2 ValidFor (c :: l) r     { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) + csize c } } ** rw [\u2190 List.reverseAux_eq, utf8Len_reverse] ** case refl l : List Char c : Char r : List Char h : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } } \u22a2 ValidFor (c :: l) r { s := { data := List.reverseAux l (c :: r) }, i := { byteIdx := utf8Len l + csize c } } ** constructor ** 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 => x ^ (p - 1) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0) ** simpa only [\u2190 integrableOn_Ioi_comp_rpow_iff f hp, mul_smul] using\n  (integrable_smul_iff (abs_pos.mpr hp).ne' _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_union_eq ** \u03b1 : Type u_1 s t : Set \u03b1 h : Disjoint s t \u22a2 encard (s \u222a t) = encard s + encard t ** classical\nhave e := (Equiv.Set.union (by rwa [subset_empty_iff, \u2190disjoint_iff_inter_eq_empty])).symm\nsimp [encard, \u2190PartENat.card_congr e, PartENat.card_sum, PartENat.withTopEquiv] ** \u03b1 : Type u_1 s t : Set \u03b1 h : Disjoint s t \u22a2 encard (s \u222a t) = encard s + encard t ** have e := (Equiv.Set.union (by rwa [subset_empty_iff, \u2190disjoint_iff_inter_eq_empty])).symm ** \u03b1 : Type u_1 s t : Set \u03b1 h : Disjoint s t e : \u2191s \u2295 \u2191t \u2243 \u2191(s \u222a t) \u22a2 encard (s \u222a t) = encard s + encard t ** simp [encard, \u2190PartENat.card_congr e, PartENat.card_sum, PartENat.withTopEquiv] ** \u03b1 : Type u_1 s t : Set \u03b1 h : Disjoint s t \u22a2 ?m.6374 \u2229 ?m.6375 \u2286 \u2205 ** rwa [subset_empty_iff, \u2190disjoint_iff_inter_eq_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.ofFunction_union_of_top_of_nonempty_inter ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s t : Set \u03b1 h : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4 \u22a2 \u2191(OuterMeasure.ofFunction m m_empty) (s \u222a t) =     \u2191(OuterMeasure.ofFunction m m_empty) s + \u2191(OuterMeasure.ofFunction m m_empty) t ** refine' le_antisymm (OuterMeasure.union _ _ _) (le_iInf fun f => le_iInf fun hf => _) ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s t : Set \u03b1 h : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4 f : \u2115 \u2192 Set \u03b1 hf : s \u222a t \u2286 \u22c3 i, f i \u22a2 \u2191(OuterMeasure.ofFunction m m_empty) s + \u2191(OuterMeasure.ofFunction m m_empty) t \u2264 \u2211' (i : \u2115), m (f i) ** set \u03bc := OuterMeasure.ofFunction m m_empty ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s t : Set \u03b1 h : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4 f : \u2115 \u2192 Set \u03b1 hf : s \u222a t \u2286 \u22c3 i, f i \u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty \u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i) ** rcases Classical.em (\u2203 i, (s \u2229 f i).Nonempty \u2227 (t \u2229 f i).Nonempty) with (\u27e8i, hs, ht\u27e9 | he) ** case inr \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s t : Set \u03b1 h : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4 f : \u2115 \u2192 Set \u03b1 hf : s \u222a t \u2286 \u22c3 i, f i \u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty he : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i) \u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i) ** set I := fun s => { i : \u2115 | (s \u2229 f i).Nonempty } ** case inr \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s t : Set \u03b1 h : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4 f : \u2115 \u2192 Set \u03b1 hf : s \u222a t \u2286 \u22c3 i, f i \u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty he : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i) I : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)} \u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i) ** have hd : Disjoint (I s) (I t) := disjoint_iff_inf_le.mpr fun i hi => he \u27e8i, hi\u27e9 ** case inr \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s t : Set \u03b1 h : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4 f : \u2115 \u2192 Set \u03b1 hf : s \u222a t \u2286 \u22c3 i, f i \u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty he : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i) I : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)} hd : Disjoint (I s) (I t) \u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i) ** have hI : \u2200 (u) (_ : u \u2286 s \u222a t), \u03bc u \u2264 \u2211' i : I u, \u03bc (f i) := fun u hu =>\n  calc\n    \u03bc u \u2264 \u03bc (\u22c3 i : I u, f i) :=\n      \u03bc.mono fun x hx =>\n        let \u27e8i, hi\u27e9 := mem_iUnion.1 (hf (hu hx))\n        mem_iUnion.2 \u27e8\u27e8i, \u27e8x, hx, hi\u27e9\u27e9, hi\u27e9\n    _ \u2264 \u2211' i : I u, \u03bc (f i) := \u03bc.iUnion _ ** case inr \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s t : Set \u03b1 h : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4 f : \u2115 \u2192 Set \u03b1 hf : s \u222a t \u2286 \u22c3 i, f i \u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty he : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i) I : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)} hd : Disjoint (I s) (I t) hI : \u2200 (u : Set \u03b1), u \u2286 s \u222a t \u2192 \u2191\u03bc u \u2264 \u2211' (i : \u2191(I u)), \u2191\u03bc (f \u2191i) \u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i) ** calc\n  \u03bc s + \u03bc t \u2264 (\u2211' i : I s, \u03bc (f i)) + \u2211' i : I t, \u03bc (f i) :=\n    add_le_add (hI _ <| subset_union_left _ _) (hI _ <| subset_union_right _ _)\n  _ = \u2211' i : \u2191(I s \u222a I t), \u03bc (f i) :=\n    (tsum_union_disjoint (f := fun i => \u03bc (f i)) hd ENNReal.summable ENNReal.summable).symm\n  _ \u2264 \u2211' i, \u03bc (f i) :=\n    (tsum_le_tsum_of_inj (\u2191) Subtype.coe_injective (fun _ _ => zero_le _) (fun _ => le_rfl)\n      ENNReal.summable ENNReal.summable)\n  _ \u2264 \u2211' i, m (f i) := ENNReal.tsum_le_tsum fun i => ofFunction_le _ ** case inl.intro.intro \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s t : Set \u03b1 h : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4 f : \u2115 \u2192 Set \u03b1 hf : s \u222a t \u2286 \u22c3 i, f i \u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty i : \u2115 hs : Set.Nonempty (s \u2229 f i) ht : Set.Nonempty (t \u2229 f i) \u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i) ** calc\n  \u03bc s + \u03bc t \u2264 \u221e := le_top\n  _ = m (f i) := (h (f i) hs ht).symm\n  _ \u2264 \u2211' i, m (f i) := ENNReal.le_tsum i ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.card_Icc_finset ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t \u22a2 card (Icc s t) = 2 ^ (card t - card s) ** rw [\u2190 card_sdiff h, \u2190 card_powerset, Icc_eq_image_powerset h, Finset.card_image_iff] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t \u22a2 Set.InjOn ((fun x x_1 => x \u222a x_1) s) \u2191(powerset (t \\ s)) ** rintro u hu v hv (huv : s \u2294 u = s \u2294 v) ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t u : Finset \u03b1 hu : u \u2208 \u2191(powerset (t \\ s)) v : Finset \u03b1 hv : v \u2208 \u2191(powerset (t \\ s)) huv : s \u2294 u = s \u2294 v \u22a2 u = v ** rw [mem_coe, mem_powerset] at hu hv ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t u : Finset \u03b1 hu : u \u2286 t \\ s v : Finset \u03b1 hv : v \u2286 t \\ s huv : s \u2294 u = s \u2294 v \u22a2 u = v ** rw [\u2190 (disjoint_sdiff.mono_right hu : Disjoint s u).sup_sdiff_cancel_left, \u2190\n  (disjoint_sdiff.mono_right hv : Disjoint s v).sup_sdiff_cancel_left, huv] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded ** \u03b9 : Type u_1 E : Type u_2 \u03bc : Measure \u211d l : Filter \u03b9 inst\u271d\u00b2 : NeBot l inst\u271d\u00b9 : IsCountablyGenerated l inst\u271d : NormedAddCommGroup E a b : \u03b9 \u2192 \u211d f : \u211d \u2192 E I a\u2080 b\u2080 : \u211d hfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) ha : Tendsto a l (\ud835\udcdd a\u2080) hb : Tendsto b l (\ud835\udcdd b\u2080) h : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I \u22a2 IntegrableOn f (Ioc a\u2080 b\u2080) ** refine (aecover_Ioc_of_Ioc ha hb).integrable_of_integral_norm_bounded I\n  (fun i => (hfi i).restrict measurableSet_Ioc) (h.mono fun i hi \u21a6 ?_) ** \u03b9 : Type u_1 E : Type u_2 \u03bc : Measure \u211d l : Filter \u03b9 inst\u271d\u00b2 : NeBot l inst\u271d\u00b9 : IsCountablyGenerated l inst\u271d : NormedAddCommGroup E a b : \u03b9 \u2192 \u211d f : \u211d \u2192 E I a\u2080 b\u2080 : \u211d hfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) ha : Tendsto a l (\ud835\udcdd a\u2080) hb : Tendsto b l (\ud835\udcdd b\u2080) h : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I i : \u03b9 hi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I \u22a2 \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2202Measure.restrict volume (Ioc a\u2080 b\u2080) \u2264 I ** rw [Measure.restrict_restrict measurableSet_Ioc] ** \u03b9 : Type u_1 E : Type u_2 \u03bc : Measure \u211d l : Filter \u03b9 inst\u271d\u00b2 : NeBot l inst\u271d\u00b9 : IsCountablyGenerated l inst\u271d : NormedAddCommGroup E a b : \u03b9 \u2192 \u211d f : \u211d \u2192 E I a\u2080 b\u2080 : \u211d hfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) ha : Tendsto a l (\ud835\udcdd a\u2080) hb : Tendsto b l (\ud835\udcdd b\u2080) h : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I i : \u03b9 hi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I \u22a2 \u222b (x : \u211d) in Ioc (a i) (b i) \u2229 Ioc a\u2080 b\u2080, \u2016f x\u2016 \u2264 I ** refine' le_trans (set_integral_mono_set (hfi i).norm _ _) hi <;> apply ae_of_all ** case refine'_1.a \u03b9 : Type u_1 E : Type u_2 \u03bc : Measure \u211d l : Filter \u03b9 inst\u271d\u00b2 : NeBot l inst\u271d\u00b9 : IsCountablyGenerated l inst\u271d : NormedAddCommGroup E a b : \u03b9 \u2192 \u211d f : \u211d \u2192 E I a\u2080 b\u2080 : \u211d hfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) ha : Tendsto a l (\ud835\udcdd a\u2080) hb : Tendsto b l (\ud835\udcdd b\u2080) h : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I i : \u03b9 hi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I \u22a2 \u2200 (a : \u211d), OfNat.ofNat 0 a \u2264 (fun x => \u2016f x\u2016) a ** simp only [Pi.zero_apply, norm_nonneg, forall_const] ** case refine'_2.a \u03b9 : Type u_1 E : Type u_2 \u03bc : Measure \u211d l : Filter \u03b9 inst\u271d\u00b2 : NeBot l inst\u271d\u00b9 : IsCountablyGenerated l inst\u271d : NormedAddCommGroup E a b : \u03b9 \u2192 \u211d f : \u211d \u2192 E I a\u2080 b\u2080 : \u211d hfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) ha : Tendsto a l (\ud835\udcdd a\u2080) hb : Tendsto b l (\ud835\udcdd b\u2080) h : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I i : \u03b9 hi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I \u22a2 \u2200 (a_1 : \u211d), (Ioc (a i) (b i) \u2229 Ioc a\u2080 b\u2080) a_1 \u2264 Ioc (a i) (b i) a_1 ** intro c hc ** case refine'_2.a \u03b9 : Type u_1 E : Type u_2 \u03bc : Measure \u211d l : Filter \u03b9 inst\u271d\u00b2 : NeBot l inst\u271d\u00b9 : IsCountablyGenerated l inst\u271d : NormedAddCommGroup E a b : \u03b9 \u2192 \u211d f : \u211d \u2192 E I a\u2080 b\u2080 : \u211d hfi : \u2200 (i : \u03b9), IntegrableOn f (Ioc (a i) (b i)) ha : Tendsto a l (\ud835\udcdd a\u2080) hb : Tendsto b l (\ud835\udcdd b\u2080) h : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I i : \u03b9 hi : \u222b (x : \u211d) in Ioc (a i) (b i), \u2016f x\u2016 \u2264 I c : \u211d hc : (Ioc (a i) (b i) \u2229 Ioc a\u2080 b\u2080) c \u22a2 Ioc (a i) (b i) c ** exact hc.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.VectorMeasure.hasSum_of_disjoint_iUnion ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) \u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i)) ** cases nonempty_encodable \u03b2 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 \u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i)) ** set g := fun i : \u2115 => \u22c3 (b : \u03b2) (_ : b \u2208 Encodable.decode\u2082 \u03b2 i), f b with hg ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b \u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i)) ** have hg\u2081 : \u2200 i, MeasurableSet (g i) :=\n  fun _ => MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => hf\u2081 b ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) \u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i)) ** have hg\u2082 : Pairwise (Disjoint on g) := Encodable.iUnion_decode\u2082_disjoint_on hf\u2082 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) \u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i)) ** have := v.of_disjoint_iUnion_nat hg\u2081 hg\u2082 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 i, g i) = \u2211' (i : \u2115), \u2191v (g i) \u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i)) ** rw [hg, Encodable.iUnion_decode\u2082] at this ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) \u22a2 HasSum (fun i => \u2191v (f i)) (\u2191v (\u22c3 i, f i)) ** rw [Summable.hasSum_iff, this, \u2190 tsum_iUnion_decode\u2082] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) \u22a2 (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) ** ext x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 \u22a2 \u2191v (f x) = \u2191v (g (Encodable.encode x)) ** rw [hg] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 \u22a2 \u2191v (f x) = \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) (Encodable.encode x)) ** simp only ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 \u22a2 \u2191v (f x) = \u2191v (\u22c3 b \u2208 Encodable.decode\u2082 \u03b2 (Encodable.encode x), f b) ** congr ** case h.e_a \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 \u22a2 f x = \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 (Encodable.encode x), f b ** ext y ** case h.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 y : \u03b1 \u22a2 y \u2208 f x \u2194 y \u2208 \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 (Encodable.encode x), f b ** simp only [exists_prop, Set.mem_iUnion, Option.mem_def] ** case h.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 y : \u03b1 \u22a2 y \u2208 f x \u2194 \u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i ** constructor ** case h.e_a.h.mp \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 y : \u03b1 \u22a2 y \u2208 f x \u2192 \u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i ** intro hy ** case h.e_a.h.mp \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 y : \u03b1 hy : y \u2208 f x \u22a2 \u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i ** refine' \u27e8x, (Encodable.decode\u2082_is_partial_inv _ _).2 rfl, hy\u27e9 ** case h.e_a.h.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 y : \u03b1 \u22a2 (\u2203 i, Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some i \u2227 y \u2208 f i) \u2192 y \u2208 f x ** rintro \u27e8b, hb\u2081, hb\u2082\u27e9 ** case h.e_a.h.mpr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 y : \u03b1 b : \u03b2 hb\u2081 : Encodable.decode\u2082 \u03b2 (Encodable.encode x) = Option.some b hb\u2082 : y \u2208 f b \u22a2 y \u2208 f x ** rw [Encodable.decode\u2082_is_partial_inv _ _] at hb\u2081 ** case h.e_a.h.mpr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) x : \u03b2 y : \u03b1 b : \u03b2 hb\u2081 : Encodable.encode b = Encodable.encode x hb\u2082 : y \u2208 f b \u22a2 y \u2208 f x ** rwa [\u2190 Encodable.encode_injective hb\u2081] ** case intro.m0 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) \u22a2 \u2191v \u2205 = 0 ** exact v.empty ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) \u22a2 Summable fun i => \u2191v (f i) ** rw [hg\u2083] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) \u22a2 Summable fun i => \u2191v (g (Encodable.encode i)) ** change Summable ((fun i => v (g i)) \u2218 Encodable.encode) ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) \u22a2 Summable ((fun i => \u2191v (g i)) \u2218 Encodable.encode) ** rw [Function.Injective.summable_iff Encodable.encode_injective] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) \u22a2 Summable fun i => \u2191v (g i) ** exact (v.m_iUnion hg\u2081 hg\u2082).summable ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) \u22a2 \u2200 (x : \u2115), \u00acx \u2208 range Encodable.encode \u2192 \u2191v (g x) = 0 ** intro x hx ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) x : \u2115 hx : \u00acx \u2208 range Encodable.encode \u22a2 \u2191v (g x) = 0 ** convert v.empty ** case h.e'_2.h.e'_7 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) x : \u2115 hx : \u00acx \u2208 range Encodable.encode \u22a2 g x = \u2205 ** simp only [Set.iUnion_eq_empty, Option.mem_def, not_exists, Set.mem_range] at hx \u22a2 ** case h.e'_2.h.e'_7 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) x : \u2115 hx : \u2200 (x_1 : \u03b2), \u00acEncodable.encode x_1 = x \u22a2 \u2200 (i : \u03b2), Encodable.decode\u2082 \u03b2 x = Option.some i \u2192 f i = \u2205 ** intro i hi ** case h.e'_2.h.e'_7 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : T2Space M v : VectorMeasure \u03b1 M f\u271d : \u2115 \u2192 Set \u03b1 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u03b2), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) val\u271d : Encodable \u03b2 g : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg : g = fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b hg\u2081 : \u2200 (i : \u2115), MeasurableSet (g i) hg\u2082 : Pairwise (Disjoint on g) this : \u2191v (\u22c3 b, f b) = \u2211' (i : \u2115), \u2191v ((fun i => \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) i) hg\u2083 : (fun i => \u2191v (f i)) = fun i => \u2191v (g (Encodable.encode i)) x : \u2115 hx : \u2200 (x_1 : \u03b2), \u00acEncodable.encode x_1 = x i : \u03b2 hi : Encodable.decode\u2082 \u03b2 x = Option.some i \u22a2 f i = \u2205 ** exact False.elim ((hx i) ((Encodable.decode\u2082_is_partial_inv _ _).1 hi)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.lintegral_eq_of_measure_preimage ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 inst\u271d : MeasurableSpace \u03b2 f : \u03b1 \u2192\u209b \u211d\u22650\u221e g : \u03b2 \u2192\u209b \u211d\u22650\u221e \u03bd : Measure \u03b2 H : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y}) \u22a2 lintegral f \u03bc = lintegral g \u03bd ** simp only [lintegral, \u2190 H] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 inst\u271d : MeasurableSpace \u03b2 f : \u03b1 \u2192\u209b \u211d\u22650\u221e g : \u03b2 \u2192\u209b \u211d\u22650\u221e \u03bd : Measure \u03b2 H : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y}) \u22a2 \u2211 x in SimpleFunc.range f, x * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x}) = \u2211 x in SimpleFunc.range g, x * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x}) ** apply lintegral_eq_of_subset ** case hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 inst\u271d : MeasurableSpace \u03b2 f : \u03b1 \u2192\u209b \u211d\u22650\u221e g : \u03b2 \u2192\u209b \u211d\u22650\u221e \u03bd : Measure \u03b2 H : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y}) \u22a2 \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 SimpleFunc.range g ** simp only [H] ** case hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 inst\u271d : MeasurableSpace \u03b2 f : \u03b1 \u2192\u209b \u211d\u22650\u221e g : \u03b2 \u2192\u209b \u211d\u22650\u221e \u03bd : Measure \u03b2 H : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y}) \u22a2 \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 SimpleFunc.range g ** intros ** case hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 inst\u271d : MeasurableSpace \u03b2 f : \u03b1 \u2192\u209b \u211d\u22650\u221e g : \u03b2 \u2192\u209b \u211d\u22650\u221e \u03bd : Measure \u03b2 H : \u2200 (y : \u211d\u22650\u221e), \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {y}) x\u271d : \u03b1 a\u271d\u00b9 : \u2191f x\u271d \u2260 0 a\u271d : \u2191\u2191\u03bd (\u2191g \u207b\u00b9' {\u2191f x\u271d}) \u2260 0 \u22a2 \u2191f x\u271d \u2208 SimpleFunc.range g ** exact mem_range_of_measure_ne_zero \u2039_\u203a ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.IsPwo.mul ** \u03b1 : Type u_1 s t : Set \u03b1 inst\u271d : OrderedCancelCommMonoid \u03b1 hs : IsPwo s ht : IsPwo t \u22a2 IsPwo (s * t) ** rw [\u2190 image_mul_prod] ** \u03b1 : Type u_1 s t : Set \u03b1 inst\u271d : OrderedCancelCommMonoid \u03b1 hs : IsPwo s ht : IsPwo t \u22a2 IsPwo ((fun x => x.1 * x.2) '' s \u00d7\u02e2 t) ** exact (hs.prod ht).image_of_monotone (monotone_fst.mul' monotone_snd) ** Qed",
        "informal": ""
    },
    {
        "formal": "Multiset.toFinset_nonempty ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t : Multiset \u03b1 \u22a2 Finset.Nonempty (toFinset s) \u2194 s \u2260 0 ** simp only [toFinset_eq_empty, Ne.def, Finset.nonempty_iff_ne_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.setToSimpleFunc_congr' ** \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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 \u22a2 setToSimpleFunc T (map Prod.fst (pair f g)) = setToSimpleFunc T (map Prod.snd (pair f g)) ** have h_pair : Integrable (f.pair g) \u03bc := integrable_pair hf hg ** \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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) \u22a2 setToSimpleFunc T (map Prod.fst (pair f g)) = setToSimpleFunc T (map Prod.snd (pair f g)) ** rw [map_setToSimpleFunc T h_add h_pair Prod.fst_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\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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) \u22a2 \u2211 x in SimpleFunc.range (pair f g), \u2191(T (\u2191(pair f g) \u207b\u00b9' {x})) x.1 = setToSimpleFunc T (map Prod.snd (pair f g)) ** rw [map_setToSimpleFunc T h_add h_pair Prod.snd_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\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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) \u22a2 \u2211 x in SimpleFunc.range (pair f g), \u2191(T (\u2191(pair f g) \u207b\u00b9' {x})) x.1 =     \u2211 x in SimpleFunc.range (pair f g), \u2191(T (\u2191(pair f g) \u207b\u00b9' {x})) x.2 ** refine' Finset.sum_congr rfl fun p hp => _ ** \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\u271d : \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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) p : E \u00d7 E hp : p \u2208 SimpleFunc.range (pair f g) \u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {p})) p.1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {p})) p.2 ** rcases mem_range.1 hp with \u27e8a, rfl\u27e9 ** case 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\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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) \u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2 ** by_cases eq : f a = g a ** 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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) eq : \u2191f a = \u2191g a \u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2 ** dsimp only [pair_apply] ** 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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) eq : \u2191f a = \u2191g a \u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)})) (\u2191f a) = \u2191(T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)})) (\u2191g a) ** rw [eq] ** 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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) eq : \u00ac\u2191f a = \u2191g a \u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2 ** have : T (pair f g \u207b\u00b9' {(f a, g a)}) = 0 := by\n  have h_eq : T ((\u21d1(f.pair g)) \u207b\u00b9' {(f a, g a)}) = T (f \u207b\u00b9' {f a} \u2229 g \u207b\u00b9' {g a}) := by\n    congr; rw [pair_preimage_singleton f g]\n  rw [h_eq]\n  exact h (f a) (g a) eq ** 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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) eq : \u00ac\u2191f a = \u2191g a this : T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = 0 \u22a2 \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).1 = \u2191(T (\u2191(pair f g) \u207b\u00b9' {\u2191(pair f g) a})) (\u2191(pair f g) a).2 ** simp only [this, ContinuousLinearMap.zero_apply, pair_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\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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) eq : \u00ac\u2191f a = \u2191g a \u22a2 T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = 0 ** have h_eq : T ((\u21d1(f.pair g)) \u207b\u00b9' {(f a, g a)}) = T (f \u207b\u00b9' {f a} \u2229 g \u207b\u00b9' {g a}) := by\n  congr; rw [pair_preimage_singleton f g] ** \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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) eq : \u00ac\u2191f a = \u2191g a h_eq : T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a}) \u22a2 T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = 0 ** rw [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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) eq : \u00ac\u2191f a = \u2191g a h_eq : T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a}) \u22a2 T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a}) = 0 ** exact h (f a) (g a) 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 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) eq : \u00ac\u2191f a = \u2191g a \u22a2 T (\u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)}) = T (\u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a}) ** congr ** case 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 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_add : FinMeasAdditive \u03bc T f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f hg : Integrable \u2191g h : \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 h_pair : Integrable \u2191(pair f g) a : \u03b1 hp : \u2191(pair f g) a \u2208 SimpleFunc.range (pair f g) eq : \u00ac\u2191f a = \u2191g a \u22a2 \u2191(pair f g) \u207b\u00b9' {(\u2191f a, \u2191g a)} = \u2191f \u207b\u00b9' {\u2191f a} \u2229 \u2191g \u207b\u00b9' {\u2191g a} ** rw [pair_preimage_singleton f g] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.Cofix.dest_corec\u2081 ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : {X : Type u} \u2192 (Cofix F \u03b1 \u2192 X) \u2192 (\u03b2 \u2192 X) \u2192 \u03b2 \u2192 F (\u03b1 ::: X) x : \u03b2 h :   \u2200 (X Y : Type u) (f : Cofix F \u03b1 \u2192 X) (f' : \u03b2 \u2192 X) (k : X \u2192 Y), g (k \u2218 f) (k \u2218 f') x = (TypeVec.id ::: k) <$$> g f f' x \u22a2 dest (corec\u2081 g x) = g _root_.id (corec\u2081 g) x ** rw [Cofix.corec\u2081, Cofix.dest_corec', \u2190 h] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : {X : Type u} \u2192 (Cofix F \u03b1 \u2192 X) \u2192 (\u03b2 \u2192 X) \u2192 \u03b2 \u2192 F (\u03b1 ::: X) x : \u03b2 h :   \u2200 (X Y : Type u) (f : Cofix F \u03b1 \u2192 X) (f' : \u03b2 \u2192 X) (k : X \u2192 Y), g (k \u2218 f) (k \u2218 f') x = (TypeVec.id ::: k) <$$> g f f' x \u22a2 g (Sum.elim _root_.id (corec' fun x => g Sum.inl Sum.inr x) \u2218 Sum.inl)       (Sum.elim _root_.id (corec' fun x => g Sum.inl Sum.inr x) \u2218 Sum.inr) x =     g _root_.id (corec\u2081 g) x ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 L : \u211d\u22650\u221e hL : L < \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** simp_rw [lintegral_eq_nnreal, lt_iSup_iff] at hL ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 L : \u211d\u22650\u221e hL : \u2203 i i_1, L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** rcases hL with \u27e8g\u2080, hg\u2080, g\u2080L\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 L : \u211d\u22650\u221e g\u2080 : \u03b1 \u2192\u209b \u211d\u22650 hg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x) g\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** have h'L : L < \u222b\u207b x, g\u2080 x \u2202\u03bc := by\n  convert g\u2080L\n  rw [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_map]\n  simp only [Function.comp_apply] ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 L : \u211d\u22650\u221e g\u2080 : \u03b1 \u2192\u209b \u211d\u22650 hg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x) g\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc h'L : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** rcases SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral h'L with \u27e8g, hg, gL, gtop\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 L : \u211d\u22650\u221e g\u2080 : \u03b1 \u2192\u209b \u211d\u22650 hg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x) g\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc h'L : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc g : \u03b1 \u2192\u209b \u211d\u22650 hg : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191g\u2080 x gL : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 gtop : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc ** exact \u27e8g, fun x => (hg x).trans (coe_le_coe.1 (hg\u2080 x)), gL, gtop\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 L : \u211d\u22650\u221e g\u2080 : \u03b1 \u2192\u209b \u211d\u22650 hg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x) g\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc \u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc ** convert g\u2080L ** case h.e'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 L : \u211d\u22650\u221e g\u2080 : \u03b1 \u2192\u209b \u211d\u22650 hg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x) g\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc = SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc ** rw [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_map] ** case h.e'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d\u22650 L : \u211d\u22650\u221e g\u2080 : \u03b1 \u2192\u209b \u211d\u22650 hg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x) g\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc = \u222b\u207b (a : \u03b1), (ENNReal.some \u2218 \u2191g\u2080) a \u2202\u03bc ** simp only [Function.comp_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.mul_inv_of_unit ** n : \u2115 a : ZMod n h : IsUnit a \u22a2 a * a\u207b\u00b9 = 1 ** rcases h with \u27e8u, rfl\u27e9 ** case intro n : \u2115 u : (ZMod n)\u02e3 \u22a2 \u2191u * (\u2191u)\u207b\u00b9 = 1 ** rw [inv_coe_unit, u.mul_inv] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.getLast?_eq_get? ** \u03b1 : Type u_1 a : \u03b1 l : List \u03b1 \u22a2 getLast? (a :: l) = get? (a :: l) (length (a :: l) - 1) ** rw [getLast?_eq_getLast (a::l) fun., getLast_eq_get, get?_eq_get] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_le_one_iff ** \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard s \u2264 1 \u2194 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b ** rw [ncard_le_one hs] ** \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b) \u2194 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b ** tauto ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Primrec'.of_prim ** n : \u2115 f : Vector \u2115 n \u2192 \u2115 this : \u2200 (f : \u2115 \u2192 \u2115), Nat.Primrec f \u2192 Primrec' fun v => f (Vector.head v) hf : Primrec f i : Vector \u2115 n \u22a2 Nat.pred ((fun m => encode (Option.map f (decode m))) (encode i)) = f i ** simp [encodek] ** n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 hf : Nat.Primrec f \u22a2 Primrec' fun v => f (Vector.head v) ** induction hf ** case zero n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun x => 0) (Vector.head v)  case succ n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => Nat.succ (Vector.head v)  case left n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)  case right n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)  case pair n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)  case comp n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)  case prec n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v) ** case zero => exact const 0 ** case succ n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => Nat.succ (Vector.head v)  case left n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)  case right n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)  case pair n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)  case comp n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)  case prec n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v) ** case succ => exact succ ** case left n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)  case right n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)  case pair n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)  case comp n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)  case prec n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v) ** case left => exact unpair\u2081 head ** case right n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)  case pair n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)  case comp n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)  case prec n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v) ** case right => exact unpair\u2082 head ** case pair n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)  case comp n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)  case prec n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v) ** case pair f g _ _ hf hg => exact natPair.comp\u2082 _ hf hg ** case comp n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)  case prec n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v) ** case comp f g _ _ hf hg => exact hf.comp\u2081 _ hg ** case prec n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f f\u271d g\u271d : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f\u271d a\u271d : Nat.Primrec g\u271d a_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v) a_ih\u271d : Primrec' fun v => g\u271d (Vector.head v) \u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v) ** case prec f g _ _ hf hg =>\n  simpa using\n    prec' (unpair\u2082 head) (hf.comp\u2081 _ (unpair\u2081 head))\n      (hg.comp\u2081 _ <|\n        natPair.comp\u2082 _ (unpair\u2081 <| tail <| tail head) (natPair.comp\u2082 _ head (tail head))) ** n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun x => 0) (Vector.head v) ** exact const 0 ** n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => Nat.succ (Vector.head v) ** exact succ ** n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v) ** exact unpair\u2081 head ** n : \u2115 f\u271d : Vector \u2115 n \u2192 \u2115 f : \u2115 \u2192 \u2115 \u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v) ** exact unpair\u2082 head ** n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f\u271d f g : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f a\u271d : Nat.Primrec g hf : Primrec' fun v => f (Vector.head v) hg : Primrec' fun v => g (Vector.head v) \u22a2 Primrec' fun v => (fun n => pair (f n) (g n)) (Vector.head v) ** exact natPair.comp\u2082 _ hf hg ** n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f\u271d f g : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f a\u271d : Nat.Primrec g hf : Primrec' fun v => f (Vector.head v) hg : Primrec' fun v => g (Vector.head v) \u22a2 Primrec' fun v => (fun n => f (g n)) (Vector.head v) ** exact hf.comp\u2081 _ hg ** n : \u2115 f\u271d\u00b9 : Vector \u2115 n \u2192 \u2115 f\u271d f g : \u2115 \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec f a\u271d : Nat.Primrec g hf : Primrec' fun v => f (Vector.head v) hg : Primrec' fun v => g (Vector.head v) \u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f z) (fun y IH => g (pair z (pair y IH))) n) (Vector.head v) ** simpa using\n  prec' (unpair\u2082 head) (hf.comp\u2081 _ (unpair\u2081 head))\n    (hg.comp\u2081 _ <|\n      natPair.comp\u2082 _ (unpair\u2081 <| tail <| tail head) (natPair.comp\u2082 _ head (tail head))) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.essSup_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 \u211d\u22650\u221e hf : Measurable f \u22a2 essSup f (Measure.trim \u03bd hm) = essSup f \u03bd ** simp_rw [essSup] ** \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 \u211d\u22650\u221e hf : Measurable f \u22a2 limsup f (Measure.ae (Measure.trim \u03bd hm)) = limsup f (Measure.ae \u03bd) ** exact limsup_trim hm hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lpMeasSubgroupToLpTrim_sub ** \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 ** rw [sub_eq_add_neg, sub_eq_add_neg, lpMeasSubgroupToLpTrim_add,\n  lpMeasSubgroupToLpTrim_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.compProdFun_eq_tsum ** \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 = \u2211' (n : \u2115) (m : \u2115), compProdFun (seq \u03ba n) (seq \u03b7 m) a s ** simp_rw [compProdFun_tsum_left \u03ba \u03b7 a s, compProdFun_tsum_right _ \u03b7 a hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrableOn_iUnion_of_summable_integral_norm ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : Countable \u03b2 f : \u03b1 \u2192 E s : \u03b2 \u2192 Set \u03b1 hs : \u2200 (b : \u03b2), MeasurableSet (s b) hi : \u2200 (b : \u03b2), IntegrableOn f (s b) h : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc \u22a2 IntegrableOn f (iUnion s) ** refine' \u27e8AEStronglyMeasurable.iUnion fun i => (hi i).1, (lintegral_iUnion_le _ _).trans_lt _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : Countable \u03b2 f : \u03b1 \u2192 E s : \u03b2 \u2192 Set \u03b1 hs : \u2200 (b : \u03b2), MeasurableSet (s b) hi : \u2200 (b : \u03b2), IntegrableOn f (s b) h : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc \u22a2 \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1) in s i, \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 ** have B := fun b : \u03b2 => lintegral_coe_eq_integral (fun a : \u03b1 => \u2016f a\u2016\u208a) (hi b).norm ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : Countable \u03b2 f : \u03b1 \u2192 E s : \u03b2 \u2192 Set \u03b1 hs : \u2200 (b : \u03b2), MeasurableSet (s b) hi : \u2200 (b : \u03b2), IntegrableOn f (s b) h : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc B : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) \u22a2 \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1) in s i, \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 ** rw [tsum_congr B] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : Countable \u03b2 f : \u03b1 \u2192 E s : \u03b2 \u2192 Set \u03b1 hs : \u2200 (b : \u03b2), MeasurableSet (s b) hi : \u2200 (b : \u03b2), IntegrableOn f (s b) h : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc B : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) \u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4 ** have S' :\n  Summable fun b : \u03b2 =>\n    (\u27e8\u222b a : \u03b1 in s b, \u2016f a\u2016\u208a \u2202\u03bc, set_integral_nonneg (hs b) fun a _ => NNReal.coe_nonneg _\u27e9 :\n      NNReal) :=\n  by rw [\u2190 NNReal.summable_coe]; exact h ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : Countable \u03b2 f : \u03b1 \u2192 E s : \u03b2 \u2192 Set \u03b1 hs : \u2200 (b : \u03b2), MeasurableSet (s b) hi : \u2200 (b : \u03b2), IntegrableOn f (s b) h : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc B : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) S' : Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) } \u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4 ** have S'' := ENNReal.tsum_coe_eq S'.hasSum ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : Countable \u03b2 f : \u03b1 \u2192 E s : \u03b2 \u2192 Set \u03b1 hs : \u2200 (b : \u03b2), MeasurableSet (s b) hi : \u2200 (b : \u03b2), IntegrableOn f (s b) h : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc B : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) S' : Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) } S'' :   \u2211' (a : \u03b2), \u2191{ val := \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc) } =     \u2191(\u2211' (b : \u03b2), { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) }) \u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4 ** simp_rw [ENNReal.coe_nnreal_eq, NNReal.coe_mk, coe_nnnorm] at S'' ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : Countable \u03b2 f : \u03b1 \u2192 E s : \u03b2 \u2192 Set \u03b1 hs : \u2200 (b : \u03b2), MeasurableSet (s b) hi : \u2200 (b : \u03b2), IntegrableOn f (s b) h : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc B : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) S' : Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) } S'' :   \u2211' (a : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s a, \u2016f a\u2016 \u2202\u03bc) =     ENNReal.ofReal       \u2191(\u2211' (b : \u03b2),           { val := \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc, property := (_ : (fun r => 0 \u2264 r) (\u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc)) }) \u22a2 \u2211' (b : \u03b2), ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) < \u22a4 ** convert ENNReal.ofReal_lt_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : Countable \u03b2 f : \u03b1 \u2192 E s : \u03b2 \u2192 Set \u03b1 hs : \u2200 (b : \u03b2), MeasurableSet (s b) hi : \u2200 (b : \u03b2), IntegrableOn f (s b) h : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc B : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) \u22a2 Summable fun b => { val := \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) } ** rw [\u2190 NNReal.summable_coe] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : Countable \u03b2 f : \u03b1 \u2192 E s : \u03b2 \u2192 Set \u03b1 hs : \u2200 (b : \u03b2), MeasurableSet (s b) hi : \u2200 (b : \u03b2), IntegrableOn f (s b) h : Summable fun b => \u222b (a : \u03b1) in s b, \u2016f a\u2016 \u2202\u03bc B : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc = ENNReal.ofReal (\u222b (a : \u03b1) in s b, \u2191\u2016f a\u2016\u208a \u2202\u03bc) \u22a2 Summable fun a => \u2191{ val := \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc, property := (_ : 0 \u2264 \u222b (a : \u03b1) in s a, \u2191\u2016f a\u2016\u208a \u2202\u03bc) } ** exact h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos ** \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 : \u2191s i < 0 \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** have hi\u2081 : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** by_cases s \u2264[i] 0 ** case neg \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** by_cases hn : \u2200 n : \u2115, \u00acs \u2264[i \\ \u22c3 l < n, restrictNonposSeq s i l] 0 ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0  case neg \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u00ac\u2200 (n : \u2115),       \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264           restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** swap ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** set A := i \\ \u22c3 l, restrictNonposSeq s i l with hA ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** set bdd : \u2115 \u2192 \u2115 := fun n => findExistsOneDivLT s (i \\ \u22c3 k \u2264 n, restrictNonposSeq s i k) ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** have h\u2081 : s i = s A + \u2211' l, s (restrictNonposSeq s i l) := by\n  rw [hA, \u2190 s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff]\n  exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _\n  exacts [hi\u2081, Set.iUnion_subset fun _ => restrictNonposSeq_subset _, fun _ =>\n    restrictNonposSeq_measurableSet _, restrictNonposSeq_disjoint] ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** have h\u2082 : s A \u2264 s i := by\n  rw [h\u2081]\n  apply le_add_of_nonneg_right\n  exact tsum_nonneg fun n => le_of_lt (measure_of_restrictNonposSeq h _ (hn n)) ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** have h\u2083 : Tendsto (fun n => (bdd n : \u211d) + 1) atTop atTop := by\n  simp only [one_div] at h\u2083'\n  exact Summable.tendsto_atTop_of_pos h\u2083' fun n => Nat.cast_add_one_pos (bdd n) ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** have h\u2084 : Tendsto (fun n => (bdd n : \u211d)) atTop atTop := by\n  convert atTop.tendsto_atTop_add_const_right (-1) h\u2083; simp ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** have A_meas : MeasurableSet A :=\n  hi\u2081.diff (MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _) ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** refine' \u27e8A, A_meas, Set.diff_subset _ _, _, h\u2082.trans_lt hi\u27e9 ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A \u22a2 restrict s A \u2264 restrict 0 A ** by_contra hnn ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A hnn : \u00acrestrict s A \u2264 restrict 0 A \u22a2 False ** rw [restrict_le_restrict_iff _ _ A_meas] at hnn ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A hnn : \u00ac\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 A \u2192 \u2191s j \u2264 \u21910 j \u22a2 False ** push_neg at hnn ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A hnn : Exists fun \u2983j\u2984 => MeasurableSet j \u2227 j \u2286 A \u2227 \u21910 j < \u2191s j \u22a2 False ** obtain \u27e8E, hE\u2081, hE\u2082, hE\u2083\u27e9 := hnn ** case pos.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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E this : \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E \u22a2 False ** obtain \u27e8k, hk\u2081, hk\u2082\u27e9 := this ** case pos.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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E \u22a2 False ** have hA' : A \u2286 i \\ \u22c3 l \u2264 k, restrictNonposSeq s i l := by\n  apply Set.diff_subset_diff_right\n  intro x; simp only [Set.mem_iUnion]\n  rintro \u27e8n, _, hn\u2082\u27e9\n  exact \u27e8n, hn\u2082\u27e9 ** case pos.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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E hA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u22a2 False ** refine'\n  findExistsOneDivLT_min (hn' k) (Nat.sub_lt hk\u2081 Nat.zero_lt_one)\n    \u27e8E, Set.Subset.trans hE\u2082 hA', hE\u2081, _\u27e9 ** case pos.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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E hA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u22a2 1 /       (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s               (i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -             1) +         1) <     \u2191s E ** convert hk\u2082 ** case h.e'_3.h.e'_6 \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E hA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u22a2 \u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s             (i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -           1) +       1 =     \u2191(bdd k) ** norm_cast ** case h.e'_3.h.e'_6 \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E hA' : A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u22a2 MeasureTheory.SignedMeasure.findExistsOneDivLT s           (i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) -         1 +       1 =     bdd k ** exact tsub_add_cancel_of_le hk\u2081 ** case pos \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : restrict s i \u2264 restrict 0 i \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** exact \u27e8i, hi\u2081, Set.Subset.refl _, h, hi\u27e9 ** case neg \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u00ac\u2200 (n : \u2115),       \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264           restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** exact exists_subset_restrict_nonpos' hi\u2081 hi hn ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u22a2 \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** intro n ** case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3 \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) n : \u2115 \u22a2 (fun l => \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) = fun l =>     \u22c3 (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ** ext l ** case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) n l : \u2115 x\u271d : \u03b1 \u22a2 x\u271d \u2208 \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2194     x\u271d \u2208 \u22c3 (_ : l < n + 1), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ** simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff] ** case h.e'_1.h.e'_4.h.e'_7.h.e'_4.h.e'_3.h.h \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) n l : \u2115 x\u271d : \u03b1 \u22a2 x\u271d \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2192 (l \u2264 n \u2194 l < n + 1) ** exact fun _ => Nat.lt_succ_iff.symm ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** rw [hA, \u2190 s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff] ** case hA \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 MeasurableSet (\u22c3 i_1, MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)  case hB \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 MeasurableSet i  case h \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u22c3 i_1, MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 \u2286 i  case hf\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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2200 (i_1 : \u2115), MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)  case hf\u2082 \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 Pairwise (Disjoint on fun l => MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _ ** case hB \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 MeasurableSet i  case h \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u22c3 i_1, MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 \u2286 i  case hf\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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2200 (i_1 : \u2115), MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1)  case hf\u2082 \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 Pairwise (Disjoint on fun l => MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** exacts [hi\u2081, Set.iUnion_subset fun _ => restrictNonposSeq_subset _, fun _ =>\n  restrictNonposSeq_measurableSet _, restrictNonposSeq_disjoint] ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2191s A \u2264 \u2191s i ** rw [h\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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2191s A \u2264 \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** apply le_add_of_nonneg_right ** case h \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 0 \u2264 \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** exact tsum_nonneg fun n => le_of_lt (measure_of_restrictNonposSeq h _ (hn n)) ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i \u22a2 Summable fun n => 1 / (\u2191(bdd n) + 1) ** have : Summable fun l => s (restrictNonposSeq s i l) :=\n  HasSum.summable\n    (s.m_iUnion (fun _ => restrictNonposSeq_measurableSet _) restrictNonposSeq_disjoint) ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i this : Summable fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 Summable fun n => 1 / (\u2191(bdd n) + 1) ** refine'\n  summable_of_nonneg_of_le (fun n => _) (fun n => _)\n    (Summable.comp_injective this Nat.succ_injective) ** case 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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i this : Summable fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) n : \u2115 \u22a2 0 \u2264 1 / (\u2191(bdd n) + 1) ** exact le_of_lt Nat.one_div_pos_of_nat ** case 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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i this : Summable fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) n : \u2115 \u22a2 1 / (\u2191(bdd n) + 1) \u2264 ((fun l => \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)) \u2218 Nat.succ) n ** exact le_of_lt (restrictNonposSeq_lt n (hn' n)) ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) \u22a2 Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop ** simp only [one_div] at h\u2083' ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' :   Summable fun n =>     (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s             (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +         1)\u207b\u00b9 \u22a2 Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop ** exact Summable.tendsto_atTop_of_pos h\u2083' fun n => Nat.cast_add_one_pos (bdd n) ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop \u22a2 Tendsto (fun n => \u2191(bdd n)) atTop atTop ** convert atTop.tendsto_atTop_add_const_right (-1) h\u2083 ** case h.e'_3.h \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop x\u271d : \u2115 \u22a2 \u2191(bdd x\u271d) = \u2191(bdd x\u271d) + 1 + -1 ** simp ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E \u22a2 \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E ** rw [tendsto_atTop_atTop] at h\u2084 ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a) A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E \u22a2 \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E ** obtain \u27e8k, hk\u27e9 := h\u2084 (max (1 / s E + 1) 1) ** case 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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a) A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a) \u22a2 \u2203 k, 1 \u2264 bdd k \u2227 1 / \u2191(bdd k) < \u2191s E ** refine' \u27e8k, _, _\u27e9 ** case 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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a) A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a) \u22a2 1 \u2264 bdd k ** have hle := le_of_max_le_right (hk k le_rfl) ** case 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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a) A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a) hle : 1 \u2264 \u2191(bdd k) \u22a2 1 \u2264 bdd k ** norm_cast at hle ** case 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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a) A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a) \u22a2 1 / \u2191(bdd k) < \u2191s E ** have : 1 / s E < bdd k := by\n  linarith only [le_of_max_le_left (hk k le_rfl)] ** case 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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a) A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a) this : 1 / \u2191s E < \u2191(bdd k) \u22a2 1 / \u2191(bdd k) < \u2191s E ** rw [one_div] at this \u22a2 ** case 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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a) A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a) this : (\u2191s E)\u207b\u00b9 < \u2191(bdd k) \u22a2 (\u2191(bdd k))\u207b\u00b9 < \u2191s E ** rwa [inv_lt (lt_trans (inv_pos.2 hE\u2083) this) hE\u2083] ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 \u2191(bdd a) A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk : \u2200 (a : \u2115), k \u2264 a \u2192 max (1 / \u2191s E + 1) 1 \u2264 \u2191(bdd a) \u22a2 1 / \u2191s E < \u2191(bdd k) ** linarith only [le_of_max_le_left (hk k le_rfl)] ** \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E \u22a2 A \u2286 i \\ \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l ** apply Set.diff_subset_diff_right ** case h \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E \u22a2 \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2286     \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l ** intro x ** case h \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E x : \u03b1 \u22a2 x \u2208 \u22c3 l, \u22c3 (_ : l \u2264 k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2192     x \u2208 \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l ** simp only [Set.mem_iUnion] ** case h \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 : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E x : \u03b1 \u22a2 (\u2203 i_1 i_2, x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) \u2192     \u2203 i_1, x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 ** rintro \u27e8n, _, hn\u2082\u27e9 ** case h.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 : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u2191s i < 0 hi\u2081 : MeasurableSet i h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) A : Set \u03b1 := i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l hA : A = i \\ \u22c3 l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l bdd : \u2115 \u2192 \u2115 :=   fun n =>     MeasureTheory.SignedMeasure.findExistsOneDivLT s       (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) hn' :   \u2200 (n : \u2115),     \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264         restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2191s i = \u2191s A + \u2211' (l : \u2115), \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2082 : \u2191s A \u2264 \u2191s i h\u2083' : Summable fun n => 1 / (\u2191(bdd n) + 1) h\u2083 : Tendsto (fun n => \u2191(bdd n) + 1) atTop atTop h\u2084 : Tendsto (fun n => \u2191(bdd n)) atTop atTop A_meas : MeasurableSet A E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : E \u2286 A hE\u2083 : \u21910 E < \u2191s E k : \u2115 hk\u2081 : 1 \u2264 bdd k hk\u2082 : 1 / \u2191(bdd k) < \u2191s E x : \u03b1 n : \u2115 w\u271d : n \u2264 k hn\u2082 : x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i n \u22a2 \u2203 i_1, x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 ** exact \u27e8n, hn\u2082\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.finSuccEquiv_coeff_coeff ** 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 m : Fin n \u2192\u2080 \u2115 f : MvPolynomial (Fin (n + 1)) R i : \u2115 \u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) = coeff (cons i m) f ** induction' f using MvPolynomial.induction_on' with j r p q hp hq generalizing i m ** case h1 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 i : \u2115 \u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (\u2191(monomial j) r)) i) = coeff (cons i m) (\u2191(monomial j) r)  case h2 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 p q : MvPolynomial (Fin (n + 1)) R hp : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) p) i) = coeff (cons i m) p hq : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) q) i) = coeff (cons i m) q m : Fin n \u2192\u2080 \u2115 i : \u2115 \u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q) ** swap ** case h1 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 i : \u2115 \u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (\u2191(monomial j) r)) i) = coeff (cons i m) (\u2191(monomial j) r) ** simp only [finSuccEquiv_apply, coe_eval\u2082Hom, eval\u2082_monomial, RingHom.coe_comp, prod_pow,\n  Polynomial.coeff_C_mul, coeff_C_mul, coeff_monomial, Fin.prod_univ_succ, Fin.cases_zero,\n  Fin.cases_succ, \u2190 map_prod, \u2190 RingHom.map_pow, Function.comp_apply] ** case h1 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 i : \u2115 \u22a2 r * coeff m (Polynomial.coeff (Polynomial.X ^ \u2191j 0 * \u2191Polynomial.C (\u220f x : Fin n, X x ^ \u2191j (Fin.succ x))) i) =     if j = cons i m then r else 0 ** rw [\u2190 mul_boole, mul_comm (Polynomial.X ^ j 0), Polynomial.coeff_C_mul_X_pow] ** case h1 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 i : \u2115 \u22a2 r * coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = r * if j = cons i m then 1 else 0 ** congr 1 ** case h1.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 : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 i : \u2115 \u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = if j = cons i m then 1 else 0 ** obtain rfl | hjmi := eq_or_ne j (m.cons i) ** case h2 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 p q : MvPolynomial (Fin (n + 1)) R hp : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) p) i) = coeff (cons i m) p hq : \u2200 (m : Fin n \u2192\u2080 \u2115) (i : \u2115), coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) q) i) = coeff (cons i m) q m : Fin n \u2192\u2080 \u2115 i : \u2115 \u22a2 coeff m (Polynomial.coeff (\u2191(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q) ** simp only [(finSuccEquiv R n).map_add, Polynomial.coeff_add, coeff_add, hp, hq] ** case h1.e_a.inl 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 r : R m : Fin n \u2192\u2080 \u2115 i : \u2115 \u22a2 coeff m (if i = \u2191(cons i m) 0 then \u220f x : Fin n, X x ^ \u2191(cons i m) (Fin.succ x) else 0) =     if cons i m = cons i m then 1 else 0 ** simpa only [cons_zero, cons_succ, if_pos rfl, monomial_eq, C_1, one_mul, prod_pow] using\n  coeff_monomial m m (1 : R) ** case h1.e_a.inr 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 i : \u2115 hjmi : j \u2260 cons i m \u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = if j = cons i m then 1 else 0 ** simp only [hjmi, if_false] ** case h1.e_a.inr 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 i : \u2115 hjmi : j \u2260 cons i m \u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = 0 ** obtain hij | rfl := ne_or_eq i (j 0) ** case h1.e_a.inr.inr 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 m\u271d : Fin n \u2192\u2080 \u2115 i : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 hjmi : j \u2260 cons (\u2191j 0) m \u22a2 coeff m (if \u2191j 0 = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = 0 ** simp only [eq_self_iff_true, if_true] ** case h1.e_a.inr.inr 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 m\u271d : Fin n \u2192\u2080 \u2115 i : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 hjmi : j \u2260 cons (\u2191j 0) m \u22a2 coeff m (\u220f x : Fin n, X x ^ \u2191j (Fin.succ x)) = 0 ** have hmj : m \u2260 j.tail := by\n  rintro rfl\n  rw [cons_tail] at hjmi\n  contradiction ** case h1.e_a.inr.inr 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 m\u271d : Fin n \u2192\u2080 \u2115 i : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 hjmi : j \u2260 cons (\u2191j 0) m hmj : m \u2260 tail j \u22a2 coeff m (\u220f x : Fin n, X x ^ \u2191j (Fin.succ x)) = 0 ** simpa only [monomial_eq, C_1, one_mul, prod_pow, Finsupp.tail_apply, if_neg hmj.symm] using\n  coeff_monomial m j.tail (1 : R) ** case h1.e_a.inr.inl 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 m\u271d : Fin n \u2192\u2080 \u2115 i\u271d : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 i : \u2115 hjmi : j \u2260 cons i m hij : i \u2260 \u2191j 0 \u22a2 coeff m (if i = \u2191j 0 then \u220f x : Fin n, X x ^ \u2191j (Fin.succ x) else 0) = 0 ** simp only [hij, if_false, coeff_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 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 m\u271d : Fin n \u2192\u2080 \u2115 i : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R m : Fin n \u2192\u2080 \u2115 hjmi : j \u2260 cons (\u2191j 0) m \u22a2 m \u2260 tail j ** rintro 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 m : Fin n \u2192\u2080 \u2115 i : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R hjmi : j \u2260 cons (\u2191j 0) (tail j) \u22a2 False ** rw [cons_tail] at hjmi ** 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 m : Fin n \u2192\u2080 \u2115 i : \u2115 j : Fin (n + 1) \u2192\u2080 \u2115 r : R hjmi : j \u2260 j \u22a2 False ** contradiction ** Qed",
        "informal": ""
    },
    {
        "formal": "StieltjesFunction.measure_Iic ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atBot (\ud835\udcdd l) x : \u211d \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Iic x) = ofReal (\u2191f x - l) ** refine' tendsto_nhds_unique (tendsto_measure_Ioc_atBot _ _) _ ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atBot (\ud835\udcdd l) x : \u211d \u22a2 Tendsto (fun x_1 => \u2191\u2191(StieltjesFunction.measure f) (Ioc x_1 x)) atBot (\ud835\udcdd (ofReal (\u2191f x - l))) ** simp_rw [measure_Ioc] ** f : StieltjesFunction l : \u211d hf : Tendsto (\u2191f) atBot (\ud835\udcdd l) x : \u211d \u22a2 Tendsto (fun x_1 => ofReal (\u2191f x - \u2191f x_1)) atBot (\ud835\udcdd (ofReal (\u2191f x - l))) ** exact ENNReal.tendsto_ofReal (Tendsto.const_sub _ hf) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.mkMetric'_isMetric ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : Set X \u2192 \u211d\u22650\u221e \u22a2 IsMetric (mkMetric' m) ** rintro s t \u27e8r, r0, hr\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 m : Set X \u2192 \u211d\u22650\u221e s t : Set X r : \u211d\u22650\u221e r0 : r \u2260 0 hr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y \u22a2 \u2191(mkMetric' m) (s \u222a t) = \u2191(mkMetric' m) s + \u2191(mkMetric' m) t ** refine' tendsto_nhds_unique_of_eventuallyEq\n  (mkMetric'.tendsto_pre _ _) ((mkMetric'.tendsto_pre _ _).add (mkMetric'.tendsto_pre _ _)) _ ** case intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : Set X \u2192 \u211d\u22650\u221e s t : Set X r : \u211d\u22650\u221e r0 : r \u2260 0 hr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y \u22a2 (fun r => \u2191(mkMetric'.pre m r) (s \u222a t)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => \u2191(mkMetric'.pre m x) s + \u2191(mkMetric'.pre m x) t ** rw [\u2190 pos_iff_ne_zero] at r0 ** case intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : Set X \u2192 \u211d\u22650\u221e s t : Set X r : \u211d\u22650\u221e r0 : 0 < r hr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y \u22a2 (fun r => \u2191(mkMetric'.pre m r) (s \u222a t)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => \u2191(mkMetric'.pre m x) s + \u2191(mkMetric'.pre m x) t ** filter_upwards [Ioo_mem_nhdsWithin_Ioi \u27e8le_rfl, r0\u27e9] ** case h \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : Set X \u2192 \u211d\u22650\u221e s t : Set X r : \u211d\u22650\u221e r0 : 0 < r hr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y \u22a2 \u2200 (a : \u211d\u22650\u221e), a \u2208 Ioo 0 r \u2192 \u2191(mkMetric'.pre m a) (s \u222a t) = \u2191(mkMetric'.pre m a) s + \u2191(mkMetric'.pre m a) t ** rintro \u03b5 \u27e8_, \u03b5r\u27e9 ** case h.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : Set X \u2192 \u211d\u22650\u221e s t : Set X r : \u211d\u22650\u221e r0 : 0 < r hr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y \u03b5 : \u211d\u22650\u221e left\u271d : 0 < \u03b5 \u03b5r : \u03b5 < r \u22a2 \u2191(mkMetric'.pre m \u03b5) (s \u222a t) = \u2191(mkMetric'.pre m \u03b5) s + \u2191(mkMetric'.pre m \u03b5) t ** refine' boundedBy_union_of_top_of_nonempty_inter _ ** case h.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : Set X \u2192 \u211d\u22650\u221e s t : Set X r : \u211d\u22650\u221e r0 : 0 < r hr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y \u03b5 : \u211d\u22650\u221e left\u271d : 0 < \u03b5 \u03b5r : \u03b5 < r \u22a2 \u2200 (u : Set X), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 extend (fun s x => m s) u = \u22a4 ** rintro u \u27e8x, hxs, hxu\u27e9 \u27e8y, hyt, hyu\u27e9 ** case h.intro.intro.intro.intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : Set X \u2192 \u211d\u22650\u221e s t : Set X r : \u211d\u22650\u221e r0 : 0 < r hr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y \u03b5 : \u211d\u22650\u221e left\u271d : 0 < \u03b5 \u03b5r : \u03b5 < r u : Set X x : X hxs : x \u2208 s hxu : x \u2208 u y : X hyt : y \u2208 t hyu : y \u2208 u \u22a2 extend (fun s x => m s) u = \u22a4 ** have : \u03b5 < diam u := \u03b5r.trans_le ((hr x hxs y hyt).trans <| edist_le_diam_of_mem hxu hyu) ** case h.intro.intro.intro.intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : Set X \u2192 \u211d\u22650\u221e s t : Set X r : \u211d\u22650\u221e r0 : 0 < r hr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y \u03b5 : \u211d\u22650\u221e left\u271d : 0 < \u03b5 \u03b5r : \u03b5 < r u : Set X x : X hxs : x \u2208 s hxu : x \u2208 u y : X hyt : y \u2208 t hyu : y \u2208 u this : \u03b5 < diam u \u22a2 extend (fun s x => m s) u = \u22a4 ** exact iInf_eq_top.2 fun h => (this.not_le h).elim ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_sum_indicator_atTop_iff' ** \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 s : \u2115 \u2192 Set \u03a9 hs : \u2200 (n : \u2115), MeasurableSet (s n) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194       Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop ** have := tendsto_sum_indicator_atTop_iff (eventually_of_forall fun \u03c9 n => ?_) (adapted_process hs)\n  (integrable_process \u03bc hs) (eventually_of_forall <| process_difference_le s) ** case refine_2 \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 s : \u2115 \u2192 Set \u03a9 hs : \u2200 (n : \u2115), MeasurableSet (s n) this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     Tendsto (fun n => process (fun n => s n) n \u03c9) atTop atTop \u2194       Tendsto (fun n => predictablePart (process fun n => s n) \u2131 \u03bc n \u03c9) atTop atTop \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194       Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop  case refine_1 \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 s : \u2115 \u2192 Set \u03a9 hs : \u2200 (n : \u2115), MeasurableSet (s n) \u03c9 : \u03a9 n : \u2115 \u22a2 process (fun n => s n) n \u03c9 \u2264 process (fun n => s n) (n + 1) \u03c9 ** swap ** case refine_2 \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 s : \u2115 \u2192 Set \u03a9 hs : \u2200 (n : \u2115), MeasurableSet (s n) this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     Tendsto (fun n => process (fun n => s n) n \u03c9) atTop atTop \u2194       Tendsto (fun n => predictablePart (process fun n => s n) \u2131 \u03bc n \u03c9) atTop atTop \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194       Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop ** simp_rw [process, predictablePart_process_ae_eq] at this ** case refine_2 \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 s : \u2115 \u2192 Set \u03a9 hs : \u2200 (n : \u2115), MeasurableSet (s n) this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     Tendsto (fun n => Finset.sum (Finset.range n) (fun k => Set.indicator (s (k + 1)) 1) \u03c9) atTop atTop \u2194       Tendsto (fun n => Finset.sum (Finset.range n) (fun k => \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194       Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop ** simpa using this ** case refine_1 \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 s : \u2115 \u2192 Set \u03a9 hs : \u2200 (n : \u2115), MeasurableSet (s n) \u03c9 : \u03a9 n : \u2115 \u22a2 process (fun n => s n) n \u03c9 \u2264 process (fun n => s n) (n + 1) \u03c9 ** rw [process, process, \u2190 sub_nonneg, Finset.sum_apply, Finset.sum_apply,\n  Finset.sum_range_succ_sub_sum] ** case refine_1 \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 s : \u2115 \u2192 Set \u03a9 hs : \u2200 (n : \u2115), MeasurableSet (s n) \u03c9 : \u03a9 n : \u2115 \u22a2 0 \u2264 Set.indicator (s (n + 1)) 1 \u03c9 ** exact Set.indicator_nonneg (fun _ _ => zero_le_one) _ ** Qed",
        "informal": ""
    },
    {
        "formal": "QPF.suppPreservation_iff_liftpPreservation ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u22a2 SuppPreservation \u2194 LiftpPreservation ** constructor <;> intro h ** case mp F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h : SuppPreservation \u22a2 LiftpPreservation ** rintro \u03b1 p \u27e8a, f\u27e9 ** case mp.mk F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h : SuppPreservation \u03b1 : Type u p : \u03b1 \u2192 Prop a : (P F).A f : PFunctor.B (P F) a \u2192 \u03b1 \u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f } ** have h' := h ** case mp.mk F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h : SuppPreservation \u03b1 : Type u p : \u03b1 \u2192 Prop a : (P F).A f : PFunctor.B (P F) a \u2192 \u03b1 h' : SuppPreservation \u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f } ** rw [suppPreservation_iff_uniform] at h' ** case mp.mk F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h : SuppPreservation \u03b1 : Type u p : \u03b1 \u2192 Prop a : (P F).A f : PFunctor.B (P F) a \u2192 \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 F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h :   \u2200 \u2983\u03b1 : Type u\u2984 (x : \u2191(P F) \u03b1), {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y} \u03b1 : Type u p : \u03b1 \u2192 Prop a : (P F).A f : PFunctor.B (P F) a \u2192 \u03b1 h' : IsUniform \u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f } ** rw [liftp_iff_of_isUniform h', supp_eq_of_isUniform h', PFunctor.liftp_iff'] ** case mp.mk F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h :   \u2200 \u2983\u03b1 : Type u\u2984 (x : \u2191(P F) \u03b1), {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y} \u03b1 : Type u p : \u03b1 \u2192 Prop a : (P F).A f : PFunctor.B (P F) a \u2192 \u03b1 h' : IsUniform \u22a2 (\u2200 (u : \u03b1), u \u2208 f '' univ \u2192 p u) \u2194 \u2200 (i : PFunctor.B (P F) a), p (f i) ** simp only [image_univ, mem_range, exists_imp] ** case mp.mk F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h :   \u2200 \u2983\u03b1 : Type u\u2984 (x : \u2191(P F) \u03b1), {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y} \u03b1 : Type u p : \u03b1 \u2192 Prop a : (P F).A f : PFunctor.B (P F) a \u2192 \u03b1 h' : IsUniform \u22a2 (\u2200 (u : \u03b1) (x : PFunctor.B (P F) a), f x = u \u2192 p u) \u2194 \u2200 (i : PFunctor.B (P F) a), p (f i) ** constructor <;> intros <;> subst_vars <;> solve_by_elim ** case mpr F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h : LiftpPreservation \u22a2 SuppPreservation ** rintro \u03b1 \u27e8a, f\u27e9 ** case mpr.mk F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h : LiftpPreservation \u03b1 : Type u a : (P F).A f : PFunctor.B (P F) a \u2192 \u03b1 \u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f } ** simp only [LiftpPreservation] at h ** case mpr.mk F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F h : \u2200 \u2983\u03b1 : Type u\u2984 (p : \u03b1 \u2192 Prop) (x : \u2191(P F) \u03b1), Liftp p (abs x) \u2194 Liftp p x \u03b1 : Type u a : (P F).A f : PFunctor.B (P F) a \u2192 \u03b1 \u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f } ** simp only [supp, h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.seq_of_forall_finite_exists ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3\u271d : Type x \u03b3 : Type u_1 P : \u03b3 \u2192 Set \u03b3 \u2192 Prop h : \u2200 (t : Set \u03b3), Set.Finite t \u2192 \u2203 c, P c t \u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n) ** haveI : Nonempty \u03b3 := (h \u2205 finite_empty).nonempty ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3\u271d : Type x \u03b3 : Type u_1 P : \u03b3 \u2192 Set \u03b3 \u2192 Prop h : \u2200 (t : Set \u03b3), Set.Finite t \u2192 \u2203 c, P c t this : Nonempty \u03b3 \u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n) ** choose! c hc using h ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3\u271d : Type x \u03b3 : Type u_1 P : \u03b3 \u2192 Set \u03b3 \u2192 Prop this : Nonempty \u03b3 c : Set \u03b3 \u2192 \u03b3 hc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t \u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n) ** set f : (n : \u2115) \u2192 (g : (m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k : Iio n => g k.1 k.2) ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3\u271d : Type x \u03b3 : Type u_1 P : \u03b3 \u2192 Set \u03b3 \u2192 Prop this : Nonempty \u03b3 c : Set \u03b3 \u2192 \u03b3 hc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t f : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n)) \u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n) ** set u : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3\u271d : Type x \u03b3 : Type u_1 P : \u03b3 \u2192 Set \u03b3 \u2192 Prop this : Nonempty \u03b3 c : Set \u03b3 \u2192 \u03b3 hc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t f : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n)) u : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f \u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n) ** refine' \u27e8u, fun n => _\u27e9 ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3\u271d : Type x \u03b3 : Type u_1 P : \u03b3 \u2192 Set \u03b3 \u2192 Prop this : Nonempty \u03b3 c : Set \u03b3 \u2192 \u03b3 hc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t f : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n)) u : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f n : \u2115 \u22a2 P (u n) (u '' Iio n) ** convert hc (u '' Iio n) ((finite_lt_nat _).image _) ** case h.e'_1 \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3\u271d : Type x \u03b3 : Type u_1 P : \u03b3 \u2192 Set \u03b3 \u2192 Prop this : Nonempty \u03b3 c : Set \u03b3 \u2192 \u03b3 hc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t f : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n)) u : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f n : \u2115 \u22a2 u n = c (u '' Iio n) ** rw [image_eq_range] ** case h.e'_1 \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3\u271d : Type x \u03b3 : Type u_1 P : \u03b3 \u2192 Set \u03b3 \u2192 Prop this : Nonempty \u03b3 c : Set \u03b3 \u2192 \u03b3 hc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t f : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n)) u : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f n : \u2115 \u22a2 u n = c (range fun x => u \u2191x) ** exact Nat.strongRecOn'_beta ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Primrec'.comp\u2082 ** f : \u2115 \u2192 \u2115 \u2192 \u2115 hf : Primrec' fun v => f (Vector.head v) (Vector.head (Vector.tail v)) n : \u2115 g h : Vector \u2115 n \u2192 \u2115 hg : Primrec' g hh : Primrec' h \u22a2 Primrec' fun v => f (g v) (h v) ** simpa using hf.comp' (hg.cons <| hh.cons Primrec'.nil) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.ext_of_Ioc_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 (Ioc a b) = \u2191\u2191\u03bd (Ioc a b) \u22a2 \u03bc = \u03bd ** refine' @ext_of_Ico_finite \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a \u03bc \u03bd _ h\u03bc\u03bd fun a b hab => _ ** \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 (Ioc a b) = \u2191\u2191\u03bd (Ioc a b) a b : \u03b1\u1d52\u1d48 hab : a < b \u22a2 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b) ** erw [dual_Ico (\u03b1 := \u03b1)] ** \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 (Ioc a b) = \u2191\u2191\u03bd (Ioc a b) a b : \u03b1\u1d52\u1d48 hab : a < b \u22a2 \u2191\u2191\u03bc (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) = \u2191\u2191\u03bd (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) ** exact h hab ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condDistrib_ae_eq_condexp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : 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 hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) =\u1d50[\u03bc]     \u03bc[indicator (Y \u207b\u00b9' s) fun \u03c9 => 1|MeasurableSpace.comap X m\u03b2] ** refine' ae_eq_condexp_of_forall_set_integral_eq hX.comap_le _ _ _ _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : 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 hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 Integrable (indicator (Y \u207b\u00b9' s) fun \u03c9 => 1) ** exact (integrable_const _).indicator (hY hs) ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : 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 hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 \u2200 (s_1 : Set \u03b1),     MeasurableSet s_1 \u2192 \u2191\u2191\u03bc s_1 < \u22a4 \u2192 IntegrableOn (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) s_1 ** exact fun t _ _ => (integrable_toReal_condDistrib hX.aemeasurable hs).integrableOn ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : 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 hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 \u2200 (s_1 : Set \u03b1),     MeasurableSet s_1 \u2192       \u2191\u2191\u03bc s_1 < \u22a4 \u2192         \u222b (x : \u03b1) in s_1, ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X x)) s) \u2202\u03bc =           \u222b (x : \u03b1) in s_1, indicator (Y \u207b\u00b9' s) (fun \u03c9 => 1) x \u2202\u03bc ** intro t ht _ ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : 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 hX : Measurable X hY : Measurable Y hs : MeasurableSet s t : Set \u03b1 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u222b (x : \u03b1) in t, ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X x)) s) \u2202\u03bc =     \u222b (x : \u03b1) in t, indicator (Y \u207b\u00b9' s) (fun \u03c9 => 1) x \u2202\u03bc ** rw [integral_toReal ((measurable_condDistrib hs).mono hX.comap_le le_rfl).aemeasurable\n  (eventually_of_forall fun \u03c9 => measure_lt_top (condDistrib Y X \u03bc (X \u03c9)) _),\n  integral_indicator_const _ (hY hs), Measure.restrict_apply (hY hs), smul_eq_mul, mul_one,\n  inter_comm, set_lintegral_condDistrib_of_measurableSet hX hY.aemeasurable hs ht] ** case refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : 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 hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 AEStronglyMeasurable' (MeasurableSpace.comap X m\u03b2) (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) \u03bc ** refine' (Measurable.stronglyMeasurable _).aeStronglyMeasurable' ** case refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : 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 hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 Measurable fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s) ** exact @Measurable.ennreal_toReal _ (m\u03b2.comap X) _ (measurable_condDistrib hs) ** Qed",
        "informal": ""
    },
    {
        "formal": "integrableOn_Icc_iff_integrableOn_Ico ** \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": "Num.castNum_and ** \u03b1 : Type u_1 \u22a2 \u2200 (m n : Num), \u2191(m &&& n) = \u2191m &&& \u2191n ** apply castNum_eq_bitwise PosNum.land <;> intros <;> (try cases_type* Bool) <;> rfl ** case pbb \u03b1 : Type u_1 a\u271d b\u271d : Bool m\u271d n\u271d : PosNum \u22a2 PosNum.land (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && b\u271d) (PosNum.land 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.land (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && b\u271d) (PosNum.land m\u271d n\u271d) ** cases_type* Bool ** Qed",
        "informal": ""
    },
    {
        "formal": "Sum.liftRel_swap_iff ** \u03b2\u271d\u00b9 : Type u_1 \u03b2\u271d : Type u_2 s : \u03b2\u271d\u00b9 \u2192 \u03b2\u271d \u2192 Prop \u03b1\u271d\u00b9 : Type u_3 \u03b1\u271d : Type u_4 r : \u03b1\u271d\u00b9 \u2192 \u03b1\u271d \u2192 Prop x : \u03b1\u271d\u00b9 \u2295 \u03b2\u271d\u00b9 y : \u03b1\u271d \u2295 \u03b2\u271d h : LiftRel s r (swap x) (swap y) \u22a2 LiftRel r s x y ** rw [\u2190 swap_swap x, \u2190 swap_swap y] ** \u03b2\u271d\u00b9 : Type u_1 \u03b2\u271d : Type u_2 s : \u03b2\u271d\u00b9 \u2192 \u03b2\u271d \u2192 Prop \u03b1\u271d\u00b9 : Type u_3 \u03b1\u271d : Type u_4 r : \u03b1\u271d\u00b9 \u2192 \u03b1\u271d \u2192 Prop x : \u03b1\u271d\u00b9 \u2295 \u03b2\u271d\u00b9 y : \u03b1\u271d \u2295 \u03b2\u271d h : LiftRel s r (swap x) (swap y) \u22a2 LiftRel r s (swap (swap x)) (swap (swap y)) ** exact h.swap ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Martingale.ae_not_tendsto_atTop_atBot ** \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 : Martingale 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, \u00acTendsto (fun n => f n \u03c9) atTop atBot ** filter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with \u03c9 h\u03c9 htop using\n  unbounded_of_tendsto_atBot htop (h\u03c9.1 <| bddAbove_range_of_tendsto_atTop_atBot htop) ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.AssocList.mapVal_toList ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b4 l : AssocList \u03b1 \u03b2 \u22a2 toList (mapVal f l) =     List.map       (fun x =>         match x with         | (a, b) => (a, f a b))       (toList l) ** induction l <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rel.preimage_id ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : Rel \u03b1 \u03b2 s : Set \u03b1 \u22a2 preimage Eq s = s ** simp only [preimage, inv_id, image_id] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded ** \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 : IsFiniteKernel \u03ba B : \u211d\u22650\u221e hB_top : B \u2260 \u22a4 hf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B \u22a2 IsFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba B : \u211d\u22650\u221e hB_top : B \u2260 \u22a4 hf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B hf : Measurable (Function.uncurry f) \u22a2 IsFiniteKernel (withDensity \u03ba f) ** exact \u27e8\u27e8B * IsFiniteKernel.bound \u03ba, ENNReal.mul_lt_top hB_top (IsFiniteKernel.bound_ne_top \u03ba),\n  fun a => by\n    rw [withDensity_apply' \u03ba hf a MeasurableSet.univ]\n    calc\n      \u222b\u207b b in Set.univ, f a b \u2202\u03ba a \u2264 \u222b\u207b _ in Set.univ, B \u2202\u03ba a := lintegral_mono (hf_B a)\n      _ = B * \u03ba a Set.univ := by\n        simp only [Measure.restrict_univ, MeasureTheory.lintegral_const]\n      _ \u2264 B * IsFiniteKernel.bound \u03ba := mul_le_mul_left' (measure_le_bound \u03ba a Set.univ) _\u27e9\u27e9 ** \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 : IsFiniteKernel \u03ba B : \u211d\u22650\u221e hB_top : B \u2260 \u22a4 hf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B hf : Measurable (Function.uncurry f) a : \u03b1 \u22a2 \u2191\u2191(\u2191(withDensity \u03ba f) a) Set.univ \u2264 B * IsFiniteKernel.bound \u03ba ** rw [withDensity_apply' \u03ba hf a MeasurableSet.univ] ** \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 : IsFiniteKernel \u03ba B : \u211d\u22650\u221e hB_top : B \u2260 \u22a4 hf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B hf : Measurable (Function.uncurry f) a : \u03b1 \u22a2 \u222b\u207b (b : \u03b2) in Set.univ, f a b \u2202\u2191\u03ba a \u2264 B * IsFiniteKernel.bound \u03ba ** calc\n  \u222b\u207b b in Set.univ, f a b \u2202\u03ba a \u2264 \u222b\u207b _ in Set.univ, B \u2202\u03ba a := lintegral_mono (hf_B a)\n  _ = B * \u03ba a Set.univ := by\n    simp only [Measure.restrict_univ, MeasureTheory.lintegral_const]\n  _ \u2264 B * IsFiniteKernel.bound \u03ba := mul_le_mul_left' (measure_le_bound \u03ba a Set.univ) _ ** \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 : IsFiniteKernel \u03ba B : \u211d\u22650\u221e hB_top : B \u2260 \u22a4 hf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B hf : Measurable (Function.uncurry f) a : \u03b1 \u22a2 \u222b\u207b (x : \u03b2) in Set.univ, B \u2202\u2191\u03ba a = B * \u2191\u2191(\u2191\u03ba a) Set.univ ** simp only [Measure.restrict_univ, MeasureTheory.lintegral_const] ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba B : \u211d\u22650\u221e hB_top : B \u2260 \u22a4 hf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B hf : \u00acMeasurable (Function.uncurry f) \u22a2 IsFiniteKernel (withDensity \u03ba f) ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba B : \u211d\u22650\u221e hB_top : B \u2260 \u22a4 hf_B : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2264 B hf : \u00acMeasurable (Function.uncurry f) \u22a2 IsFiniteKernel 0 ** infer_instance ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.reaches_eval ** \u03c3 : Type u_1 f : \u03c3 \u2192 Option \u03c3 a b : \u03c3 ab : Reaches f a b \u22a2 eval f a = eval f b ** refine' Part.ext fun _ \u21a6 \u27e8fun h \u21a6 _, fun h \u21a6 _\u27e9 ** case refine'_1 \u03c3 : Type u_1 f : \u03c3 \u2192 Option \u03c3 a b : \u03c3 ab : Reaches f a b x\u271d : \u03c3 h : x\u271d \u2208 eval f a \u22a2 x\u271d \u2208 eval f b ** have \u27e8ac, c0\u27e9 := mem_eval.1 h ** case refine'_1 \u03c3 : Type u_1 f : \u03c3 \u2192 Option \u03c3 a b : \u03c3 ab : Reaches f a b x\u271d : \u03c3 h : x\u271d \u2208 eval f a ac : Reaches f a x\u271d c0 : f x\u271d = none \u22a2 x\u271d \u2208 eval f b ** exact mem_eval.2 \u27e8(or_iff_left_of_imp fun cb \u21a6 (eval_maximal h).1 cb \u25b8 ReflTransGen.refl).1\n  (reaches_total ab ac), c0\u27e9 ** case refine'_2 \u03c3 : Type u_1 f : \u03c3 \u2192 Option \u03c3 a b : \u03c3 ab : Reaches f a b x\u271d : \u03c3 h : x\u271d \u2208 eval f b \u22a2 x\u271d \u2208 eval f a ** have \u27e8bc, c0\u27e9 := mem_eval.1 h ** case refine'_2 \u03c3 : Type u_1 f : \u03c3 \u2192 Option \u03c3 a b : \u03c3 ab : Reaches f a b x\u271d : \u03c3 h : x\u271d \u2208 eval f b bc : Reaches f b x\u271d c0 : f x\u271d = none \u22a2 x\u271d \u2208 eval f a ** exact mem_eval.2 \u27e8ab.trans bc, c0\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_add_ncard_compl ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d hsc : autoParam (Set.Finite s\u1d9c) _auto\u271d \u22a2 ncard s + ncard s\u1d9c = Nat.card \u03b1 ** rw [\u2190 ncard_univ, \u2190 ncard_union_eq (@disjoint_compl_right _ _ s) hs hsc, union_compl_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.mapAccumr\u2082_bisim_tail ** \u03b1 : Type n : \u2115 xs : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type ys : Vector \u03b2 n f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 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) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 \u22a2 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2 ** rcases h with \u27e8R, h\u2080, hR\u27e9 ** case intro.intro \u03b1 : Type n : \u2115 xs : Vector \u03b1 n \u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type ys : Vector \u03b2 n f\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 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) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).1 (f\u2082 a b q).1 \u2227 (f\u2081 a b s).2 = (f\u2082 a b q).2 \u22a2 (mapAccumr\u2082 f\u2081 xs ys s\u2081).2 = (mapAccumr\u2082 f\u2082 xs ys s\u2082).2 ** exact (mapAccumr\u2082_bisim R h\u2080 hR).2 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Content.outerMeasure_caratheodory ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G A : Set G \u22a2 MeasurableSet A \u2194     \u2200 (U : Opens G),       \u2191(Content.outerMeasure \u03bc) (\u2191U \u2229 A) + \u2191(Content.outerMeasure \u03bc) (\u2191U \\ A) \u2264 \u2191(Content.outerMeasure \u03bc) \u2191U ** rw [Opens.forall] ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G A : Set G \u22a2 MeasurableSet A \u2194     \u2200 (U : Set G) (hU : IsOpen U),       \u2191(Content.outerMeasure \u03bc) (\u2191{ carrier := U, is_open' := hU } \u2229 A) +           \u2191(Content.outerMeasure \u03bc) (\u2191{ carrier := U, is_open' := hU } \\ A) \u2264         \u2191(Content.outerMeasure \u03bc) \u2191{ carrier := U, is_open' := hU } ** apply inducedOuterMeasure_caratheodory ** case msU G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G A : Set G \u22a2 \u2200 \u2983f : \u2115 \u2192 Set G\u2984 (hm : \u2200 (i : \u2115), IsOpen (f i)),     innerContent \u03bc { carrier := \u22c3 i, f i, is_open' := (_ : ?P (\u22c3 i, f i)) } \u2264       \u2211' (i : \u2115), innerContent \u03bc { carrier := f i, is_open' := (_ : IsOpen (f i)) }  case m_mono G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G A : Set G \u22a2 \u2200 \u2983s\u2081 s\u2082 : Set G\u2984 (hs\u2081 : IsOpen s\u2081) (hs\u2082 : IsOpen s\u2082),     s\u2081 \u2286 s\u2082 \u2192 innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 } \u2264 innerContent \u03bc { carrier := s\u2082, is_open' := hs\u2082 }  case PU G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G A : Set G \u22a2 \u2200 \u2983f : \u2115 \u2192 Set G\u2984, (\u2200 (i : \u2115), IsOpen (f i)) \u2192 IsOpen (\u22c3 i, f i) ** apply innerContent_iUnion_nat ** case m_mono G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G A : Set G \u22a2 \u2200 \u2983s\u2081 s\u2082 : Set G\u2984 (hs\u2081 : IsOpen s\u2081) (hs\u2082 : IsOpen s\u2082),     s\u2081 \u2286 s\u2082 \u2192 innerContent \u03bc { carrier := s\u2081, is_open' := hs\u2081 } \u2264 innerContent \u03bc { carrier := s\u2082, is_open' := hs\u2082 } ** apply innerContent_mono' ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pi_update_of_mem ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \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 : \u03b9 inst\u271d : DecidableEq \u03b9 hi : i \u2208 s f : (j : \u03b9) \u2192 \u03b1 j a : \u03b1 i t : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j) \u22a2 (pi s fun j => t j (update f i a j)) = pi ({i} \u222a s \\ {i}) fun j => t j (update f i a j) ** rw [union_diff_self, union_eq_self_of_subset_left (singleton_subset_iff.2 hi)] ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \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 : \u03b9 inst\u271d : DecidableEq \u03b9 hi : i \u2208 s f : (j : \u03b9) \u2192 \u03b1 j a : \u03b1 i t : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j) \u22a2 (pi ({i} \u222a s \\ {i}) fun j => t j (update f i a j)) = {x | x i \u2208 t i a} \u2229 pi (s \\ {i}) fun j => t j (f j) ** rw [union_pi, singleton_pi', update_same, pi_update_of_not_mem] ** case hi \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \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 : \u03b9 inst\u271d : DecidableEq \u03b9 hi : i \u2208 s f : (j : \u03b9) \u2192 \u03b1 j a : \u03b1 i t : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j) \u22a2 \u00aci \u2208 s \\ {i} ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "VitaliFamily.fineSubfamilyOn_of_frequently ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v\u271d v : VitaliFamily \u03bc f : \u03b1 \u2192 Set (Set \u03b1) s : Set \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x \u22a2 FineSubfamilyOn v f s ** intro x hx \u03b5 \u03b5pos ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v\u271d v : VitaliFamily \u03bc f : \u03b1 \u2192 Set (Set \u03b1) s : Set \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x x : \u03b1 hx : x \u2208 s \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** obtain \u27e8a, av, ha, af\u27e9 : \u2203 (a : Set \u03b1) , a \u2208 v.setsAt x \u2227 a \u2286 closedBall x \u03b5 \u2227 a \u2208 f x :=\n  v.frequently_filterAt_iff.1 (h x hx) \u03b5 \u03b5pos ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v\u271d v : VitaliFamily \u03bc f : \u03b1 \u2192 Set (Set \u03b1) s : Set \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x x : \u03b1 hx : x \u2208 s \u03b5 : \u211d \u03b5pos : \u03b5 > 0 a : Set \u03b1 av : a \u2208 setsAt v x ha : a \u2286 closedBall x \u03b5 af : a \u2208 f x \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** exact \u27e8a, \u27e8av, af\u27e9, ha\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.inf_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 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (t : \u211a), \u2a05 r, preCdf \u03c1 (\u2191r) a = preCdf \u03c1 t a ** rw [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 : \u211a), \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2a05 r, preCdf \u03c1 (\u2191r) a = preCdf \u03c1 i a ** refine' fun t => ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite _ 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 \u22a2 \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2191\u2191(Measure.fst \u03c1) s < \u22a4 \u2192         \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1 ** intro s hs _ ** 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 a\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1 ** rw [set_lintegral_iInf_gt_preCdf \u03c1 t hs, set_lintegral_preCdf_fst \u03c1 t hs] ** 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 \u22a2 Measurable fun a => \u2a05 r, preCdf \u03c1 (\u2191r) a ** exact measurable_iInf fun i => measurable_preCdf ** Qed",
        "informal": ""
    },
    {
        "formal": "List.count_filter ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 p : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 h : p a = true \u22a2 count a (filter p l) = count a l ** rw [count, countP_filter] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 p : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 h : p a = true \u22a2 countP (fun a_1 => decide ((a_1 == a) = true \u2227 p a_1 = true)) l = countP (fun x => x == a) l ** congr ** case e_p \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 p : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 h : p a = true \u22a2 (fun a_1 => decide ((a_1 == a) = true \u2227 p a_1 = true)) = fun x => x == a ** funext b ** case e_p.h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 p : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 h : p a = true b : \u03b1 \u22a2 decide ((b == a) = true \u2227 p b = true) = (b == a) ** rw [(by rfl : (b == a) = decide (b = a)), decide_eq_decide] ** case e_p.h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 p : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 h : p a = true b : \u03b1 \u22a2 decide (b = a) = true \u2227 p b = true \u2194 b = a ** simp ** case e_p.h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 p : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 h : p a = true b : \u03b1 \u22a2 b = a \u2192 p b = true ** rintro rfl ** case e_p.h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 p : \u03b1 \u2192 Bool l : List \u03b1 b : \u03b1 h : p b = true \u22a2 p b = true ** exact h ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 p : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 h : p a = true b : \u03b1 \u22a2 (b == a) = decide (b = a) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.diag_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 \u22a2 diag (insert a s) = insert (a, a) (diag s) ** rw [insert_eq, insert_eq, diag_union, diag_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "VitaliFamily.withDensity_limRatioMeas_eq ** \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 h\u03c1 : \u03c1 \u226a \u03bc \u22a2 withDensity \u03bc (limRatioMeas v h\u03c1) = \u03c1 ** ext1 s 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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s = \u2191\u2191\u03c1 s ** refine' le_antisymm _ _ ** case h.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s this : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (1 ^ 2 * \u2191\u2191\u03c1 s)) \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191\u2191\u03c1 s ** simp only [one_pow, one_mul, ENNReal.coe_one] at this ** case h.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s this : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (\u2191\u2191\u03c1 s)) \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191\u2191\u03c1 s ** refine' ge_of_tendsto this _ ** case h.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s this : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (\u2191\u2191\u03c1 s)) \u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 1] 1, \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191c ^ 2 * \u2191\u2191\u03c1 s ** filter_upwards [self_mem_nhdsWithin] with _ ht ** 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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s this : Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (\u2191\u2191\u03c1 s)) a\u271d : \u211d\u22650 ht : a\u271d \u2208 Ioi 1 \u22a2 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2264 \u2191a\u271d ^ 2 * \u2191\u2191\u03c1 s ** exact v.withDensity_le_mul h\u03c1 hs ht ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s \u22a2 Tendsto (fun t => \u2191t ^ 2 * \u2191\u2191\u03c1 s) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (1 ^ 2 * \u2191\u2191\u03c1 s)) ** refine' ENNReal.Tendsto.mul _ _ tendsto_const_nhds _ ** 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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s \u22a2 Tendsto (fun t => \u2191t ^ 2) (\ud835\udcdd[Ioi 1] 1) (\ud835\udcdd (1 ^ 2)) ** exact ENNReal.Tendsto.pow (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) ** 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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s \u22a2 1 ^ 2 \u2260 0 \u2228 \u2191\u2191\u03c1 s \u2260 \u22a4 ** simp only [one_pow, ENNReal.coe_one, true_or_iff, Ne.def, not_false_iff, one_ne_zero] ** case refine'_3 \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191\u03c1 s \u2260 0 \u2228 1 ^ 2 \u2260 \u22a4 ** simp only [one_pow, ENNReal.coe_one, Ne.def, or_true_iff, ENNReal.one_ne_top, not_false_iff] ** case h.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s ** have :\n  Tendsto (fun t : \u211d\u22650 => (t : \u211d\u22650\u221e) * \u03bc.withDensity (v.limRatioMeas h\u03c1) s) (\ud835\udcdd[>] 1)\n    (\ud835\udcdd ((1 : \u211d\u22650\u221e) * \u03bc.withDensity (v.limRatioMeas h\u03c1) s)) := by\n  refine' ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) _\n  simp only [ENNReal.coe_one, true_or_iff, Ne.def, not_false_iff, one_ne_zero] ** case h.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s this :   Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)     (\ud835\udcdd (1 * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s)) \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s ** simp only [one_mul, ENNReal.coe_one] at this ** case h.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s this :   Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)     (\ud835\udcdd (\u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s)) \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s ** refine' ge_of_tendsto this _ ** case h.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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s this :   Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)     (\ud835\udcdd (\u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s)) \u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 1] 1, \u2191\u2191\u03c1 s \u2264 \u2191c * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s ** filter_upwards [self_mem_nhdsWithin] with _ ht ** 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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s this :   Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)     (\ud835\udcdd (\u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s)) a\u271d : \u211d\u22650 ht : a\u271d \u2208 Ioi 1 \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191a\u271d * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s ** exact v.le_mul_withDensity h\u03c1 hs ht ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s \u22a2 Tendsto (fun t => \u2191t * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s) (\ud835\udcdd[Ioi 1] 1)     (\ud835\udcdd (1 * \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s)) ** refine' ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) _ ** \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 h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : MeasurableSet s \u22a2 1 \u2260 0 \u2228 \u2191\u2191(withDensity \u03bc (limRatioMeas v h\u03c1)) s \u2260 \u22a4 ** simp only [ENNReal.coe_one, true_or_iff, Ne.def, not_false_iff, one_ne_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_eq_of_ae_le_of_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 g : \u03b1 \u2192 \u211d\u22650\u221e hfg : f \u2264\u1d50[\u03bc] g hf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u22a2 f =\u1d50[\u03bc] g ** have : \u2200 n : \u2115, \u2200\u1d50 x \u2202\u03bc, g x < f x + (n : \u211d\u22650\u221e)\u207b\u00b9 := by\n  intro n\n  simp only [ae_iff, not_lt]\n  have : \u222b\u207b x, f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u03bc { x : \u03b1 | f x + (n : \u211d\u22650\u221e)\u207b\u00b9 \u2264 g x } \u2264 \u222b\u207b x, f x \u2202\u03bc :=\n    (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n\u207b\u00b9).trans hgf\n  rw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this\n  exact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.nat_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 f g : \u03b1 \u2192 \u211d\u22650\u221e hfg : f \u2264\u1d50[\u03bc] g hf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc this : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9 \u22a2 f =\u1d50[\u03bc] g ** refine' hfg.mp ((ae_all_iff.2 this).mono fun x hlt hle => hle.antisymm _) ** \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 hfg : f \u2264\u1d50[\u03bc] g hf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc this : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9 x : \u03b1 hlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9 hle : f x \u2264 g x \u22a2 g x \u2264 f x ** suffices Tendsto (fun n : \u2115 => f x + (n : \u211d\u22650\u221e)\u207b\u00b9) atTop (\ud835\udcdd (f x)) from\n  ge_of_tendsto' this fun i => (hlt i).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 g : \u03b1 \u2192 \u211d\u22650\u221e hfg : f \u2264\u1d50[\u03bc] g hf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc this : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9 x : \u03b1 hlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9 hle : f x \u2264 g x \u22a2 Tendsto (fun n => f x + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd (f x)) ** simpa only [inv_top, add_zero] using\n  tendsto_const_nhds.add (ENNReal.tendsto_inv_iff.2 ENNReal.tendsto_nat_nhds_top) ** \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 hfg : f \u2264\u1d50[\u03bc] g hf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9 ** 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 f g : \u03b1 \u2192 \u211d\u22650\u221e hfg : f \u2264\u1d50[\u03bc] g hf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc n : \u2115 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9 ** simp only [ae_iff, not_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 g : \u03b1 \u2192 \u211d\u22650\u221e hfg : f \u2264\u1d50[\u03bc] g hf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc n : \u2115 \u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0 ** have : \u222b\u207b x, f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u03bc { x : \u03b1 | f x + (n : \u211d\u22650\u221e)\u207b\u00b9 \u2264 g x } \u2264 \u222b\u207b x, f x \u2202\u03bc :=\n  (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n\u207b\u00b9).trans hgf ** \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 hfg : f \u2264\u1d50[\u03bc] g hf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc n : \u2115 this : \u222b\u207b (x : \u03b1), f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u2191\u2191\u03bc {x | f x + (\u2191n)\u207b\u00b9 \u2264 g x} \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0 ** rw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] 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 g : \u03b1 \u2192 \u211d\u22650\u221e hfg : f \u2264\u1d50[\u03bc] g hf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc n : \u2115 this : (\u2191n)\u207b\u00b9 = 0 \u2228 \u2191\u2191\u03bc {x | f x + (\u2191n)\u207b\u00b9 \u2264 g x} = 0 \u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0 ** exact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_mul_Iio_of_neg ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a c : \u03b1 h : c < 0 \u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Iio a = Ioi (a / c) ** simpa only [mul_comm] using preimage_mul_const_Iio_of_neg a h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measureUnivNNReal_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 \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bc h\u03bc : \u03bc \u2260 0 \u22a2 0 < measureUnivNNReal \u03bc ** contrapose! h\u03bc ** \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 : IsFiniteMeasure \u03bc h\u03bc : measureUnivNNReal \u03bc \u2264 0 \u22a2 \u03bc = 0 ** simpa [measureUnivNNReal_eq_zero, le_zero_iff] using h\u03bc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.exists_positive_of_not_mutuallySingular ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** have :\n  \u2200 n : \u2115, \u2203 i : Set \u03b1,\n    MeasurableSet i \u2227\n      0 \u2264[i] \u03bc.toSignedMeasure - ((1 / (n + 1) : \u211d\u22650) \u2022 \u03bd).toSignedMeasure \u2227\n        \u03bc.toSignedMeasure - ((1 / (n + 1) : \u211d\u22650) \u2022 \u03bd).toSignedMeasure \u2264[i\u1d9c] 0 := by\n  intro; exact exists_compl_positive_negative _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd this :   \u2200 (n : \u2115),     \u2203 i,       MeasurableSet i \u2227         VectorMeasure.restrict 0 i \u2264             VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i \u2227           VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i\u1d9c \u2264             VectorMeasure.restrict 0 i\u1d9c \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** choose f hf\u2081 hf\u2082 hf\u2083 using this ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** set A := \u22c2 n, (f n)\u1d9c with hA\u2081 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** have hAmeas : MeasurableSet A := MeasurableSet.iInter fun n => (hf\u2081 n).compl ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** have hA\u2082 : \u2200 n : \u2115, \u03bc.toSignedMeasure - ((1 / (n + 1) : \u211d\u22650) \u2022 \u03bd).toSignedMeasure \u2264[A] 0 := by\n  intro n; exact restrict_le_restrict_subset _ _ (hf\u2081 n).compl (hf\u2083 n) (iInter_subset _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** have hA\u2083 : \u2200 n : \u2115, \u03bc A \u2264 (1 / (n + 1) : \u211d\u22650) * \u03bd A := by\n  intro n\n  have := nonpos_of_restrict_le_zero _ (hA\u2082 n)\n  rwa [toSignedMeasure_sub_apply hAmeas, sub_nonpos, ENNReal.toReal_le_toReal] at this\n  exacts [ne_of_lt (measure_lt_top _ _), ne_of_lt (measure_lt_top _ _)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A hA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A h\u03bc : \u2191\u2191\u03bc A = 0 \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** rw [MutuallySingular] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u2203 s, MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0 \u2227 \u2191\u2191\u03bd s\u1d9c = 0 f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A hA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A h\u03bc : \u2191\u2191\u03bc A = 0 \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** push_neg at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A hA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A h\u03bc : \u2191\u2191\u03bc A = 0 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0 \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** have := h _ hAmeas h\u03bc ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A hA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A h\u03bc : \u2191\u2191\u03bc A = 0 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0 this : \u2191\u2191\u03bd A\u1d9c \u2260 0 \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** simp_rw [compl_iInter, compl_compl] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A hA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A h\u03bc : \u2191\u2191\u03bc A = 0 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0 this : \u2191\u2191\u03bd (\u22c3 i, f i) \u2260 0 \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** obtain \u27e8n, hn\u27e9 := exists_measure_pos_of_not_measure_iUnion_null this ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A hA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A h\u03bc : \u2191\u2191\u03bc A = 0 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0 this : \u2191\u2191\u03bd (\u22c3 i, f i) \u2260 0 n : \u2115 hn : 0 < \u2191\u2191\u03bd (f n) \u22a2 \u2203 \u03b5,     0 < \u03b5 \u2227       \u2203 E,         MeasurableSet E \u2227           0 < \u2191\u2191\u03bd E \u2227             VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure (\u03b5 \u2022 \u03bd)) E ** exact \u27e81 / (n + 1), by simp, f n, hf\u2081 n, hn, hf\u2082 n\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd \u22a2 \u2200 (n : \u2115),     \u2203 i,       MeasurableSet i \u2227         VectorMeasure.restrict 0 i \u2264             VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i \u2227           VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) i\u1d9c \u2264             VectorMeasure.restrict 0 i\u1d9c ** intro ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd n\u271d : \u2115 \u22a2 \u2203 i,     MeasurableSet i \u2227       VectorMeasure.restrict 0 i \u2264           VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n\u271d + 1)) \u2022 \u03bd)) i \u2227         VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n\u271d + 1)) \u2022 \u03bd)) i\u1d9c \u2264           VectorMeasure.restrict 0 i\u1d9c ** exact exists_compl_positive_negative _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A \u22a2 \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A n : \u2115 \u22a2 VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A ** exact restrict_le_restrict_subset _ _ (hf\u2081 n).compl (hf\u2083 n) (iInter_subset _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A \u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A n : \u2115 \u22a2 \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A ** have := nonpos_of_restrict_le_zero _ (hA\u2082 n) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A n : \u2115 this : \u2191(toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 0 \u22a2 \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A ** rwa [toSignedMeasure_sub_apply hAmeas, sub_nonpos, ENNReal.toReal_le_toReal] at this ** case ha \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A n : \u2115 this : ENNReal.toReal (\u2191\u2191\u03bc A) \u2264 ENNReal.toReal (\u2191\u2191((1 / (\u2191n + 1)) \u2022 \u03bd) A) \u22a2 \u2191\u2191\u03bc A \u2260 \u22a4  case hb \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A n : \u2115 this : ENNReal.toReal (\u2191\u2191\u03bc A) \u2264 ENNReal.toReal (\u2191\u2191((1 / (\u2191n + 1)) \u2022 \u03bd) A) \u22a2 \u2191\u2191((1 / (\u2191n + 1)) \u2022 \u03bd) A \u2260 \u22a4 ** exacts [ne_of_lt (measure_lt_top _ _), ne_of_lt (measure_lt_top _ _)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A hA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A \u22a2 \u2191\u2191\u03bc A = 0 ** lift \u03bc A to \u211d\u22650 using ne_of_lt (measure_lt_top _ _) with \u03bcA ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u22a2 \u2191\u03bcA = 0 ** lift \u03bd A to \u211d\u22650 using ne_of_lt (measure_lt_top _ _) with \u03bdA ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA \u22a2 \u2191\u03bcA = 0 ** rw [ENNReal.coe_eq_zero] ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA \u22a2 \u03bcA = 0 ** by_cases hb : 0 < \u03bdA ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA \u22a2 \u03bcA = 0 ** suffices \u2200 b, 0 < b \u2192 \u03bcA \u2264 b by\n  by_contra h\n  have h' := this (\u03bcA / 2) (half_pos (zero_lt_iff.2 h))\n  rw [\u2190 @Classical.not_not (\u03bcA \u2264 \u03bcA / 2)] at h'\n  exact h' (not_le.2 (NNReal.half_lt_self h)) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA \u22a2 \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b ** intro c hc ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c \u22a2 \u03bcA \u2264 c ** have : \u2203 n : \u2115, 1 / (n + 1 : \u211d) < c * (\u03bdA : \u211d)\u207b\u00b9 ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c \u22a2 \u2203 n, 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9  case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c this : \u2203 n, 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 \u22a2 \u03bcA \u2264 c ** refine' exists_nat_one_div_lt _ ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c this : \u2203 n, 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 \u22a2 \u03bcA \u2264 c ** rcases this with \u27e8n, hn\u27e9 ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c n : \u2115 hn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 \u22a2 \u03bcA \u2264 c ** have hb\u2081 : (0 : \u211d) < (\u03bdA : \u211d)\u207b\u00b9 := by rw [_root_.inv_pos]; exact hb ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c n : \u2115 hn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 hb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9 h' : 1 / (\u2191n + 1) * \u03bdA < c \u22a2 \u03bcA \u2264 c ** refine' le_trans _ (le_of_lt h') ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c n : \u2115 hn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 hb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9 h' : 1 / (\u2191n + 1) * \u03bdA < c \u22a2 \u03bcA \u2264 1 / (\u2191n + 1) * \u03bdA ** rw [\u2190 ENNReal.coe_le_coe, ENNReal.coe_mul] ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c n : \u2115 hn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 hb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9 h' : 1 / (\u2191n + 1) * \u03bdA < c \u22a2 \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA ** exact hA\u2083 n ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA this : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b \u22a2 \u03bcA = 0 ** by_contra h ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h\u271d : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA this : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b h : \u00ac\u03bcA = 0 \u22a2 False ** have h' := this (\u03bcA / 2) (half_pos (zero_lt_iff.2 h)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h\u271d : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA this : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b h : \u00ac\u03bcA = 0 h' : \u03bcA \u2264 \u03bcA / 2 \u22a2 False ** rw [\u2190 @Classical.not_not (\u03bcA \u2264 \u03bcA / 2)] at h' ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h\u271d : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA this : \u2200 (b : \u211d\u22650), 0 < b \u2192 \u03bcA \u2264 b h : \u00ac\u03bcA = 0 h' : \u00ac\u00ac\u03bcA \u2264 \u03bcA / 2 \u22a2 False ** exact h' (not_le.2 (NNReal.half_lt_self h)) ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c \u22a2 0 < \u2191c * (\u2191\u03bdA)\u207b\u00b9 ** refine' mul_pos hc _ ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c \u22a2 0 < (\u2191\u03bdA)\u207b\u00b9 ** rw [_root_.inv_pos] ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c \u22a2 0 < \u2191\u03bdA ** exact hb ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c n : \u2115 hn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 \u22a2 0 < (\u2191\u03bdA)\u207b\u00b9 ** rw [_root_.inv_pos] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c n : \u2115 hn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 \u22a2 0 < \u2191\u03bdA ** exact hb ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c n : \u2115 hn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 hb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9 \u22a2 1 / (\u2191n + 1) * \u03bdA < c ** rw [\u2190 NNReal.coe_lt_coe, \u2190 mul_lt_mul_right hb\u2081, NNReal.coe_mul, mul_assoc, \u2190\n  NNReal.coe_inv, \u2190 NNReal.coe_mul, _root_.mul_inv_cancel, \u2190 NNReal.coe_mul, mul_one,\n  NNReal.coe_inv] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c n : \u2115 hn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 hb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9 \u22a2 \u2191(1 / (\u2191n + 1)) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 ** exact hn ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : 0 < \u03bdA c : \u211d\u22650 hc : 0 < c n : \u2115 hn : 1 / (\u2191n + 1) < \u2191c * (\u2191\u03bdA)\u207b\u00b9 hb\u2081 : 0 < (\u2191\u03bdA)\u207b\u00b9 \u22a2 \u03bdA \u2260 0 ** exact Ne.symm (ne_of_lt hb) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : \u00ac0 < \u03bdA \u22a2 \u03bcA = 0 ** rw [not_lt, le_zero_iff] at hb ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d hA\u2083 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : \u03bdA = 0 \u22a2 \u03bcA = 0 ** specialize hA\u2083 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : \u03bdA = 0 hA\u2083 : \u2191\u03bcA \u2264 \u2191(1 / (\u21910 + 1)) * \u2191\u03bdA \u22a2 \u03bcA = 0 ** simp [hb, le_zero_iff] at hA\u2083 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h : \u00ac\u03bc \u27c2\u2098 \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet (\u22c2 n, (f n)\u1d9c) hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (\u22c2 n, (f n)\u1d9c) \u2264       VectorMeasure.restrict 0 (\u22c2 n, (f n)\u1d9c) \u03bcA : \u211d\u22650 hA\u2081\u271d hA\u2081 : True hA\u2083\u271d\u00b9 : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd (\u22c2 n, (f n)\u1d9c) \u03bdA : \u211d\u22650 hA\u2083\u271d : \u2200 (n : \u2115), \u2191\u03bcA \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u03bdA hb : \u03bdA = 0 hA\u2083 : \u03bcA = 0 \u22a2 \u03bcA = 0 ** assumption ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (n : \u2115), MeasurableSet (f n) hf\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict 0 (f n) \u2264       VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n) hf\u2083 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) (f n)\u1d9c \u2264       VectorMeasure.restrict 0 (f n)\u1d9c A : Set \u03b1 := \u22c2 n, (f n)\u1d9c hA\u2081 : A = \u22c2 n, (f n)\u1d9c hAmeas : MeasurableSet A hA\u2082 :   \u2200 (n : \u2115),     VectorMeasure.restrict (toSignedMeasure \u03bc - toSignedMeasure ((1 / (\u2191n + 1)) \u2022 \u03bd)) A \u2264 VectorMeasure.restrict 0 A hA\u2083 : \u2200 (n : \u2115), \u2191\u2191\u03bc A \u2264 \u2191(1 / (\u2191n + 1)) * \u2191\u2191\u03bd A h\u03bc : \u2191\u2191\u03bc A = 0 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bd s\u1d9c \u2260 0 this : \u2191\u2191\u03bd (\u22c3 i, f i) \u2260 0 n : \u2115 hn : 0 < \u2191\u2191\u03bd (f n) \u22a2 0 < 1 / (\u2191n + 1) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "generatePiSystem_mono ** \u03b1 : Type u_1 S T : Set (Set \u03b1) hST : S \u2286 T t : Set \u03b1 ht : t \u2208 generatePiSystem S \u22a2 t \u2208 generatePiSystem T ** induction' ht with s h_s s u _ _ h_nonempty h_s h_u ** case base \u03b1 : Type u_1 S T : Set (Set \u03b1) hST : S \u2286 T t s : Set \u03b1 h_s : s \u2208 S \u22a2 s \u2208 generatePiSystem T ** exact generatePiSystem.base (Set.mem_of_subset_of_mem hST h_s) ** case inter \u03b1 : Type u_1 S T : Set (Set \u03b1) hST : S \u2286 T t s u : Set \u03b1 h_s\u271d : generatePiSystem S s h_t\u271d : generatePiSystem S u h_nonempty : Set.Nonempty (s \u2229 u) h_s : s \u2208 generatePiSystem T h_u : u \u2208 generatePiSystem T \u22a2 s \u2229 u \u2208 generatePiSystem T ** exact isPiSystem_generatePiSystem T _ h_s _ h_u h_nonempty ** Qed",
        "informal": ""
    },
    {
        "formal": "AddCircle.isAddFundamentalDomain_of_ae_ball ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u hI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u))) \u22a2 IsAddFundamentalDomain { x // x \u2208 AddSubgroup.zmultiples u } I ** set G := AddSubgroup.zmultiples u ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u hI : I =\u1da0[ae volume] ball x (T / (2 * \u2191(addOrderOf u))) G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u \u22a2 IsAddFundamentalDomain { x // x \u2208 G } I ** set n := addOrderOf u ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u hI : I =\u1da0[ae volume] ball x (T / (2 * \u2191n)) \u22a2 IsAddFundamentalDomain { x // x \u2208 G } I ** set B := ball x (T / (2 * n)) ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B \u22a2 IsAddFundamentalDomain { x // x \u2208 G } I ** have hn : 1 \u2264 (n : \u211d) := by norm_cast; linarith [addOrderOf_pos' hu] ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n \u22a2 IsAddFundamentalDomain { x // x \u2208 G } I ** refine' IsAddFundamentalDomain.mk_of_measure_univ_le _ _ _ _ ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B \u22a2 1 \u2264 \u2191n ** norm_cast ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B \u22a2 1 \u2264 n ** linarith [addOrderOf_pos' hu] ** case refine'_1 T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n \u22a2 NullMeasurableSet I ** exact measurableSet_ball.nullMeasurableSet.congr hI.symm ** case refine'_2 T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n \u22a2 \u2200 (g : { x // x \u2208 G }), g \u2260 0 \u2192 AEDisjoint volume (g +\u1d65 I) I ** rintro \u27e8g, hg\u27e9 hg' ** case refine'_2.mk T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : { val := g, property := hg } \u2260 0 \u22a2 AEDisjoint volume ({ val := g, property := hg } +\u1d65 I) I ** replace hg' : g \u2260 0 ** case refine'_2.mk T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 \u22a2 AEDisjoint volume ({ val := g, property := hg } +\u1d65 I) I ** change AEDisjoint volume (g +\u1d65 I) I ** case refine'_2.mk T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 \u22a2 AEDisjoint volume (g +\u1d65 I) I ** refine' AEDisjoint.congr (Disjoint.aedisjoint _)\n  ((quasiMeasurePreserving_add_left volume (-g)).vadd_ae_eq_of_ae_eq g hI) hI ** case refine'_2.mk T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 \u22a2 Disjoint (g +\u1d65 B) B ** have hBg : g +\u1d65 B = ball (g + x) (T / (2 * n)) := by\n  rw [add_comm g x, \u2190 singleton_add_ball _ x g, add_ball, thickening_singleton] ** case refine'_2.mk T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 hBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n)) \u22a2 Disjoint (g +\u1d65 B) B ** rw [hBg] ** case refine'_2.mk T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 hBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n)) \u22a2 Disjoint (ball (g + x) (T / (2 * \u2191n))) B ** apply ball_disjoint_ball ** case refine'_2.mk.h T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 hBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n)) \u22a2 T / (2 * \u2191n) + T / (2 * \u2191n) \u2264 dist (g + x) x ** rw [dist_eq_norm, add_sub_cancel, div_mul_eq_div_div, \u2190 add_div, \u2190 add_div, add_self_div_two,\n  div_le_iff' (by positivity : 0 < (n : \u211d)), \u2190 nsmul_eq_mul] ** case refine'_2.mk.h T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 hBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n)) \u22a2 T \u2264 n \u2022 \u2016g\u2016 ** refine' (le_add_order_smul_norm_of_isOfFinAddOrder (hu.of_mem_zmultiples hg) hg').trans\n  (nsmul_le_nsmul (norm_nonneg g) _) ** case refine'_2.mk.h T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 hBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n)) \u22a2 addOrderOf g \u2264 n ** exact Nat.le_of_dvd (addOrderOf_pos_iff.mpr hu) (addOrderOf_dvd_of_mem_zmultiples hg) ** case hg' T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : { val := g, property := hg } \u2260 0 \u22a2 g \u2260 0 ** simpa only [Ne.def, AddSubgroup.mk_eq_zero_iff] using hg' ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 \u22a2 g +\u1d65 B = ball (g + x) (T / (2 * \u2191n)) ** rw [add_comm g x, \u2190 singleton_add_ball _ x g, add_ball, thickening_singleton] ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n g : AddCircle T hg : g \u2208 G hg' : g \u2260 0 hBg : g +\u1d65 B = ball (g + x) (T / (2 * \u2191n)) \u22a2 0 < \u2191n ** positivity ** case refine'_3 T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n \u22a2 \u2200 (g : { x // x \u2208 G }), QuasiMeasurePreserving ((fun x x_1 => x +\u1d65 x_1) g) ** exact fun g => quasiMeasurePreserving_add_left (G := AddCircle T) volume g ** case refine'_4 T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hI : I =\u1da0[ae volume] B hn : 1 \u2264 \u2191n \u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I) ** replace hI := hI.trans closedBall_ae_eq_ball.symm ** case refine'_4 T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hn : 1 \u2264 \u2191n hI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n)) \u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I) ** haveI : Fintype G := @Fintype.ofFinite _ hu.finite_zmultiples ** case refine'_4 T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hn : 1 \u2264 \u2191n hI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n)) this : Fintype { x // x \u2208 G } \u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I) ** have hG_card : (Finset.univ : Finset G).card = n := by\n  show _ = addOrderOf u\n  rw [add_order_eq_card_zmultiples', Nat.card_eq_fintype_card]; rfl ** case refine'_4 T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hn : 1 \u2264 \u2191n hI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n)) this : Fintype { x // x \u2208 G } hG_card : Finset.card Finset.univ = n \u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 G }), \u2191\u2191volume (g +\u1d65 I) ** simp_rw [measure_vadd] ** case refine'_4 T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hn : 1 \u2264 \u2191n hI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n)) this : Fintype { x // x \u2208 G } hG_card : Finset.card Finset.univ = n \u22a2 \u2191\u2191volume univ \u2264 \u2211' (g : { x // x \u2208 AddSubgroup.zmultiples u }), \u2191\u2191volume I ** rw [AddCircle.measure_univ, tsum_fintype, Finset.sum_const, measure_congr hI,\n  volume_closedBall, \u2190 ENNReal.ofReal_nsmul, mul_div, mul_div_mul_comm,\n  div_self, one_mul, min_eq_right (div_le_self hT.out.le hn), hG_card,\n  nsmul_eq_mul, mul_div_cancel' T (lt_of_lt_of_le zero_lt_one hn).ne.symm] ** case refine'_4 T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hn : 1 \u2264 \u2191n hI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n)) this : Fintype { x // x \u2208 G } hG_card : Finset.card Finset.univ = n \u22a2 2 \u2260 0 ** exact two_ne_zero ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hn : 1 \u2264 \u2191n hI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n)) this : Fintype { x // x \u2208 G } \u22a2 Finset.card Finset.univ = n ** show _ = addOrderOf u ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hn : 1 \u2264 \u2191n hI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n)) this : Fintype { x // x \u2208 G } \u22a2 Finset.card Finset.univ = addOrderOf u ** rw [add_order_eq_card_zmultiples', Nat.card_eq_fintype_card] ** T : \u211d hT : Fact (0 < T) I : Set (AddCircle T) u x : AddCircle T hu : IsOfFinAddOrder u G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u n : \u2115 := addOrderOf u B : Set (AddCircle T) := ball x (T / (2 * \u2191n)) hn : 1 \u2264 \u2191n hI : I =\u1da0[ae volume] closedBall x (T / (2 * \u2191n)) this : Fintype { x // x \u2208 G } \u22a2 Finset.card Finset.univ = Fintype.card { x // x \u2208 AddSubgroup.zmultiples u } ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.pderiv_monomial ** R : Type u \u03c3 : Type v a a' a\u2081 a\u2082 : R s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R i : \u03c3 \u22a2 \u2191(pderiv i) (\u2191(monomial s) a) = \u2191(monomial (s - fun\u2080 | i => 1)) (a * \u2191(\u2191s i)) ** simp only [pderiv_def, mkDerivation_monomial, Finsupp.smul_sum, smul_eq_mul, \u2190 smul_mul_assoc,\n  \u2190 (monomial _).map_smul] ** R : Type u \u03c3 : Type v a a' a\u2081 a\u2082 : R s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R i : \u03c3 \u22a2 (sum s fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) =     \u2191(monomial (s - fun\u2080 | i => 1)) (a * \u2191(\u2191s i)) ** refine' (Finset.sum_eq_single i (fun j _ hne => _) fun hi => _).trans _ ** case refine'_1 R : Type u \u03c3 : Type v a a' a\u2081 a\u2082 : R s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R i j : \u03c3 x\u271d : j \u2208 s.support hne : j \u2260 i \u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) j (\u2191s j) = 0 ** simp [Pi.single_eq_of_ne hne] ** case refine'_2 R : Type u \u03c3 : Type v a a' a\u2081 a\u2082 : R s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R i : \u03c3 hi : \u00aci \u2208 s.support \u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) i (\u2191s i) = 0 ** rw [Finsupp.not_mem_support_iff] at hi ** case refine'_2 R : Type u \u03c3 : Type v a a' a\u2081 a\u2082 : R s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R i : \u03c3 hi : \u2191s i = 0 \u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) i (\u2191s i) = 0 ** simp [hi] ** case refine'_3 R : Type u \u03c3 : Type v a a' a\u2081 a\u2082 : R s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R i : \u03c3 \u22a2 (fun a_1 b => \u2191(monomial (s - fun\u2080 | a_1 => 1)) (a * \u2191b) * Pi.single i 1 a_1) i (\u2191s i) =     \u2191(monomial (s - fun\u2080 | i => 1)) (a * \u2191(\u2191s i)) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.coe_le_one ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 \u22a2 \u2191p a \u2264 1 ** refine' hasSum_le (fun b => _) (hasSum_ite_eq a (p a)) (hasSum_coe_one p) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a b : \u03b1 \u22a2 (if b = a then \u2191p a else 0) \u2264 \u2191p b ** split_ifs with h <;> simp only [h, zero_le', le_rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.bind_eq_bindTR ** \u22a2 @List.bind = @bindTR ** funext \u03b1 \u03b2 as f ** case h.h.h.h \u03b1 : Type u_2 \u03b2 : Type u_1 as : List \u03b1 f : \u03b1 \u2192 List \u03b2 \u22a2 List.bind as f = bindTR as f ** exact (go as #[]).symm ** \u03b1 : Type u_2 \u03b2 : Type u_1 as : List \u03b1 f : \u03b1 \u2192 List \u03b2 acc : Array \u03b2 \u22a2 bindTR.go f [] acc = acc.data ++ List.bind [] f ** simp [bindTR.go, bind] ** \u03b1 : Type u_2 \u03b2 : Type u_1 as : List \u03b1 f : \u03b1 \u2192 List \u03b2 x : \u03b1 xs : List \u03b1 acc : Array \u03b2 \u22a2 bindTR.go f (x :: xs) acc = acc.data ++ List.bind (x :: xs) f ** simp [bindTR.go, bind, go xs] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.lookup_insert ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b : \u03b2 a s\u271d : Finmap \u03b2 s : AList \u03b2 \u22a2 lookup a (insert a b \u27e6s\u27e7) = some b ** simp only [insert_toFinmap, lookup_toFinmap, AList.lookup_insert] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.IsMetric.finset_iUnion_of_pairwise_separated ** \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 I : Finset \u03b9 s : \u03b9 \u2192 Set X hI : \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j) \u22a2 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i) ** induction' I using Finset.induction_on with i I hiI ihI hI ** case insert \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 I\u271d : Finset \u03b9 s : \u03b9 \u2192 Set X hI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j) i : \u03b9 I : Finset \u03b9 hiI : \u00aci \u2208 I ihI :   (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i) hI : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 \u2200 (j : \u03b9), j \u2208 insert i I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j) \u22a2 \u2191\u03bc (\u22c3 i_1 \u2208 insert i I, s i_1) = \u2211 i in insert i I, \u2191\u03bc (s i) ** simp only [Finset.mem_insert] at hI ** case insert \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 I\u271d : Finset \u03b9 s : \u03b9 \u2192 Set X hI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j) i : \u03b9 I : Finset \u03b9 hiI : \u00aci \u2208 I ihI :   (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i) hI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j) \u22a2 \u2191\u03bc (\u22c3 i_1 \u2208 insert i I, s i_1) = \u2211 i in insert i I, \u2191\u03bc (s i) ** rw [Finset.set_biUnion_insert, hm, ihI, Finset.sum_insert hiI] ** case insert \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 I\u271d : Finset \u03b9 s : \u03b9 \u2192 Set X hI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j) i : \u03b9 I : Finset \u03b9 hiI : \u00aci \u2208 I ihI :   (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i) hI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j) \u22a2 \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)  case insert.a \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 I\u271d : Finset \u03b9 s : \u03b9 \u2192 Set X hI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j) i : \u03b9 I : Finset \u03b9 hiI : \u00aci \u2208 I ihI :   (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192 \u2191\u03bc (\u22c3 i \u2208 I, s i) = \u2211 i in I, \u2191\u03bc (s i) hI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j) \u22a2 IsMetricSeparated (s i) (\u22c3 x \u2208 I, s x) ** exacts [fun i hi j hj hij => hI i (Or.inr hi) j (Or.inr hj) hij,\n  IsMetricSeparated.finset_iUnion_right fun j hj =>\n    hI i (Or.inl rfl) j (Or.inr hj) (ne_of_mem_of_not_mem hj hiI).symm] ** case empty \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 I : Finset \u03b9 s : \u03b9 \u2192 Set X hI\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j) hI : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (j : \u03b9), j \u2208 \u2205 \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j) \u22a2 \u2191\u03bc (\u22c3 i \u2208 \u2205, s i) = \u2211 i in \u2205, \u2191\u03bc (s i) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "generateFrom_generatePiSystem_eq ** \u03b1 : Type u_1 g : Set (Set \u03b1) \u22a2 generateFrom (generatePiSystem g) = generateFrom g ** apply le_antisymm <;> apply generateFrom_le ** case a.h \u03b1 : Type u_1 g : Set (Set \u03b1) \u22a2 \u2200 (t : Set \u03b1), t \u2208 generatePiSystem g \u2192 MeasurableSet t ** exact fun t h_t => generateFrom_measurableSet_of_generatePiSystem t h_t ** case a.h \u03b1 : Type u_1 g : Set (Set \u03b1) \u22a2 \u2200 (t : Set \u03b1), t \u2208 g \u2192 MeasurableSet t ** exact fun t h_t => measurableSet_generateFrom (generatePiSystem.base h_t) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.iUnion_smul_eq_setOf_exists ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b9 : Group \u03b1 inst\u271d : MulAction \u03b1 \u03b2 s\u271d t A B : Set \u03b2 a : \u03b1 x : \u03b2 s : Set \u03b2 \u22a2 \u22c3 g, g \u2022 s = {a | \u2203 g, g \u2022 a \u2208 s} ** simp_rw [\u2190 iUnion_setOf, \u2190 iUnion_inv_smul, \u2190 preimage_smul, preimage] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.neg_mkRat ** n : Int d : Nat \u22a2 -mkRat n d = mkRat (-n) d ** if z : d = 0 then simp [z] else simp [\u2190 normalize_eq_mkRat z, neg_normalize] ** n : Int d : Nat z : d = 0 \u22a2 -mkRat n d = mkRat (-n) d ** simp [z] ** n : Int d : Nat z : \u00acd = 0 \u22a2 -mkRat n d = mkRat (-n) d ** simp [\u2190 normalize_eq_mkRat z, neg_normalize] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.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' \u22a2 \u2200 {f : \u03b2 \u00d7 \u03b3 \u2192 E}, Integrable f \u2192 \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** apply Integrable.induction ** case h_ind \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 (c : E) \u2983s : Set (\u03b2 \u00d7 \u03b3)\u2984,     MeasurableSet s \u2192       \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u2192         \u222b (z : \u03b2 \u00d7 \u03b3), indicator s (fun x => c) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a =           \u222b (x : \u03b2), \u222b (y : \u03b3), indicator s (fun x => c) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** intro c s hs h2s ** case h_ind \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), indicator s (fun x => c) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a =     \u222b (x : \u03b2), \u222b (y : \u03b3), indicator s (fun x => c) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** simp_rw [integral_indicator hs, \u2190 indicator_comp_right, Function.comp,\n  integral_indicator (measurable_prod_mk_left hs), MeasureTheory.set_integral_const,\n  integral_smul_const] ** case h_ind \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) \u2022 c = (\u222b (x : \u03b2), ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)) \u2202\u2191\u03ba a) \u2022 c ** congr 1 ** case h_ind.e_a \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) = \u222b (x : \u03b2), ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)) \u2202\u2191\u03ba a ** rw [integral_toReal] ** case h_ind.e_a \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) = ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s) \u2202\u2191\u03ba a)  case h_ind.e_a.hfm \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 AEMeasurable fun x => \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s)  case h_ind.e_a.hf \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s) < \u22a4 ** rotate_left ** case h_ind.e_a \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s) = ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s) \u2202\u2191\u03ba a) ** rw [kernel.compProd_apply _ _ _ hs] ** case h_ind.e_a \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 ENNReal.toReal (\u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a) =     ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s) \u2202\u2191\u03ba a) ** rfl ** case h_ind.e_a.hfm \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 AEMeasurable fun x => \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s) ** exact (kernel.measurable_kernel_prod_mk_left' hs _).aemeasurable ** case h_ind.e_a.hf \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' c : E s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, \u2191\u2191(\u2191\u03b7 (a, x)) (Prod.mk x \u207b\u00b9' s) < \u22a4 ** exact ae_kernel_lt_top a h2s.ne ** case h_add \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 \u2983f g : \u03b2 \u00d7 \u03b3 \u2192 E\u2984,     Disjoint (support f) (support g) \u2192       Integrable f \u2192         Integrable g \u2192           \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u2192             \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u2192               \u222b (z : \u03b2 \u00d7 \u03b3), (f + g) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), (f + g) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** intro f g _ i_f i_g hf hg ** case h_add \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' f g : \u03b2 \u00d7 \u03b3 \u2192 E a\u271d : Disjoint (support f) (support g) i_f : Integrable f i_g : Integrable g hf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a hg : \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), (f + g) z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), (f + g) (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** simp_rw [integral_add' i_f i_g, kernel.integral_integral_add' i_f i_g, hf, hg] ** case h_closed \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 IsClosed {f | \u222b (z : \u03b2 \u00d7 \u03b3), \u2191\u2191f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a} ** exact isClosed_eq continuous_integral kernel.continuous_integral_integral ** case h_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\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 \u2983f g : \u03b2 \u00d7 \u03b3 \u2192 E\u2984,     f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g \u2192       Integrable f \u2192         \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u2192           \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** intro f g hfg _ hf ** case h_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\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' f g : \u03b2 \u00d7 \u03b3 \u2192 E hfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g a\u271d : Integrable f hf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** convert hf using 1 ** case h.e'_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' f g : \u03b2 \u00d7 \u03b3 \u2192 E hfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g a\u271d : Integrable f hf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u22a2 \u222b (z : \u03b2 \u00d7 \u03b3), g z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a ** exact integral_congr_ae hfg.symm ** case h.e'_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' f g : \u03b2 \u00d7 \u03b3 \u2192 E hfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g a\u271d : Integrable f hf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u22a2 \u222b (x : \u03b2), \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** refine' integral_congr_ae _ ** case h.e'_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' f g : \u03b2 \u00d7 \u03b3 \u2192 E hfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g a\u271d : Integrable f hf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u22a2 (fun x => \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x)) =\u1d50[\u2191\u03ba a] fun x => \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) ** refine' (ae_ae_of_ae_compProd hfg).mp (eventually_of_forall _) ** case h.e'_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' f g : \u03b2 \u00d7 \u03b3 \u2192 E hfg : f =\u1d50[\u2191(\u03ba \u2297\u2096 \u03b7) a] g a\u271d : Integrable f hf : \u222b (z : \u03b2 \u00d7 \u03b3), f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b (x : \u03b2), \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a \u22a2 \u2200 (x : \u03b2),     (\u2200\u1d50 (c : \u03b3) \u2202\u2191\u03b7 (a, x), f (x, c) = g (x, c)) \u2192       (fun x => \u222b (y : \u03b3), g (x, y) \u2202\u2191\u03b7 (a, x)) x = (fun x => \u222b (y : \u03b3), f (x, y) \u2202\u2191\u03b7 (a, x)) x ** exact fun x hfgx => integral_congr_ae (ae_eq_symm hfgx) ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.neg_lt_sub_left_of_lt_add ** a b c : Int h : c < a + b \u22a2 -a < b - c ** have h := Int.lt_neg_add_of_add_lt (Int.sub_left_lt_of_lt_add h) ** a b c : Int h\u271d : c < a + b h : -a < -c + b \u22a2 -a < b - c ** rwa [Int.add_comm] at h ** Qed",
        "informal": ""
    },
    {
        "formal": "List.range_loop_range' ** s n : Nat \u22a2 range.loop (s + 1) (range' (s + 1) n) = range' 0 (n + (s + 1)) ** rw [\u2190 Nat.add_assoc, Nat.add_right_comm n s 1] ** s n : Nat \u22a2 range.loop (s + 1) (range' (s + 1) n) = range' 0 (n + 1 + s) ** exact range_loop_range' s (n + 1) ** Qed",
        "informal": ""
    },
    {
        "formal": "List.Sublist.erase ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List \u03b1 h : l\u2081 <+ l\u2082 \u22a2 List.erase l\u2081 a <+ List.erase l\u2082 a ** simp [erase_eq_eraseP] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List \u03b1 h : l\u2081 <+ l\u2082 \u22a2 List.eraseP (fun b => decide (a = b)) l\u2081 <+ List.eraseP (fun b => decide (a = b)) l\u2082 ** exact Sublist.eraseP h ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Balanced.le_size ** \u03b1 : Type u_1 t : RBNode \u03b1 c : RBColor n\u271d n : Nat x\u271d y\u271d : RBNode \u03b1 v\u271d : \u03b1 hl : Balanced x\u271d black n hr : Balanced y\u271d black n \u22a2 2 ^ depthLB red n \u2264 size (node red x\u271d v\u271d y\u271d) + 1 ** rw [size, Nat.add_right_comm (size _), Nat.add_assoc, depthLB, Nat.pow_succ, Nat.mul_two] ** \u03b1 : Type u_1 t : RBNode \u03b1 c : RBColor n\u271d n : Nat x\u271d y\u271d : RBNode \u03b1 v\u271d : \u03b1 hl : Balanced x\u271d black n hr : Balanced y\u271d black n \u22a2 2 ^ n + 2 ^ n \u2264 size x\u271d + 1 + (size y\u271d + 1) ** exact Nat.add_le_add hl.le_size hr.le_size ** \u03b1 : Type u_1 t : RBNode \u03b1 c : RBColor n : Nat 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 \u22a2 2 ^ depthLB black (n\u271d + 1) \u2264 size (node black x\u271d v\u271d y\u271d) + 1 ** rw [size, Nat.add_right_comm (size _), Nat.add_assoc, depthLB, Nat.pow_succ, Nat.mul_two] ** \u03b1 : Type u_1 t : RBNode \u03b1 c : RBColor n : Nat 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 \u22a2 2 ^ n\u271d + 2 ^ n\u271d \u2264 size x\u271d + 1 + (size y\u271d + 1) ** refine Nat.add_le_add (Nat.le_trans ?_ hl.le_size) (Nat.le_trans ?_ hr.le_size) <;>\n  exact Nat.pow_le_pow_of_le_right (by decide) (depthLB_le ..) ** \u03b1 : Type u_1 t : RBNode \u03b1 c : RBColor n : Nat 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 \u22a2 2 > 0 ** decide ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.aemeasurable_fderivWithin ** 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 AEMeasurable f' ** refine' aemeasurable_of_unif_approx fun \u03b5 \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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5 ** let \u03b4 : \u211d\u22650 := \u27e8\u03b5, le_of_lt \u03b5pos\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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5 ** have \u03b4pos : 0 < \u03b4 := \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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 \u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5 ** obtain \u27e8t, A, t_disj, t_meas, t_cover, ht, _\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) \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' (fun _ => \u03b4) fun _ =>\n    \u03b4pos.ne' ** case 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 : 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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5 ** obtain \u27e8g, g_meas, hg\u27e9 :\n    \u2203 g : E \u2192 E \u2192L[\u211d] E, Measurable g \u2227 \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n :=\n  exists_measurable_piecewise t t_meas (fun n _ => A n) (fun n => measurable_const) <|\n    t_disj.mono fun i j h => by simp only [h.inter_eq, eqOn_empty] ** case intro.intro.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 : 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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n \u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5 ** refine' \u27e8g, g_meas.aemeasurable, _\u27e9 ** case intro.intro.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 : 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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5 ** suffices H : \u2200\u1d50 x : E \u2202sum fun n => \u03bc.restrict (s \u2229 t n), dist (g x) (f' x) \u2264 \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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n \u22a2 \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 ** refine' ae_sum_iff.2 fun 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 ** have E\u2081 : \u2200\u1d50 x : E \u2202\u03bc.restrict (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 :=\n  (ht n).norm_fderiv_sub_le \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n    (hf' x hx.1).mono (inter_subset_left _ _) ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 ** have E\u2082 : \u2200\u1d50 x : E \u2202\u03bc.restrict (s \u2229 t n), g x = A n := by\n  suffices H : \u2200\u1d50 x : E \u2202\u03bc.restrict (t n), g x = A n\n  exact ae_mono (restrict_mono (inter_subset_right _ _) le_rfl) H\n  filter_upwards [ae_restrict_mem (t_meas n)]\n  exact hg 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 E\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 ** filter_upwards [E\u2081, E\u2082] with x hx1 hx2 ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 E\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n x : E hx1 : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 hx2 : g x = A n \u22a2 dist (g x) (f' x) \u2264 \u03b5 ** rw [\u2190 nndist_eq_nnnorm] at hx1 ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 E\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n x : E hx1 : nndist (f' x) (A n) \u2264 \u03b4 hx2 : g x = A n \u22a2 dist (g x) (f' x) \u2264 \u03b5 ** rw [hx2, dist_comm] ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 E\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n x : E hx1 : nndist (f' x) (A n) \u2264 \u03b4 hx2 : g x = A n \u22a2 dist (f' x) (A n) \u2264 \u03b5 ** exact hx1 ** 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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y i j : \u2115 h : (Disjoint on t) i j \u22a2 EqOn ((fun n x => A n) i) ((fun n x => A n) j) (t i \u2229 t j) ** simp only [h.inter_eq, eqOn_empty] ** case intro.intro.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 : 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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n H : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5 ** have : \u03bc.restrict s \u2264 sum fun n => \u03bc.restrict (s \u2229 t n) := by\n  have : s = \u22c3 n, s \u2229 t n := by\n    rw [\u2190 inter_iUnion]\n    exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n  conv_lhs => rw [this]\n  exact restrict_iUnion_le ** case intro.intro.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 : 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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n H : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 this : Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n) \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5 ** exact ae_mono this 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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n H : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 \u22a2 Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n) ** have : s = \u22c3 n, s \u2229 t n := by\n  rw [\u2190 inter_iUnion]\n  exact Subset.antisymm (subset_inter Subset.rfl t_cover) (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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n H : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 this : s = \u22c3 n, s \u2229 t n \u22a2 Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n) ** conv_lhs => rw [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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n H : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 this : s = \u22c3 n, s \u2229 t n \u22a2 Measure.restrict \u03bc (\u22c3 n, s \u2229 t n) \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n) ** exact restrict_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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n H : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 \u22a2 s = \u22c3 n, s \u2229 t n ** 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n H : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5 \u22a2 s = s \u2229 \u22c3 i, t i ** exact Subset.antisymm (subset_inter Subset.rfl t_cover) (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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n ** suffices H : \u2200\u1d50 x : E \u2202\u03bc.restrict (t n), g x = A 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 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 H : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n ** exact ae_mono (restrict_mono (inter_subset_right _ _) le_rfl) H ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n ** filter_upwards [ae_restrict_mem (t_meas 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u03b4pos : 0 < \u03b4 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 right\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y g : E \u2192 E \u2192L[\u211d] E g_meas : Measurable g hg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n n : \u2115 E\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 \u22a2 \u2200 (a : E), a \u2208 t n \u2192 g a = A n ** exact hg n ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.completeSpace_lp_of_cauchy_complete_\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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) \u22a2 CompleteSpace { x // x \u2208 Lp E p } ** let B := fun n : \u2115 => ((1 : \u211d) / 2) ^ n ** \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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n \u22a2 CompleteSpace { x // x \u2208 Lp E p } ** have hB_pos : \u2200 n, 0 < B n := fun n => pow_pos (div_pos zero_lt_one zero_lt_two) n ** \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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n \u22a2 CompleteSpace { x // x \u2208 Lp E p } ** refine' Metric.complete_of_convergent_controlled_sequences B hB_pos fun f hf => _ ** \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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N \u22a2 \u2203 x, Tendsto f atTop (\ud835\udcdd x) ** rsuffices \u27e8f_lim, hf_lim_meas, h_tendsto\u27e9 :\n  \u2203 (f_lim : \u03b1 \u2192 E), Mem\u2112p f_lim p \u03bc \u2227\n    atTop.Tendsto (fun n => snorm (\u21d1(f n) - f_lim) p \u03bc) (\ud835\udcdd 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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N \u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** obtain \u27e8M, hB\u27e9 : Summable B := summable_geometric_two ** 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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M \u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** let B1 n := ENNReal.ofReal (B n) ** 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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) \u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** have hB1_has : HasSum B1 (ENNReal.ofReal M) := by\n  have h_tsum_B1 : \u2211' i, B1 i = ENNReal.ofReal M := by\n    change (\u2211' n : \u2115, ENNReal.ofReal (B n)) = ENNReal.ofReal M\n    rw [\u2190 hB.tsum_eq]\n    exact (ENNReal.ofReal_tsum_of_nonneg (fun n => le_of_lt (hB_pos n)) hB.summable).symm\n  have h_sum := (@ENNReal.summable _ B1).hasSum\n  rwa [h_tsum_B1] at h_sum ** 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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) \u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** have hB1 : \u2211' i, B1 i < \u221e := by\n  rw [hB1_has.tsum_eq]\n  exact ENNReal.ofReal_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 : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) hB1 : \u2211' (i : \u2115), B1 i < \u22a4 \u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** let f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => f n ** 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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) hB1 : \u2211' (i : \u2115), B1 i < \u22a4 f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n) \u22a2 \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0) ** refine' H f1 (fun n => Lp.mem\u2112p (f n)) B1 hB1 fun N n m hn hm => _ ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) hB1 : \u2211' (i : \u2115), B1 i < \u22a4 f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n) N n m : \u2115 hn : N \u2264 n hm : N \u2264 m \u22a2 snorm (f1 n - f1 m) p \u03bc < B1 N ** specialize hf N n m hn hm ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) hB1 : \u2211' (i : \u2115), B1 i < \u22a4 f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n) N n m : \u2115 hn : N \u2264 n hm : N \u2264 m hf : dist (f n) (f m) < B N \u22a2 snorm (f1 n - f1 m) p \u03bc < B1 N ** rw [dist_def] at hf ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) hB1 : \u2211' (i : \u2115), B1 i < \u22a4 f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n) N n m : \u2115 hn : N \u2264 n hm : N \u2264 m hf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N \u22a2 snorm (f1 n - f1 m) p \u03bc < B1 N ** dsimp only ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) hB1 : \u2211' (i : \u2115), B1 i < \u22a4 f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n) N n m : \u2115 hn : N \u2264 n hm : N \u2264 m hf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N \u22a2 snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc < ENNReal.ofReal ((1 / 2) ^ N) ** rwa [ENNReal.lt_ofReal_iff_toReal_lt] ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) hB1 : \u2211' (i : \u2115), B1 i < \u22a4 f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n) N n m : \u2115 hn : N \u2264 n hm : N \u2264 m hf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N \u22a2 snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc \u2260 \u22a4 ** rw [snorm_congr_ae (Lp.coeFn_sub _ _).symm] ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) hB1 : \u2211' (i : \u2115), B1 i < \u22a4 f1 : \u2115 \u2192 \u03b1 \u2192 E := fun n => \u2191\u2191(f n) N n m : \u2115 hn : N \u2264 n hm : N \u2264 m hf : ENNReal.toReal (snorm (\u2191\u2191(f n) - \u2191\u2191(f m)) p \u03bc) < B N \u22a2 snorm (\u2191\u2191(f n - f m)) p \u03bc \u2260 \u22a4 ** exact Lp.snorm_ne_top _ ** case intro.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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N f_lim : \u03b1 \u2192 E hf_lim_meas : Mem\u2112p f_lim p h_tendsto : Tendsto (fun n => snorm (\u2191\u2191(f n) - f_lim) p \u03bc) atTop (\ud835\udcdd 0) \u22a2 \u2203 x, Tendsto f atTop (\ud835\udcdd x) ** exact \u27e8hf_lim_meas.toLp f_lim, tendsto_Lp_of_tendsto_\u2112p f_lim hf_lim_meas h_tendsto\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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) \u22a2 HasSum B1 (ENNReal.ofReal M) ** have h_tsum_B1 : \u2211' i, B1 i = ENNReal.ofReal M := by\n  change (\u2211' n : \u2115, ENNReal.ofReal (B n)) = ENNReal.ofReal M\n  rw [\u2190 hB.tsum_eq]\n  exact (ENNReal.ofReal_tsum_of_nonneg (fun n => le_of_lt (hB_pos n)) hB.summable).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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) h_tsum_B1 : \u2211' (i : \u2115), B1 i = ENNReal.ofReal M \u22a2 HasSum B1 (ENNReal.ofReal M) ** have h_sum := (@ENNReal.summable _ B1).hasSum ** \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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) h_tsum_B1 : \u2211' (i : \u2115), B1 i = ENNReal.ofReal M h_sum : HasSum B1 (\u2211' (b : \u2115), B1 b) \u22a2 HasSum B1 (ENNReal.ofReal M) ** rwa [h_tsum_B1] at h_sum ** \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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) \u22a2 \u2211' (i : \u2115), B1 i = ENNReal.ofReal M ** change (\u2211' n : \u2115, ENNReal.ofReal (B n)) = ENNReal.ofReal M ** \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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) \u22a2 \u2211' (n : \u2115), ENNReal.ofReal (B n) = ENNReal.ofReal M ** rw [\u2190 hB.tsum_eq] ** \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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) \u22a2 \u2211' (n : \u2115), ENNReal.ofReal (B n) = ENNReal.ofReal (\u2211' (b : \u2115), B b) ** exact (ENNReal.ofReal_tsum_of_nonneg (fun n => le_of_lt (hB_pos n)) hB.summable).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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) \u22a2 \u2211' (i : \u2115), B1 i < \u22a4 ** rw [hB1_has.tsum_eq] ** \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 hp : Fact (1 \u2264 p) H :   \u2200 (f : \u2115 \u2192 \u03b1 \u2192 E),     (\u2200 (n : \u2115), Mem\u2112p (f n) p) \u2192       \u2200 (B : \u2115 \u2192 \u211d\u22650\u221e),         \u2211' (i : \u2115), B i < \u22a4 \u2192           (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) p \u03bc < B N) \u2192             \u2203 f_lim, Mem\u2112p f_lim p \u2227 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) atTop (\ud835\udcdd 0) B : \u2115 \u2192 \u211d := fun n => (1 / 2) ^ n hB_pos : \u2200 (n : \u2115), 0 < B n f : \u2115 \u2192 { x // x \u2208 Lp E p } hf : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) < B N M : \u211d hB : HasSum B M B1 : \u2115 \u2192 \u211d\u22650\u221e := fun n => ENNReal.ofReal (B n) hB1_has : HasSum B1 (ENNReal.ofReal M) \u22a2 ENNReal.ofReal M < \u22a4 ** exact ENNReal.ofReal_lt_top ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.Buckets.update_update ** \u03b1 : Type u_1 \u03b2 : Type u_2 self : Buckets \u03b1 \u03b2 i : USize d d' : AssocList \u03b1 \u03b2 h : USize.toNat i < Array.size self.val h' : USize.toNat i < Array.size (update self i d h).val \u22a2 update (update self i d h) i d' h' = update self i d' h ** simp [update] ** \u03b1 : Type u_1 \u03b2 : Type u_2 self : Buckets \u03b1 \u03b2 i : USize d d' : AssocList \u03b1 \u03b2 h : USize.toNat i < Array.size self.val h' : USize.toNat i < Array.size (update self i d h).val \u22a2 { val := Array.set (Array.set self.val { val := USize.toNat i, isLt := h } d) { val := USize.toNat i, isLt := h' } d',       property :=         (_ :           (fun b => 0 < Array.size b)             (Array.set (Array.set self.val { val := USize.toNat i, isLt := h } d) { val := USize.toNat i, isLt := h' }               d')) } =     { val := Array.set self.val { val := USize.toNat i, isLt := h } d',       property := (_ : (fun b => 0 < Array.size b) (Array.set self.val { val := USize.toNat i, isLt := h } d')) } ** congr 1 ** case e_val \u03b1 : Type u_1 \u03b2 : Type u_2 self : Buckets \u03b1 \u03b2 i : USize d d' : AssocList \u03b1 \u03b2 h : USize.toNat i < Array.size self.val h' : USize.toNat i < Array.size (update self i d h).val \u22a2 Array.set (Array.set self.val { val := USize.toNat i, isLt := h } d) { val := USize.toNat i, isLt := h' } d' =     Array.set self.val { val := USize.toNat i, isLt := h } d' ** rw [Array.set_set] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.MeasurablySeparable.iUnion ** \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 h : \u2200 (m n : \u03b9), MeasurablySeparable (s m) (t n) \u22a2 MeasurablySeparable (\u22c3 n, s n) (\u22c3 m, t m) ** choose u hsu htu hu using h ** \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 u : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1 hsu : \u2200 (m n : \u03b9), s m \u2286 u m n htu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n) hu : \u2200 (m n : \u03b9), MeasurableSet (u m n) \u22a2 MeasurablySeparable (\u22c3 n, s n) (\u22c3 m, t m) ** refine' \u27e8\u22c3 m, \u22c2 n, u m n, _, _, _\u27e9 ** case refine'_1 \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 u : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1 hsu : \u2200 (m n : \u03b9), s m \u2286 u m n htu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n) hu : \u2200 (m n : \u03b9), MeasurableSet (u m n) \u22a2 \u22c3 n, s n \u2286 \u22c3 m, \u22c2 n, u m n ** refine' iUnion_subset fun m => subset_iUnion_of_subset m _ ** case refine'_1 \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 u : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1 hsu : \u2200 (m n : \u03b9), s m \u2286 u m n htu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n) hu : \u2200 (m n : \u03b9), MeasurableSet (u m n) m : \u03b9 \u22a2 s m \u2286 \u22c2 n, u m n ** exact subset_iInter fun n => hsu m n ** case refine'_2 \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 u : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1 hsu : \u2200 (m n : \u03b9), s m \u2286 u m n htu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n) hu : \u2200 (m n : \u03b9), MeasurableSet (u m n) \u22a2 Disjoint (\u22c3 m, t m) (\u22c3 m, \u22c2 n, u m n) ** simp_rw [disjoint_iUnion_left, disjoint_iUnion_right] ** case refine'_2 \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 u : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1 hsu : \u2200 (m n : \u03b9), s m \u2286 u m n htu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n) hu : \u2200 (m n : \u03b9), MeasurableSet (u m n) \u22a2 \u2200 (i i_1 : \u03b9), Disjoint (t i) (\u22c2 n, u i_1 n) ** intro n m ** case refine'_2 \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 u : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1 hsu : \u2200 (m n : \u03b9), s m \u2286 u m n htu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n) hu : \u2200 (m n : \u03b9), MeasurableSet (u m n) n m : \u03b9 \u22a2 Disjoint (t n) (\u22c2 n, u m n) ** apply Disjoint.mono_right _ (htu m n) ** \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 u : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1 hsu : \u2200 (m n : \u03b9), s m \u2286 u m n htu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n) hu : \u2200 (m n : \u03b9), MeasurableSet (u m n) n m : \u03b9 \u22a2 \u22c2 n, u m n \u2264 u m n ** apply iInter_subset ** case refine'_3 \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 u : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1 hsu : \u2200 (m n : \u03b9), s m \u2286 u m n htu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n) hu : \u2200 (m n : \u03b9), MeasurableSet (u m n) \u22a2 MeasurableSet (\u22c3 m, \u22c2 n, u m n) ** refine' MeasurableSet.iUnion fun m => _ ** case refine'_3 \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_3 inst\u271d : MeasurableSpace \u03b1 s t : \u03b9 \u2192 Set \u03b1 u : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1 hsu : \u2200 (m n : \u03b9), s m \u2286 u m n htu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n) hu : \u2200 (m n : \u03b9), MeasurableSet (u m n) m : \u03b9 \u22a2 MeasurableSet (\u22c2 n, u m n) ** exact MeasurableSet.iInter fun n => hu m n ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.pi_caratheodory ** \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) \u22a2 MeasurableSpace.pi \u2264 OuterMeasure.caratheodory (OuterMeasure.pi fun i => \u2191(\u03bc i)) ** refine' iSup_le _ ** \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) \u22a2 \u2200 (i : \u03b9),     MeasurableSpace.comap (fun b => b i) ((fun a => inst\u271d a) i) \u2264       OuterMeasure.caratheodory (OuterMeasure.pi fun i => \u2191(\u03bc i)) ** 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) i : \u03b9 s : Set ((a : \u03b9) \u2192 \u03b1 a) hs : MeasurableSet s \u22a2 MeasurableSet s ** rcases hs with \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s \u22a2 MeasurableSet ((fun b => b i) \u207b\u00b9' s) ** apply boundedBy_caratheodory ** case intro.intro.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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s \u22a2 \u2200 (t : Set ((a : \u03b9) \u2192 \u03b1 a)),     piPremeasure (fun i => \u2191(\u03bc i)) (t \u2229 (fun b => b i) \u207b\u00b9' s) +         piPremeasure (fun i => \u2191(\u03bc i)) (t \\ (fun b => b i) \u207b\u00b9' s) \u2264       piPremeasure (fun i => \u2191(\u03bc i)) t ** intro t ** case intro.intro.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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) \u22a2 piPremeasure (fun i => \u2191(\u03bc i)) (t \u2229 (fun b => b i) \u207b\u00b9' s) +       piPremeasure (fun i => \u2191(\u03bc i)) (t \\ (fun b => b i) \u207b\u00b9' s) \u2264     piPremeasure (fun i => \u2191(\u03bc i)) t ** simp_rw [piPremeasure] ** case intro.intro.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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) \u22a2 \u220f x : \u03b9, \u2191\u2191(\u03bc x) (eval x '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) + \u220f x : \u03b9, \u2191\u2191(\u03bc x) (eval x '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264     \u220f x : \u03b9, \u2191\u2191(\u03bc x) (eval x '' t) ** refine' Finset.prod_add_prod_le' (Finset.mem_univ i) _ _ _ ** case intro.intro.hs.refine'_1 \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) \u22a2 \u2191\u2191(\u03bc i) (eval i '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) + \u2191\u2191(\u03bc i) (eval i '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264     \u2191\u2191(\u03bc i) (eval i '' t) ** simp [image_inter_preimage, image_diff_preimage, measure_inter_add_diff _ hs, le_refl] ** case intro.intro.hs.refine'_2 \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) \u22a2 \u2200 (j : \u03b9), j \u2208 Finset.univ \u2192 j \u2260 i \u2192 \u2191\u2191(\u03bc j) (eval j '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t) ** rintro j - _ ** case intro.intro.hs.refine'_2 \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) j : \u03b9 a\u271d : j \u2260 i \u22a2 \u2191\u2191(\u03bc j) (eval j '' (t \u2229 (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t) ** apply mono' ** case intro.intro.hs.refine'_2.h \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) j : \u03b9 a\u271d : j \u2260 i \u22a2 eval j '' (t \u2229 (fun b => b i) \u207b\u00b9' s) \u2286 eval j '' t ** apply image_subset ** case intro.intro.hs.refine'_2.h.h \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) j : \u03b9 a\u271d : j \u2260 i \u22a2 t \u2229 (fun b => b i) \u207b\u00b9' s \u2286 t ** apply inter_subset_left ** case intro.intro.hs.refine'_3 \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) \u22a2 \u2200 (j : \u03b9), j \u2208 Finset.univ \u2192 j \u2260 i \u2192 \u2191\u2191(\u03bc j) (eval j '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t) ** rintro j - _ ** case intro.intro.hs.refine'_3 \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) j : \u03b9 a\u271d : j \u2260 i \u22a2 \u2191\u2191(\u03bc j) (eval j '' (t \\ (fun b => b i) \u207b\u00b9' s)) \u2264 \u2191\u2191(\u03bc j) (eval j '' t) ** apply mono' ** case intro.intro.hs.refine'_3.h \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) j : \u03b9 a\u271d : j \u2260 i \u22a2 eval j '' (t \\ (fun b => b i) \u207b\u00b9' s) \u2286 eval j '' t ** apply image_subset ** case intro.intro.hs.refine'_3.h.h \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) i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : MeasurableSet s t : Set ((a : \u03b9) \u2192 \u03b1 a) j : \u03b9 a\u271d : j \u2260 i \u22a2 t \\ (fun b => b i) \u207b\u00b9' s \u2286 t ** apply diff_subset ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.exists_disjoint_closedBall_covering_ae ** \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R 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 \u2229 Ioo 0 (R x)) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x) ** let g x := f x \u2229 Ioo 0 (R x) ** \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R 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 \u2229 Ioo 0 (R x)) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x) ** have hg : \u2200 x \u2208 s, \u2200 \u03b4 > 0, (g x \u2229 Ioo 0 \u03b4).Nonempty := by\n  intro x hx \u03b4 \u03b4pos\n  rcases hf x hx (min \u03b4 (R x)) (lt_min \u03b4pos (hR x hx)) with \u27e8r, hr\u27e9\n  exact\n    \u27e8r,\n      \u27e8\u27e8hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)\u27e9,\n        \u27e8hr.2.1, hr.2.2.trans_le (min_le_left _ _)\u27e9\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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g 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 \u2229 Ioo 0 (R x)) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x) ** rcases exists_disjoint_closedBall_covering_ae_aux \u03bc g s hg with \u27e8v, v_count, vs, vg, \u03bcv, v_disj\u27e9 ** case intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.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 \u2229 Ioo 0 (R x)) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x) ** let t := Prod.fst '' v ** case intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' 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 \u2229 Ioo 0 (R x)) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x) ** have : \u2200 x \u2208 t, \u2203 r : \u211d, (x, r) \u2208 v := by\n  intro x hx\n  rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9\n  exact \u27e8q, hp\u27e9 ** case intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v this : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 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 \u2229 Ioo 0 (R x)) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x) ** choose! r hr using this ** case intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = 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 \u2229 Ioo 0 (R x)) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x) ** refine' \u27e8t, r, v_count.image _, _, _, _, _\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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) \u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) ** intro x hx \u03b4 \u03b4pos ** \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) x : \u03b1 hx : x \u2208 s \u03b4 : \u211d \u03b4pos : \u03b4 > 0 \u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) ** rcases hf x hx (min \u03b4 (R x)) (lt_min \u03b4pos (hR x hx)) with \u27e8r, hr\u27e9 ** case 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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) x : \u03b1 hx : x \u2208 s \u03b4 : \u211d \u03b4pos : \u03b4 > 0 r : \u211d hr : r \u2208 f x \u2229 Ioo 0 (min \u03b4 (R x)) \u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) ** exact\n  \u27e8r,\n    \u27e8\u27e8hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)\u27e9,\n      \u27e8hr.2.1, hr.2.2.trans_le (min_le_left _ _)\u27e9\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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v \u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 v ** 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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v x : \u03b1 hx : x \u2208 t \u22a2 \u2203 r, (x, r) \u2208 v ** rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9 ** case intro.mk.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v p : \u03b1 q : \u211d hp : (p, q) \u2208 v hx : (p, q).1 \u2208 t \u22a2 \u2203 r, ((p, q).1, r) \u2208 v ** exact \u27e8q, hp\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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v \u22a2 (fun x => (x, r x)) '' t = v ** have I : \u2200 p : \u03b1 \u00d7 \u211d, p \u2208 v \u2192 0 \u2264 p.2 := fun p hp => (vg p hp).2.1.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 \u22a2 (fun x => (x, r x)) '' t = v ** apply Subset.antisymm ** case h\u2081 \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 \u22a2 (fun x => (x, r x)) '' t \u2286 v ** simp only [image_subset_iff] ** case h\u2081 \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 \u22a2 v \u2286 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v) ** rintro \u27e8x, p\u27e9 hxp ** case h\u2081.mk \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v \u22a2 (x, p) \u2208 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v) ** simp only [mem_preimage] ** case h\u2081.mk \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v \u22a2 (x, r x) \u2208 v ** exact hr _ (mem_image_of_mem _ hxp) ** case h\u2082 \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 \u22a2 v \u2286 (fun x => (x, r x)) '' t ** rintro \u27e8x, p\u27e9 hxp ** case h\u2082.mk \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v \u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t ** have hxrx : (x, r x) \u2208 v := hr _ (mem_image_of_mem _ hxp) ** case h\u2082.mk \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v \u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t ** have : p = r x := by\n  by_contra h\n  have A : (x, p) \u2260 (x, r x) := by\n    simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h\n  have H := v_disj hxp hxrx A\n  contrapose H\n  rw [not_disjoint_iff_nonempty_inter]\n  refine' \u27e8x, by simp (config := { proj := false }) [I _ hxp, I _ hxrx]\u27e9 ** case h\u2082.mk \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v this : p = r x \u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t ** rw [this] ** case h\u2082.mk \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v this : p = r x \u22a2 (x, r x) \u2208 (fun x => (x, r x)) '' t ** apply mem_image_of_mem ** case h\u2082.mk.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v this : p = r x \u22a2 x \u2208 t ** exact mem_image_of_mem _ hxp ** \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v \u22a2 p = r x ** by_contra 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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v h : \u00acp = r x \u22a2 False ** have A : (x, p) \u2260 (x, r x) := by\n  simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using 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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v h : \u00acp = r x A : (x, p) \u2260 (x, r x) \u22a2 False ** have H := v_disj hxp hxrx A ** \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v h : \u00acp = r x A : (x, p) \u2260 (x, r x) H : (Disjoint on fun p => closedBall p.1 p.2) (x, p) (x, r x) \u22a2 False ** contrapose 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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v h : \u00acp = r x A : (x, p) \u2260 (x, r x) H : \u00acFalse \u22a2 \u00ac(Disjoint on fun p => closedBall p.1 p.2) (x, p) (x, r x) ** rw [not_disjoint_iff_nonempty_inter] ** \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v h : \u00acp = r x A : (x, p) \u2260 (x, r x) H : \u00acFalse \u22a2 Set.Nonempty ((fun p => closedBall p.1 p.2) (x, p) \u2229 (fun p => closedBall p.1 p.2) (x, r x)) ** refine' \u27e8x, by simp (config := { proj := false }) [I _ hxp, I _ hxrx]\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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v h : \u00acp = r x \u22a2 (x, p) \u2260 (x, r x) ** simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using 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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v I : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.2 x : \u03b1 p : \u211d hxp : (x, p) \u2208 v hxrx : (x, r x) \u2208 v h : \u00acp = r x A : (x, p) \u2260 (x, r x) H : \u00acFalse \u22a2 x \u2208 (fun p => closedBall p.1 p.2) (x, p) \u2229 (fun p => closedBall p.1 p.2) (x, r x) ** simp (config := { proj := false }) [I _ hxp, I _ hxrx] ** case intro.intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v \u22a2 t \u2286 s ** intro x hx ** case intro.intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v x : \u03b1 hx : x \u2208 t \u22a2 x \u2208 s ** rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9 ** case intro.intro.intro.intro.intro.refine'_1.intro.mk.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v p : \u03b1 q : \u211d hp : (p, q) \u2208 v hx : (p, q).1 \u2208 t \u22a2 (p, q).1 \u2208 s ** exact vs _ hp ** case intro.intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v \u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x) ** intro x hx ** case intro.intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v x : \u03b1 hx : x \u2208 t \u22a2 r x \u2208 f x \u2229 Ioo 0 (R x) ** rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, _, rfl\u27e9 ** case intro.intro.intro.intro.intro.refine'_2.intro.mk.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v p : \u03b1 q : \u211d left\u271d : (p, q) \u2208 v hx : (p, q).1 \u2208 t \u22a2 r (p, q).1 \u2208 f (p, q).1 \u2229 Ioo 0 (R (p, q).1) ** exact vg _ (hr _ hx) ** case intro.intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 ** have :\n  \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) =\n    \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.1 p.2 :=\n  by conv_rhs => rw [biUnion_image] ** case intro.intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v this : \u22c3 x \u2208 t, closedBall x (r x) = \u22c3 p \u2208 (fun x => (x, r x)) '' t, closedBall p.1 p.2 \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 ** rw [this, im_t] ** case intro.intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v this : \u22c3 x \u2208 t, closedBall x (r x) = \u22c3 p \u2208 (fun x => (x, r x)) '' t, closedBall p.1 p.2 \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 ** exact \u03bcv ** \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v \u22a2 \u22c3 x \u2208 t, closedBall x (r x) = \u22c3 p \u2208 (fun x => (x, r x)) '' t, closedBall p.1 p.2 ** conv_rhs => rw [biUnion_image] ** case intro.intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v \u22a2 PairwiseDisjoint t fun x => closedBall x (r x) ** have A : InjOn (fun x : \u03b1 => (x, r x)) t := by\n  simp (config := { contextual := true }) only [InjOn, Prod.mk.inj_iff, imp_true_iff,\n    eq_self_iff_true] ** case intro.intro.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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v A : InjOn (fun x => (x, r x)) t \u22a2 PairwiseDisjoint t fun x => closedBall x (r x) ** rwa [\u2190 im_t, A.pairwiseDisjoint_image] at v_disj ** \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 : SigmaFinite \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) R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x g : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x) hg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4) v : Set (\u03b1 \u00d7 \u211d) v_count : Set.Countable v vs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.1 \u2208 s vg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.2 \u2208 g p.1 \u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 v, closedBall p.1 p.2) = 0 v_disj : PairwiseDisjoint v fun p => closedBall p.1 p.2 t : Set \u03b1 := Prod.fst '' v r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v im_t : (fun x => (x, r x)) '' t = v \u22a2 InjOn (fun x => (x, r x)) t ** simp (config := { contextual := true }) only [InjOn, Prod.mk.inj_iff, imp_true_iff,\n  eq_self_iff_true] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.tr_respects_aux\u2081 ** 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 o : StAct k q : Stmt\u2082 v : \u03c3 S : List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) n : \u2115 H : n \u2264 List.length S \u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (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 k : K o : StAct k q : Stmt\u2082 v : \u03c3 S : List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) n\u271d : \u2115 H\u271d : n\u271d \u2264 List.length S n : \u2115 IH :   n \u2264 List.length S \u2192     Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }       { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } H : Nat.succ n \u2264 List.length S \u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) } ** apply (IH (le_of_lt H)).tail ** 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 k : K o : StAct k q : Stmt\u2082 v : \u03c3 S : List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) n\u271d : \u2115 H\u271d : n\u271d \u2264 List.length S n : \u2115 IH :   n \u2264 List.length S \u2192     Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }       { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } H : Nat.succ n \u2264 List.length S \u22a2 { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) } \u2208     TM1.step (tr M) { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } ** rw [iterate_succ_apply'] ** 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 k : K o : StAct k q : Stmt\u2082 v : \u03c3 S : List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) n\u271d : \u2115 H\u271d : n\u271d \u2264 List.length S n : \u2115 IH :   n \u2264 List.length S \u2192     Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }       { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } H : Nat.succ n \u2264 List.length S \u22a2 { l := some (go k o q), var := v,       Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) } \u2208     TM1.step (tr M) { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } ** simp only [TM1.step, TM1.stepAux, tr, Tape.mk'_nth_nat, Tape.move_right_n_head,\n  addBottom_nth_snd, Option.mem_def] ** 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 k : K o : StAct k q : Stmt\u2082 v : \u03c3 S : List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) n\u271d : \u2115 H\u271d : n\u271d \u2264 List.length S n : \u2115 IH :   n \u2264 List.length S \u2192     Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }       { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } H : Nat.succ n \u2264 List.length S \u22a2 some       (bif Option.isNone (ListBlank.nth L n k) then         TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))       else         { l := some (go k o q), var := v,           Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }) =     some       { l := some (go k o q), var := v,         Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) } ** rw [stk_nth_val _ hL, List.get?_eq_get] ** 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 k : K o : StAct k q : Stmt\u2082 v : \u03c3 S : List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) n\u271d : \u2115 H\u271d : n\u271d \u2264 List.length S n : \u2115 IH :   n \u2264 List.length S \u2192     Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }       { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } H : Nat.succ n \u2264 List.length S \u22a2 some       (bif Option.isNone (some (List.get (List.reverse S) { val := n, isLt := ?succ })) then         TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))       else         { l := some (go k o q), var := v,           Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }) =     some       { l := some (go k o q), var := v,         Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }  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 k : K o : StAct k q : Stmt\u2082 v : \u03c3 S : List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) n\u271d : \u2115 H\u271d : n\u271d \u2264 List.length S n : \u2115 IH :   n \u2264 List.length S \u2192     Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }       { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } H : Nat.succ n \u2264 List.length S \u22a2 n < List.length (List.reverse S) ** rfl ** 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 k : K o : StAct k q : Stmt\u2082 v : \u03c3 S : List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) n\u271d : \u2115 H\u271d : n\u271d \u2264 List.length S n : \u2115 IH :   n \u2264 List.length S \u2192     Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }       { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } H : Nat.succ n \u2264 List.length S \u22a2 n < List.length (List.reverse S) ** rwa [List.length_reverse] ** 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 k : K o : StAct k q : Stmt\u2082 v : \u03c3 S : List (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) n : \u2115 H\u271d : n \u2264 List.length S H : Nat.zero \u2264 List.length S \u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.zero] (Tape.mk' \u2205 (addBottom L)) } ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.set_lintegral_preCdf_fst ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) r : \u211a s : Set \u03b1 hs : MeasurableSet s inst\u271d : IsFiniteMeasure \u03c1 \u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s ** have : \u2200 r, \u222b\u207b x in s, preCdf \u03c1 r x \u2202\u03c1.fst = \u222b\u207b x in s, (preCdf \u03c1 r * 1) x \u2202\u03c1.fst := by\n  simp only [mul_one, eq_self_iff_true, forall_const] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) r : \u211a s : Set \u03b1 hs : MeasurableSet s inst\u271d : IsFiniteMeasure \u03c1 this : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1 \u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s ** rw [this, \u2190 set_lintegral_withDensity_eq_set_lintegral_mul _ measurable_preCdf _ hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) r : \u211a s : Set \u03b1 hs : MeasurableSet s inst\u271d : IsFiniteMeasure \u03c1 \u22a2 \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1 ** simp only [mul_one, eq_self_iff_true, forall_const] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) r : \u211a s : Set \u03b1 hs : MeasurableSet s inst\u271d : IsFiniteMeasure \u03c1 this : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1 \u22a2 \u222b\u207b (x : \u03b1) in s, OfNat.ofNat 1 x \u2202Measure.withDensity (Measure.fst \u03c1) (preCdf \u03c1 r) = \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s ** simp only [withDensity_preCdf \u03c1 r, Pi.one_apply, lintegral_one, Measure.restrict_apply,\n  MeasurableSet.univ, univ_inter] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) r : \u211a s : Set \u03b1 hs : MeasurableSet s inst\u271d : IsFiniteMeasure \u03c1 this : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1 \u22a2 Measurable 1 ** rw [(_ : (1 : \u03b1 \u2192 \u211d\u22650\u221e) = fun _ => 1)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) r : \u211a s : Set \u03b1 hs : MeasurableSet s inst\u271d : IsFiniteMeasure \u03c1 this : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1 \u22a2 Measurable fun x => 1  \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) r : \u211a s : Set \u03b1 hs : MeasurableSet s inst\u271d : IsFiniteMeasure \u03c1 this : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1 \u22a2 1 = fun x => 1 ** exacts [measurable_const, rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.coeFn_compQuasiMeasurePreserving ** \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 \u2191(compQuasiMeasurePreserving g f hf) =\u1d50[\u03bc] \u2191g \u2218 f ** rw [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 \u2191(mk (\u2191g \u2218 f) (_ : AEStronglyMeasurable (\u2191g \u2218 f) \u03bc)) =\u1d50[\u03bc] \u2191g \u2218 f ** apply coeFn_mk ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.smul_extend ** \u03b1 : Type u_1 P : \u03b1 \u2192 Prop m : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e R : Type u_2 inst\u271d\u00b3 : Zero R inst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e c : R hc : c \u2260 0 \u22a2 c \u2022 extend m = extend fun s h => c \u2022 m s h ** ext1 s ** case h \u03b1 : Type u_1 P : \u03b1 \u2192 Prop m : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e R : Type u_2 inst\u271d\u00b3 : Zero R inst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e c : R hc : c \u2260 0 s : \u03b1 \u22a2 (c \u2022 extend m) s = extend (fun s h => c \u2022 m s h) s ** dsimp [extend] ** case h \u03b1 : Type u_1 P : \u03b1 \u2192 Prop m : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e R : Type u_2 inst\u271d\u00b3 : Zero R inst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e c : R hc : c \u2260 0 s : \u03b1 \u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h ** by_cases h : P s ** case pos \u03b1 : Type u_1 P : \u03b1 \u2192 Prop m : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e R : Type u_2 inst\u271d\u00b3 : Zero R inst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e c : R hc : c \u2260 0 s : \u03b1 h : P s \u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h ** simp [h] ** case neg \u03b1 : Type u_1 P : \u03b1 \u2192 Prop m : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e R : Type u_2 inst\u271d\u00b3 : Zero R inst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e c : R hc : c \u2260 0 s : \u03b1 h : \u00acP s \u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h ** simp [h, ENNReal.smul_top, hc] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpIndSMul_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\u2079 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2078 : NormedAddCommGroup E inst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2076 : CompleteSpace E inst\u271d\u00b9\u2075 : NormedAddCommGroup E' inst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b3 : CompleteSpace E' inst\u271d\u00b9\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedAddCommGroup G' inst\u271d\u2077 : NormedSpace \u211d G' inst\u271d\u2076 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2075 : IsROrC \ud835\udd5c' inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b2 : CompleteSpace E'' inst\u271d\u00b9 : NormedSpace \u211d E'' inst\u271d : NormedSpace \u211d G hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \u211d x : G \u22a2 condexpIndSMul hm hs h\u03bcs (c \u2022 x) = c \u2022 condexpIndSMul hm hs h\u03bcs x ** simp_rw [condexpIndSMul] ** \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\u2079 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2078 : NormedAddCommGroup E inst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2076 : CompleteSpace E inst\u271d\u00b9\u2075 : NormedAddCommGroup E' inst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b3 : CompleteSpace E' inst\u271d\u00b9\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedAddCommGroup G' inst\u271d\u2077 : NormedSpace \u211d G' inst\u271d\u2076 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2075 : IsROrC \ud835\udd5c' inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b2 : CompleteSpace E'' inst\u271d\u00b9 : NormedSpace \u211d E'' inst\u271d : NormedSpace \u211d G hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \u211d x : G \u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d (c \u2022 x))) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =     c \u2022 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) ** rw [toSpanSingleton_smul, smul_compLpL, smul_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "TypeVec.diag_sub_val ** n\u271d n : \u2115 \u03b1 : TypeVec.{u} n \u22a2 subtypeVal (repeatEq \u03b1) \u229a diagSub = prod.diag ** ext i x ** case a.h n\u271d n : \u2115 \u03b1 : TypeVec.{u} n i : Fin2 n x : \u03b1 i \u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) i x = prod.diag i x ** induction' i with _ _ _ i_ih ** case a.h.fz n\u271d\u00b9 n : \u2115 \u03b1\u271d : TypeVec.{u} n i : Fin2 n x\u271d : \u03b1\u271d i n\u271d : \u2115 \u03b1 : TypeVec.{u} (succ n\u271d) x : \u03b1 Fin2.fz \u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) Fin2.fz x = prod.diag Fin2.fz x  case a.h.fs n\u271d\u00b9 n : \u2115 \u03b1\u271d : TypeVec.{u} n i : Fin2 n x\u271d : \u03b1\u271d i n\u271d : \u2115 a\u271d : Fin2 n\u271d i_ih : \u2200 {\u03b1 : TypeVec.{u} n\u271d} (x : \u03b1 a\u271d), (subtypeVal (repeatEq \u03b1) \u229a diagSub) a\u271d x = prod.diag a\u271d x \u03b1 : TypeVec.{u} (succ n\u271d) x : \u03b1 (Fin2.fs a\u271d) \u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) (Fin2.fs a\u271d) x = prod.diag (Fin2.fs a\u271d) x ** simp only [comp, subtypeVal, repeatEq._eq_2, diagSub, prod.diag] ** case a.h.fs n\u271d\u00b9 n : \u2115 \u03b1\u271d : TypeVec.{u} n i : Fin2 n x\u271d : \u03b1\u271d i n\u271d : \u2115 a\u271d : Fin2 n\u271d i_ih : \u2200 {\u03b1 : TypeVec.{u} n\u271d} (x : \u03b1 a\u271d), (subtypeVal (repeatEq \u03b1) \u229a diagSub) a\u271d x = prod.diag a\u271d x \u03b1 : TypeVec.{u} (succ n\u271d) x : \u03b1 (Fin2.fs a\u271d) \u22a2 (subtypeVal (repeatEq \u03b1) \u229a diagSub) (Fin2.fs a\u271d) x = prod.diag (Fin2.fs a\u271d) x ** apply @i_ih (drop \u03b1) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_smul_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 c : \u211d\u22650\u221e \u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f 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\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 c : \u211d\u22650\u221e hG : CompleteSpace G \u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f 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\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 c : \u211d\u22650\u221e hG : \u00acCompleteSpace G \u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f 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\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 c : \u211d\u22650\u221e hG : CompleteSpace G \u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** rcases eq_or_ne c \u221e with (rfl | hc) ** case pos.inr \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 c : \u211d\u22650\u221e hG : CompleteSpace G hc : c \u2260 \u22a4 \u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** simp_rw [integral_eq_setToFun, \u2190 setToFun_smul_left] ** case pos.inr \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 c : \u211d\u22650\u221e hG : CompleteSpace G hc : c \u2260 \u22a4 \u22a2 (setToFun (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) (_ : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1) fun a =>       f a) =     setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)       (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>       f a ** have hdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc) : Set \u03b1 \u2192 G \u2192L[\u211d] G) c.toReal :=\n  mul_one c.toReal \u25b8 (dominatedFinMeasAdditive_weightedSMul (c \u2022 \u03bc)).of_smul_measure c hc ** case pos.inr \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 c : \u211d\u22650\u221e hG : CompleteSpace G hc : c \u2260 \u22a4 hdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc)) (ENNReal.toReal c) \u22a2 (setToFun (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) (_ : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1) fun a =>       f a) =     setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)       (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>       f a ** have hdfma_smul := dominatedFinMeasAdditive_weightedSMul (F := G) (c \u2022 \u03bc) ** case pos.inr \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 c : \u211d\u22650\u221e hG : CompleteSpace G hc : c \u2260 \u22a4 hdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc)) (ENNReal.toReal c) hdfma_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1 \u22a2 (setToFun (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) (_ : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1) fun a =>       f a) =     setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)       (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>       f a ** rw [\u2190 setToFun_congr_smul_measure c hc hdfma hdfma_smul f] ** case pos.inr \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 c : \u211d\u22650\u221e hG : CompleteSpace G hc : c \u2260 \u22a4 hdfma : DominatedFinMeasAdditive \u03bc (weightedSMul (c \u2022 \u03bc)) (ENNReal.toReal c) hdfma_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) (weightedSMul (c \u2022 \u03bc)) 1 \u22a2 setToFun \u03bc (weightedSMul (c \u2022 \u03bc)) hdfma f =     setToFun \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s)       (_ : DominatedFinMeasAdditive \u03bc (fun s => ENNReal.toReal c \u2022 weightedSMul \u03bc s) (\u2016ENNReal.toReal c\u2016 * 1)) fun a =>       f a ** exact setToFun_congr_left' _ _ (fun s _ _ => weightedSMul_smul_measure \u03bc c) f ** 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 c : \u211d\u22650\u221e hG : \u00acCompleteSpace G \u22a2 \u222b (x : \u03b1), f x \u2202c \u2022 \u03bc = ENNReal.toReal c \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** simp [integral, hG] ** case pos.inl \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 hG : CompleteSpace G \u22a2 \u222b (x : \u03b1), f x \u2202\u22a4 \u2022 \u03bc = ENNReal.toReal \u22a4 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** rw [ENNReal.top_toReal, zero_smul, integral_eq_setToFun, setToFun_top_smul_measure] ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_measurable_piecewise ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) \u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n) ** inhabit \u03b9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 \u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n) ** set g' : (i : \u03b9) \u2192 t i \u2192 \u03b2 := fun i => g i \u2218 (\u2191) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val \u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n) ** have ht' : \u2200 (i j) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j), g' i \u27e8x, hxi\u27e9 = g' j \u27e8x, hxj\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val ht' :   \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),     g' i { val := x, property := hxi } = g' j { val := x, property := hxj } \u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n) ** set f : (\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' _ Subset.rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val ht' :   \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),     g' i { val := x, property := hxi } = g' j { val := x, property := hxj } f : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i) \u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n) ** have hfm : Measurable f := measurable_iUnionLift _ _ t_meas\n  (fun i => (hg i).comp measurable_subtype_coe) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val ht' :   \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),     g' i { val := x, property := hxi } = g' j { val := x, property := hxj } f : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i) hfm : Measurable f \u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n) ** classical\n  refine \u27e8fun x => if hx : x \u2208 \u22c3 i, t i then f \u27e8x, hx\u27e9 else g default x,\n    hfm.dite ((hg default).comp measurable_subtype_coe) (.iUnion t_meas), fun i x hx => ?_\u27e9\n  simp only [dif_pos (mem_iUnion.2 \u27e8i, hx\u27e9)]\n  exact iUnionLift_of_mem \u27e8x, mem_iUnion.2 \u27e8i, hx\u27e9\u27e9 hx ** case ht' \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val \u22a2 \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),     g' i { val := x, property := hxi } = g' j { val := x, property := hxj } ** intro i j x hxi hxj ** case ht' \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val i j : \u03b9 x : \u03b1 hxi : x \u2208 t i hxj : x \u2208 t j \u22a2 g' i { val := x, property := hxi } = g' j { val := x, property := hxj } ** rcases eq_or_ne i j with rfl | hij ** case ht'.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val i : \u03b9 x : \u03b1 hxi hxj : x \u2208 t i \u22a2 g' i { val := x, property := hxi } = g' i { val := x, property := hxj } ** rfl ** case ht'.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val i j : \u03b9 x : \u03b1 hxi : x \u2208 t i hxj : x \u2208 t j hij : i \u2260 j \u22a2 g' i { val := x, property := hxi } = g' j { val := x, property := hxj } ** exact ht hij \u27e8hxi, hxj\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val ht' :   \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),     g' i { val := x, property := hxi } = g' j { val := x, property := hxj } f : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i) hfm : Measurable f \u22a2 \u2203 f, Measurable f \u2227 \u2200 (n : \u03b9), EqOn f (g n) (t n) ** refine \u27e8fun x => if hx : x \u2208 \u22c3 i, t i then f \u27e8x, hx\u27e9 else g default x,\n  hfm.dite ((hg default).comp measurable_subtype_coe) (.iUnion t_meas), fun i x hx => ?_\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val ht' :   \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),     g' i { val := x, property := hxi } = g' j { val := x, property := hxj } f : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i) hfm : Measurable f i : \u03b9 x : \u03b1 hx : x \u2208 t i \u22a2 (fun x => if hx : x \u2208 \u22c3 i, t i then f { val := x, property := hx } else g default x) x = g i x ** simp only [dif_pos (mem_iUnion.2 \u27e8i, hx\u27e9)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u\u03b9 s t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03b9 : Type u_6 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : Nonempty \u03b9 t : \u03b9 \u2192 Set \u03b1 t_meas : \u2200 (n : \u03b9), MeasurableSet (t n) g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hg : \u2200 (n : \u03b9), Measurable (g n) ht : Pairwise fun i j => EqOn (g i) (g j) (t i \u2229 t j) inhabited_h : Inhabited \u03b9 g' : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 := fun i => g i \u2218 Subtype.val ht' :   \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),     g' i { val := x, property := hxi } = g' j { val := x, property := hxj } f : \u2191(\u22c3 i, t i) \u2192 \u03b2 := iUnionLift t g' ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i) hfm : Measurable f i : \u03b9 x : \u03b1 hx : x \u2208 t i \u22a2 iUnionLift t (fun i => g i \u2218 Subtype.val) ht' (\u22c3 i, t i) (_ : \u22c3 i, t i \u2286 \u22c3 i, t i)       { val := x, property := (_ : x \u2208 \u22c3 i, t i) } =     g i x ** exact iUnionLift_of_mem \u27e8x, mem_iUnion.2 \u27e8i, hx\u27e9\u27e9 hx ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.get_push_lt ** \u03b1 : Type ?u.45557 a : Array \u03b1 x : \u03b1 i : Nat h : i < size a \u22a2 i < size (push a x) ** simp [*, Nat.lt_succ_of_le, Nat.le_of_lt] ** \u03b1 : Type u_1 a : Array \u03b1 x : \u03b1 i : Nat h : i < size a \u22a2 (push a x)[i] = a[i] ** simp only [push, getElem_eq_data_get, List.concat_eq_append, List.get_append_left, h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.martingale_zero ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b3 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u03b9 \u2192 \u03a9 \u2192 E \u2131\u271d \u2131 : Filtration \u03b9 m0 \u03bc : Measure \u03a9 i j : \u03b9 x\u271d : i \u2264 j \u22a2 \u03bc[OfNat.ofNat 0 j|\u2191\u2131 i] =\u1d50[\u03bc] OfNat.ofNat 0 i ** rw [Pi.zero_apply, condexp_zero] ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b3 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u03b9 \u2192 \u03a9 \u2192 E \u2131\u271d \u2131 : Filtration \u03b9 m0 \u03bc : Measure \u03a9 i j : \u03b9 x\u271d : i \u2264 j \u22a2 0 =\u1d50[\u03bc] OfNat.ofNat 0 i ** simp ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b3 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u03b9 \u2192 \u03a9 \u2192 E \u2131\u271d \u2131 : Filtration \u03b9 m0 \u03bc : Measure \u03a9 i j : \u03b9 x\u271d : i \u2264 j \u22a2 0 =\u1d50[\u03bc] 0 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.comp_snd_map_prod_id ** \u03a9 : Type u_1 F : Type u_2 m m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F inst\u271d : NormedAddCommGroup F hm : m \u2264 m\u03a9 hf : Integrable f \u22a2 Integrable fun x => f x.2 ** rw [\u2190 integrable_comp_snd_map_prod_mk_iff (measurable_id'' hm)] at hf ** \u03a9 : Type u_1 F : Type u_2 m m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F inst\u271d : NormedAddCommGroup F hm : m \u2264 m\u03a9 hf : Integrable fun x => f x.2 \u22a2 Integrable fun x => f x.2 ** exact hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2.stmts_trans ** K : Type u_1 inst\u271d : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 \u03c3 : Type u_4 M : \u039b \u2192 Stmt\u2082 S : Finset \u039b q\u2081 q\u2082 : Stmt\u2082 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] ** K : Type u_1 inst\u271d : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 \u03c3 : Type u_4 M : \u039b \u2192 Stmt\u2082 S : Finset \u039b q\u2081 q\u2082 : Stmt\u2082 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": "List.range_add ** a b : Nat \u22a2 range (a + b) = range a ++ map (fun x => a + x) (range b) ** rw [\u2190 range'_eq_map_range] ** a b : Nat \u22a2 range (a + b) = range a ++ range' a b ** simpa [range_eq_range', Nat.add_comm] using (range'_append_1 0 a b).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "circleIntegrable_sub_zpow_iff ** E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 R : \u211d n : \u2124 \u22a2 CircleIntegrable (fun z => (z - w) ^ n) c R \u2194 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R| ** constructor ** case mp E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 R : \u211d n : \u2124 \u22a2 CircleIntegrable (fun z => (z - w) ^ n) c R \u2192 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R| ** intro h ** case mp E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 R : \u211d n : \u2124 h : CircleIntegrable (fun z => (z - w) ^ n) c R \u22a2 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R| ** contrapose! h ** case mp E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 R : \u211d n : \u2124 h : R \u2260 0 \u2227 n < 0 \u2227 w \u2208 sphere c |R| \u22a2 \u00acCircleIntegrable (fun z => (z - w) ^ n) c R ** rcases h with \u27e8hR, hn, hw\u27e9 ** case mp.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 hw : w \u2208 sphere c |R| \u22a2 \u00acCircleIntegrable (fun z => (z - w) ^ n) c R ** simp only [circleIntegrable_iff R, deriv_circleMap] ** case mp.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 hw : w \u2208 sphere c |R| \u22a2 \u00acIntervalIntegrable (fun \u03b8 => (circleMap 0 R \u03b8 * I) \u2022 (circleMap c R \u03b8 - w) ^ n) volume 0 (2 * \u03c0) ** rw [\u2190 image_circleMap_Ioc] at hw ** case mp.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 hw : w \u2208 circleMap c R '' Ioc 0 (2 * \u03c0) \u22a2 \u00acIntervalIntegrable (fun \u03b8 => (circleMap 0 R \u03b8 * I) \u2022 (circleMap c R \u03b8 - w) ^ n) volume 0 (2 * \u03c0) ** rcases hw with \u27e8\u03b8, h\u03b8, rfl\u27e9 ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 Ioc 0 (2 * \u03c0) \u22a2 \u00acIntervalIntegrable (fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n) volume 0 (2 * \u03c0) ** replace h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] ** case h\u03b8 E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 Ioc 0 (2 * \u03c0) \u22a2 \u03b8 \u2208 [[0, 2 * \u03c0]]  case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] \u22a2 \u00acIntervalIntegrable (fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n) volume 0 (2 * \u03c0) ** exact Icc_subset_uIcc (Ioc_subset_Icc_self h\u03b8) ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] \u22a2 \u00acIntervalIntegrable (fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n) volume 0 (2 * \u03c0) ** refine' not_intervalIntegrable_of_sub_inv_isBigO_punctured _ Real.two_pi_pos.ne h\u03b8 ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] \u22a2 (fun x => (x - \u03b8)\u207b\u00b9) =O[\ud835\udcdd[{\u03b8}\u1d9c] \u03b8] fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n ** set f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 \u22a2 (fun x => (x - \u03b8)\u207b\u00b9) =O[\ud835\udcdd[{\u03b8}\u1d9c] \u03b8] fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n ** have : \u2200\u1da0 \u03b8' in \ud835\udcdd[\u2260] \u03b8, f \u03b8' \u2208 ball (0 : \u2102) 1 \\ {0} := by\n  suffices : \u2200\u1da0 z in \ud835\udcdd[\u2260] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball (0 : \u2102) 1 \\ {0}\n  exact ((differentiable_circleMap c R \u03b8).hasDerivAt.tendsto_punctured_nhds\n    (deriv_circleMap_ne_zero hR)).eventually this\n  filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (ball_mem_nhds _ zero_lt_one)]\n  simp_all [dist_eq, sub_eq_zero] ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u22a2 (fun x => (x - \u03b8)\u207b\u00b9) =O[\ud835\udcdd[{\u03b8}\u1d9c] \u03b8] fun \u03b8_1 => (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n ** refine' (((hasDerivAt_circleMap c R \u03b8).isBigO_sub.mono inf_le_left).inv_rev\n  (this.mono fun \u03b8' h\u2081 h\u2082 => absurd h\u2082 h\u2081.2)).trans _ ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u22a2 (fun x => (circleMap c R x - circleMap c R \u03b8)\u207b\u00b9) =O[\ud835\udcdd \u03b8 \u2293 \ud835\udcdf {\u03b8}\u1d9c] fun \u03b8_1 =>     (circleMap 0 R \u03b8_1 * I) \u2022 (circleMap c R \u03b8_1 - circleMap c R \u03b8) ^ n ** refine' IsBigO.of_bound |R|\u207b\u00b9 (this.mono fun \u03b8' h\u03b8' => _) ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u03b8' : \u211d h\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0} \u22a2 \u2016(circleMap c R \u03b8' - circleMap c R \u03b8)\u207b\u00b9\u2016 \u2264 |R|\u207b\u00b9 * \u2016(circleMap 0 R \u03b8' * I) \u2022 (circleMap c R \u03b8' - circleMap c R \u03b8) ^ n\u2016 ** set x := abs (f \u03b8') ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u03b8' : \u211d h\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0} x : \u211d := \u2191Complex.abs (f \u03b8') \u22a2 \u2016(circleMap c R \u03b8' - circleMap c R \u03b8)\u207b\u00b9\u2016 \u2264 |R|\u207b\u00b9 * \u2016(circleMap 0 R \u03b8' * I) \u2022 (circleMap c R \u03b8' - circleMap c R \u03b8) ^ n\u2016 ** suffices x\u207b\u00b9 \u2264 x ^ n by\n  simpa only [inv_mul_cancel_left\u2080, abs_eq_zero.not.2 hR, norm_eq_abs, map_inv\u2080,\n    Algebra.id.smul_eq_mul, map_mul, abs_circleMap_zero, abs_I, mul_one, abs_zpow, Ne.def,\n    not_false_iff] using this ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u03b8' : \u211d h\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0} x : \u211d := \u2191Complex.abs (f \u03b8') \u22a2 x\u207b\u00b9 \u2264 x ^ n ** have : x \u2208 Ioo (0 : \u211d) 1 := by simpa [and_comm] using h\u03b8' ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u03b8' : \u211d h\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0} x : \u211d := \u2191Complex.abs (f \u03b8') this : x \u2208 Ioo 0 1 \u22a2 x\u207b\u00b9 \u2264 x ^ n ** rw [\u2190 zpow_neg_one] ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u03b8' : \u211d h\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0} x : \u211d := \u2191Complex.abs (f \u03b8') this : x \u2208 Ioo 0 1 \u22a2 x ^ (-1) \u2264 x ^ n ** refine' (zpow_strictAnti this.1 this.2).le_iff_le.2 (Int.lt_add_one_iff.1 _) ** case mp.intro.intro.intro.intro E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u03b8' : \u211d h\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0} x : \u211d := \u2191Complex.abs (f \u03b8') this : x \u2208 Ioo 0 1 \u22a2 n < -1 + 1 ** exact hn ** E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 \u22a2 \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} ** suffices : \u2200\u1da0 z in \ud835\udcdd[\u2260] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball (0 : \u2102) 1 \\ {0} ** E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this : \u2200\u1da0 (z : \u2102) in \ud835\udcdd[{circleMap c R \u03b8}\u1d9c] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0} \u22a2 \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0}  case this E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 \u22a2 \u2200\u1da0 (z : \u2102) in \ud835\udcdd[{circleMap c R \u03b8}\u1d9c] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0} ** exact ((differentiable_circleMap c R \u03b8).hasDerivAt.tendsto_punctured_nhds\n  (deriv_circleMap_ne_zero hR)).eventually this ** case this E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 \u22a2 \u2200\u1da0 (z : \u2102) in \ud835\udcdd[{circleMap c R \u03b8}\u1d9c] circleMap c R \u03b8, z - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0} ** filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (ball_mem_nhds _ zero_lt_one)] ** case h E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 \u22a2 \u2200 (a : \u2102), a \u2208 {circleMap c R \u03b8}\u1d9c \u2192 a \u2208 ball (circleMap c R \u03b8) 1 \u2192 a - circleMap c R \u03b8 \u2208 ball 0 1 \\ {0} ** simp_all [dist_eq, sub_eq_zero] ** E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this\u271d : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u03b8' : \u211d h\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0} x : \u211d := \u2191Complex.abs (f \u03b8') this : x\u207b\u00b9 \u2264 x ^ n \u22a2 \u2016(circleMap c R \u03b8' - circleMap c R \u03b8)\u207b\u00b9\u2016 \u2264 |R|\u207b\u00b9 * \u2016(circleMap 0 R \u03b8' * I) \u2022 (circleMap c R \u03b8' - circleMap c R \u03b8) ^ n\u2016 ** simpa only [inv_mul_cancel_left\u2080, abs_eq_zero.not.2 hR, norm_eq_abs, map_inv\u2080,\n  Algebra.id.smul_eq_mul, map_mul, abs_circleMap_zero, abs_I, mul_one, abs_zpow, Ne.def,\n  not_false_iff] using this ** E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d n : \u2124 hR : R \u2260 0 hn : n < 0 \u03b8 : \u211d h\u03b8 : \u03b8 \u2208 [[0, 2 * \u03c0]] f : \u211d \u2192 \u2102 := fun \u03b8' => circleMap c R \u03b8' - circleMap c R \u03b8 this : \u2200\u1da0 (\u03b8' : \u211d) in \ud835\udcdd[{\u03b8}\u1d9c] \u03b8, f \u03b8' \u2208 ball 0 1 \\ {0} \u03b8' : \u211d h\u03b8' : f \u03b8' \u2208 ball 0 1 \\ {0} x : \u211d := \u2191Complex.abs (f \u03b8') \u22a2 x \u2208 Ioo 0 1 ** simpa [and_comm] using h\u03b8' ** case mpr E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 R : \u211d n : \u2124 \u22a2 R = 0 \u2228 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R| \u2192 CircleIntegrable (fun z => (z - w) ^ n) c R ** rintro (rfl | H) ** case mpr.inl E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 n : \u2124 \u22a2 CircleIntegrable (fun z => (z - w) ^ n) c 0  case mpr.inr E : Type u_1 inst\u271d : NormedAddCommGroup E c w : \u2102 R : \u211d n : \u2124 H : 0 \u2264 n \u2228 \u00acw \u2208 sphere c |R| \u22a2 CircleIntegrable (fun z => (z - w) ^ n) c R ** exacts [circleIntegrable_zero_radius,\n  ((continuousOn_id.sub continuousOn_const).zpow\u2080 _ fun z hz =>\n    H.symm.imp_left fun (hw : w \u2209 sphere c |R|) =>\n      sub_ne_zero.2 <| ne_of_mem_of_not_mem hz hw).circleIntegrable'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Option.elim_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 f : \u03b3 \u2192 \u03b1 \u2192 \u03b2 x : \u03b1 \u2192 \u03b2 i : Option \u03b3 y : \u03b1 \u22a2 Option.elim i x f y = Option.elim i (x y) fun j => f j y ** rw [elim_comp fun f : \u03b1 \u2192 \u03b2 => f y] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.hasPDF_iff_of_measurable ** \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 map X \u2119 \u226a volume ** rw [hasPDF_iff_of_measurable hX] ** \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 HaveLebesgueDecomposition (map X \u2119) volume \u2227 map X \u2119 \u226a volume \u2194 map X \u2119 \u226a volume ** exact and_iff_right inferInstance ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.valMinAbs_eq_zero ** n : \u2115 x : ZMod n \u22a2 valMinAbs x = 0 \u2194 x = 0 ** cases' n with n ** case succ n : \u2115 x : ZMod (Nat.succ n) \u22a2 valMinAbs x = 0 \u2194 x = 0 ** rw [\u2190 valMinAbs_zero n.succ] ** case succ n : \u2115 x : ZMod (Nat.succ n) \u22a2 valMinAbs x = valMinAbs 0 \u2194 x = 0 ** apply injective_valMinAbs.eq_iff ** case zero x : ZMod Nat.zero \u22a2 valMinAbs x = 0 \u2194 x = 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.strong_law_ae ** \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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** have A : \u2200 i, Integrable (X i) := fun i \u21a6 (hident i).integrable_iff.2 hint ** \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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** let Y : \u2115 \u2192 \u03a9 \u2192 E := fun i \u21a6 (A i).1.mk (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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** have B : \u2200\u1d50 \u03c9, \u2200 n, X n \u03c9 = Y n \u03c9 :=\n  ae_all_iff.2 (fun i \u21a6 AEStronglyMeasurable.ae_eq_mk (A i).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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) B : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** have Yint: Integrable (Y 0) := Integrable.congr hint (AEStronglyMeasurable.ae_eq_mk (A 0).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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) B : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 Yint : Integrable (Y 0) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** have C : \u2200\u1d50 \u03c9, Tendsto (fun n : \u2115 \u21a6 (n : \u211d) \u207b\u00b9 \u2022 (\u2211 i in range n, Y i \u03c9)) atTop (\ud835\udcdd \ud835\udd3c[Y 0]) := by\n  apply strong_law_ae_of_measurable Y Yint ((A 0).1.stronglyMeasurable_mk)\n    (fun i j hij \u21a6 IndepFun.ae_eq (hindep hij) (A i).1.ae_eq_mk (A j).1.ae_eq_mk)\n    (fun i \u21a6 ((A i).1.identDistrib_mk.symm.trans (hident i)).trans (A 0).1.identDistrib_mk) ** \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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) B : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 Yint : Integrable (Y 0) C : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** filter_upwards [B, C] with \u03c9 h\u2081 h\u2082 ** case 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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) B : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 Yint : Integrable (Y 0) C : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) \u03c9 : \u03a9 h\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 h\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) \u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** have : \ud835\udd3c[X 0] = \ud835\udd3c[Y 0] := integral_congr_ae (AEStronglyMeasurable.ae_eq_mk (A 0).1) ** case 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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) B : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 Yint : Integrable (Y 0) C : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) \u03c9 : \u03a9 h\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 h\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) this : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a \u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** rw [this] ** case 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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) B : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 Yint : Integrable (Y 0) C : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) \u03c9 : \u03a9 h\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 h\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) this : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a \u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) ** apply Tendsto.congr (fun n \u21a6 ?_) h\u2082 ** \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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) B : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 Yint : Integrable (Y 0) C : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) \u03c9 : \u03a9 h\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 h\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) this : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a n : \u2115 \u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9 = (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, X i \u03c9 ** congr with i ** case e_a.e_f.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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) B : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 Yint : Integrable (Y 0) C : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) \u03c9 : \u03a9 h\u2081 : \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 h\u2082 : Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) this : \u222b (a : \u03a9), X 0 a = \u222b (a : \u03a9), Y 0 a n i : \u2115 \u22a2 Y i \u03c9 = X i \u03c9 ** exact (h\u2081 i).symm ** \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 X : \u2115 \u2192 \u03a9 \u2192 E hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) A : \u2200 (i : \u2115), Integrable (X i) Y : \u2115 \u2192 \u03a9 \u2192 E := fun i => AEStronglyMeasurable.mk (X i) (_ : AEStronglyMeasurable (X i) \u2119) B : \u2200\u1d50 (\u03c9 : \u03a9), \u2200 (n : \u2115), X n \u03c9 = Y n \u03c9 Yint : Integrable (Y 0) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, Y i \u03c9) atTop (\ud835\udcdd (\u222b (a : \u03a9), Y 0 a)) ** apply strong_law_ae_of_measurable Y Yint ((A 0).1.stronglyMeasurable_mk)\n  (fun i j hij \u21a6 IndepFun.ae_eq (hindep hij) (A i).1.ae_eq_mk (A j).1.ae_eq_mk)\n  (fun i \u21a6 ((A i).1.identDistrib_mk.symm.trans (hident i)).trans (A 0).1.identDistrib_mk) ** Qed",
        "informal": ""
    },
    {
        "formal": "StrictMonoOn.Iic_id_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 h\u03c6 : StrictMonoOn \u03c6 (Iic n) \u22a2 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m ** revert h\u03c6 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 \u22a2 StrictMonoOn \u03c6 (Iic n) \u2192 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m ** refine'\n  Succ.rec_bot (fun n => StrictMonoOn \u03c6 (Set.Iic n) \u2192 \u2200 m \u2264 n, m \u2264 \u03c6 m)\n    (fun _ _ hm => hm.trans bot_le) _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 \u22a2 \u2200 (a : \u03b1),     (fun n => StrictMonoOn \u03c6 (Iic n) \u2192 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m) a \u2192       (fun n => StrictMonoOn \u03c6 (Iic n) \u2192 \u2200 (m : \u03b1), m \u2264 n \u2192 m \u2264 \u03c6 m) (succ a) ** rintro k ih h\u03c6 m hm ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 k : \u03b1 ih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m h\u03c6 : StrictMonoOn \u03c6 (Iic (succ k)) m : \u03b1 hm : m \u2264 succ k \u22a2 m \u2264 \u03c6 m ** by_cases hk : IsMax k ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 k : \u03b1 ih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m h\u03c6 : StrictMonoOn \u03c6 (Iic (succ k)) m : \u03b1 hm : m \u2264 succ k hk : \u00acIsMax k \u22a2 m \u2264 \u03c6 m ** obtain rfl | h := le_succ_iff_eq_or_le.1 hm ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 k : \u03b1 ih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m h\u03c6 : StrictMonoOn \u03c6 (Iic (succ k)) m : \u03b1 hm : m \u2264 succ k hk : IsMax k \u22a2 m \u2264 \u03c6 m ** rw [succ_eq_iff_isMax.2 hk] at hm ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 k : \u03b1 ih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m h\u03c6 : StrictMonoOn \u03c6 (Iic (succ k)) m : \u03b1 hm : m \u2264 k hk : IsMax k \u22a2 m \u2264 \u03c6 m ** exact ih (h\u03c6.mono <| Iic_subset_Iic.2 (le_succ _)) _ hm ** case neg.inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 k : \u03b1 ih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m h\u03c6 : StrictMonoOn \u03c6 (Iic (succ k)) hk : \u00acIsMax k hm : succ k \u2264 succ k \u22a2 succ k \u2264 \u03c6 (succ k) ** specialize ih (StrictMonoOn.mono h\u03c6 fun x hx => le_trans hx (le_succ _)) k le_rfl ** case neg.inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 k : \u03b1 h\u03c6 : StrictMonoOn \u03c6 (Iic (succ k)) hk : \u00acIsMax k hm : succ k \u2264 succ k ih : k \u2264 \u03c6 k \u22a2 succ k \u2264 \u03c6 (succ k) ** refine' le_trans (succ_mono ih) (succ_le_of_lt (h\u03c6 (le_succ _) le_rfl _)) ** case neg.inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 k : \u03b1 h\u03c6 : StrictMonoOn \u03c6 (Iic (succ k)) hk : \u00acIsMax k hm : succ k \u2264 succ k ih : k \u2264 \u03c6 k \u22a2 k < succ k ** rw [lt_succ_iff_eq_or_lt_of_not_isMax hk] ** case neg.inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 k : \u03b1 h\u03c6 : StrictMonoOn \u03c6 (Iic (succ k)) hk : \u00acIsMax k hm : succ k \u2264 succ k ih : k \u2264 \u03c6 k \u22a2 k = k \u2228 k < k ** exact Or.inl rfl ** case neg.inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : SuccOrder \u03b1 inst\u271d\u00b9 : IsSuccArchimedean \u03b1 inst\u271d : OrderBot \u03b1 n : \u03b1 \u03c6 : \u03b1 \u2192 \u03b1 k : \u03b1 ih : StrictMonoOn \u03c6 (Iic k) \u2192 \u2200 (m : \u03b1), m \u2264 k \u2192 m \u2264 \u03c6 m h\u03c6 : StrictMonoOn \u03c6 (Iic (succ k)) m : \u03b1 hm : m \u2264 succ k hk : \u00acIsMax k h : m \u2264 k \u22a2 m \u2264 \u03c6 m ** exact ih (StrictMonoOn.mono h\u03c6 fun x hx => le_trans hx (le_succ _)) _ h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.QuasiMeasurePreserving.limsup_preimage_iterate_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 s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f\u271d : \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b1 hf : QuasiMeasurePreserving f hs : f \u207b\u00b9' s =\u1d50[\u03bc] s \u22a2 \u2200 (n : \u2115), (preimage f)^[n] s =\u1d50[\u03bc] s ** intro 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 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 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f\u271d : \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b1 hf : QuasiMeasurePreserving f hs : f \u207b\u00b9' s =\u1d50[\u03bc] s n : \u2115 \u22a2 (preimage f)^[n] s =\u1d50[\u03bc] s ** induction' n with n ih ** case succ \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 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f\u271d : \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b1 hf : QuasiMeasurePreserving f hs : f \u207b\u00b9' s =\u1d50[\u03bc] s n : \u2115 ih : (preimage f)^[n] s =\u1d50[\u03bc] s \u22a2 (preimage f)^[Nat.succ n] s =\u1d50[\u03bc] s ** simpa only [iterate_succ', comp_apply] using ae_eq_trans (hf.ae_eq ih) hs ** case 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 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 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f\u271d : \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b1 hf : QuasiMeasurePreserving f hs : f \u207b\u00b9' s =\u1d50[\u03bc] s \u22a2 (preimage f)^[Nat.zero] s =\u1d50[\u03bc] s ** rfl ** 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\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E f : \u03b2 \u2192 E 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 (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 f _ s] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Icc_subset_Ico_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 h\u2081 : a\u2081 \u2264 b\u2081 \u22a2 Icc a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2194 a\u2082 \u2264 a\u2081 \u2227 b\u2081 < b\u2082 ** rw [\u2190 coe_subset, coe_Icc, coe_Ico, Set.Icc_subset_Ico_iff h\u2081] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.neg_divInt ** n d : Int \u22a2 -(n /. d) = -n /. d ** rcases Int.eq_nat_or_neg d with \u27e8_, rfl | rfl\u27e9 <;> simp [divInt_neg', neg_mkRat] ** Qed",
        "informal": ""
    },
    {
        "formal": "ApproximatesLinearOn.norm_fderiv_sub_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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, \u2016f' x - A\u2016\u208a \u2264 \u03b4 ** filter_upwards [Besicovitch.ae_tendsto_measure_inter_div \u03bc s, ae_restrict_mem hs] ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u22a2 \u2200 (a : E),     Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall a r) / \u2191\u2191\u03bc (closedBall a r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) \u2192 a \u2208 s \u2192 \u2016f' a - A\u2016\u208a \u2264 \u03b4 ** intro x hx xs ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s \u22a2 \u2016f' x - A\u2016\u208a \u2264 \u03b4 ** apply ContinuousLinearMap.op_norm_le_bound _ \u03b4.2 fun z => ?_ ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016 ** suffices H : \u2200 \u03b5, 0 < \u03b5 \u2192 \u2016(f' x - A) z\u2016 \u2264 (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** intro \u03b5 \u03b5pos ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** have B\u2081 : \u2200\u1da0 r in \ud835\udcdd[>] (0 : \u211d), (s \u2229 ({x} + r \u2022 closedBall z \u03b5)).Nonempty :=\n  eventually_nonempty_inter_smul_of_density_one \u03bc s x hx _ measurableSet_closedBall\n    (measure_closedBall_pos \u03bc z \u03b5pos).ne' ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** obtain \u27e8\u03c1, \u03c1pos, h\u03c1\u27e9 :\n  \u2203 \u03c1 > 0, ball x \u03c1 \u2229 s \u2286 {y : E | \u2016f y - f x - (f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} :=\n  mem_nhdsWithin_iff.1 (IsLittleO.def (hf' x xs) \u03b5pos) ** case 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** have B\u2082 : \u2200\u1da0 r in \ud835\udcdd[>] (0 : \u211d), {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 := by\n  apply nhdsWithin_le_nhds\n  exact eventually_singleton_add_smul_subset isBounded_closedBall (ball_mem_nhds x \u03c1pos) ** case 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** obtain \u27e8r, \u27e8y, \u27e8ys, hy\u27e9\u27e9, r\u03c1, rpos\u27e9 :\n  \u2203 r : \u211d,\n    (s \u2229 ({x} + r \u2022 closedBall z \u03b5)).Nonempty \u2227 {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 \u2227 0 < r :=\n  (B\u2081.and (B\u2082.and self_mem_nhdsWithin)).exists ** case H.intro.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 : Set E f : 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 A : E \u2192L[\u211d] E \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** obtain \u27e8a, az, ya\u27e9 : \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a := by\n  simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy\n  rcases hy with \u27e8a, az, ha\u27e9\n  exact \u27e8a, az, by simp only [ha, add_neg_cancel_left]\u27e9 ** case H.intro.intro.intro.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 : Set E f : 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 A : E \u2192L[\u211d] E \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** have norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 :=\n  calc\n    \u2016a\u2016 = \u2016z + (a - z)\u2016 := by simp only [add_sub_cancel'_right]\n    _ \u2264 \u2016z\u2016 + \u2016a - z\u2016 := (norm_add_le _ _)\n    _ \u2264 \u2016z\u2016 + \u03b5 := add_le_add_left (mem_closedBall_iff_norm.1 az) _ ** case H.intro.intro.intro.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 : Set E f : 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 A : E \u2192L[\u211d] E \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** have I : r * \u2016(f' x - A) a\u2016 \u2264 r * (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) :=\n  calc\n    r * \u2016(f' x - A) a\u2016 = \u2016(f' x - A) (r \u2022 a)\u2016 := by\n      simp only [ContinuousLinearMap.map_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]\n    _ = \u2016f y - f x - A (y - x) - (f y - f x - (f' x) (y - x))\u2016 := by\n      congr 1\n      simp only [ya, add_sub_cancel', sub_sub_sub_cancel_left, ContinuousLinearMap.coe_sub',\n        eq_self_iff_true, sub_left_inj, Pi.sub_apply, ContinuousLinearMap.map_smul, smul_sub]\n    _ \u2264 \u2016f y - f x - A (y - x)\u2016 + \u2016f y - f x - (f' x) (y - x)\u2016 := (norm_sub_le _ _)\n    _ \u2264 \u03b4 * \u2016y - x\u2016 + \u03b5 * \u2016y - x\u2016 := (add_le_add (hf _ ys _ xs) (h\u03c1 \u27e8r\u03c1 hy, ys\u27e9))\n    _ = r * (\u03b4 + \u03b5) * \u2016a\u2016 := by\n      simp only [ya, add_sub_cancel', norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]\n      ring\n    _ \u2264 r * (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) :=\n      mul_le_mul_of_nonneg_left norm_a (mul_nonneg rpos.le (add_nonneg \u03b4.2 \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 : Set E f : 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 A : E \u2192L[\u211d] E \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E H : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016 ** have :\n  Tendsto (fun \u03b5 : \u211d => ((\u03b4 : \u211d) + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd ((\u03b4 + 0) * (\u2016z\u2016 + 0) + \u2016f' x - A\u2016 * 0)) :=\n  Tendsto.mono_left (Continuous.tendsto (by continuity) 0) 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E H : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 this : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd ((\u2191\u03b4 + 0) * (\u2016z\u2016 + 0) + \u2016f' x - A\u2016 * 0)) \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016 ** simp only [add_zero, mul_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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E H : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 this : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016)) \u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016 ** apply le_of_tendsto_of_tendsto tendsto_const_nhds 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E H : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 this : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016)) \u22a2 (fun x_1 => \u2016\u2191(f' x - A) z\u2016) \u2264\u1da0[\ud835\udcdd[Ioi 0] 0] fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E H : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 this : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016)) \u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + a) * (\u2016z\u2016 + a) + \u2016f' x - A\u2016 * a ** exact 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E H : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 \u22a2 Continuous fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 ** continuity ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} \u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} \u22a2 {x_1 | (fun r => {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1) x_1} \u2208 \ud835\udcdd 0 ** exact eventually_singleton_add_smul_subset isBounded_closedBall (ball_mem_nhds x \u03c1pos) ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r \u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a ** simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r hy : \u2203 y_1, y_1 \u2208 closedBall z \u03b5 \u2227 r \u2022 y_1 = -x + y \u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a ** rcases hy with \u27e8a, az, ha\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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ha : r \u2022 a = -x + y \u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a ** exact \u27e8a, az, by simp only [ha, add_neg_cancel_left]\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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ha : r \u2022 a = -x + y \u22a2 y = x + r \u2022 a ** simp only [ha, add_neg_cancel_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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a \u22a2 \u2016a\u2016 = \u2016z + (a - z)\u2016 ** simp only [add_sub_cancel'_right] ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 \u22a2 r * \u2016\u2191(f' x - A) a\u2016 = \u2016\u2191(f' x - A) (r \u2022 a)\u2016 ** simp only [ContinuousLinearMap.map_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 \u22a2 \u2016\u2191(f' x - A) (r \u2022 a)\u2016 = \u2016f y - f x - \u2191A (y - x) - (f y - f x - \u2191(f' x) (y - x))\u2016 ** congr 1 ** case e_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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 \u22a2 \u2191(f' x - A) (r \u2022 a) = f y - f x - \u2191A (y - x) - (f y - f x - \u2191(f' x) (y - x)) ** simp only [ya, add_sub_cancel', sub_sub_sub_cancel_left, ContinuousLinearMap.coe_sub',\n  eq_self_iff_true, sub_left_inj, Pi.sub_apply, ContinuousLinearMap.map_smul, smul_sub] ** 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 \u22a2 \u2191\u03b4 * \u2016y - x\u2016 + \u03b5 * \u2016y - x\u2016 = r * (\u2191\u03b4 + \u03b5) * \u2016a\u2016 ** simp only [ya, add_sub_cancel', norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 \u22a2 \u2191\u03b4 * (r * \u2016a\u2016) + \u03b5 * (r * \u2016a\u2016) = r * (\u2191\u03b4 + \u03b5) * \u2016a\u2016 ** 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 : Set E f : 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 A : E \u2192L[\u211d] E \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 I : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) \u22a2 \u2016\u2191(f' x - A) z\u2016 = \u2016\u2191(f' x - A) a + \u2191(f' x - A) (z - a)\u2016 ** congr 1 ** case e_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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 I : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) \u22a2 \u2191(f' x - A) z = \u2191(f' x - A) a + \u2191(f' x - A) (z - a) ** simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply] ** case e_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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 I : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) \u22a2 \u2191(f' x) z - \u2191A z = \u2191(f' x) a - \u2191A a + (\u2191(f' x) z - \u2191A z - (\u2191(f' x) a - \u2191A a)) ** 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 : Set E f : 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 A : E \u2192L[\u211d] E \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 I : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) \u22a2 \u2016\u2191(f' x - A) a\u2016 + \u2016\u2191(f' x - A) (z - a)\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u2016z - a\u2016 ** apply add_le_add ** case h\u2081 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 I : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) \u22a2 \u2016\u2191(f' x - A) a\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) ** rw [mul_assoc] at I ** case h\u2081 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 I : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * ((\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)) \u22a2 \u2016\u2191(f' x - A) a\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) ** exact (mul_le_mul_left rpos).1 I ** case h\u2082 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'\u271d : 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 \u03b4 : \u211d\u22650 hf : ApproximatesLinearOn f A s \u03b4 hs : MeasurableSet s f' : E \u2192 E \u2192L[\u211d] E hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x x : E hx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) xs : x \u2208 s z : E \u03b5 : \u211d \u03b5pos : 0 < \u03b5 B\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5)) \u03c1 : \u211d \u03c1pos : \u03c1 > 0 h\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} B\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 r : \u211d y : E ys : y \u2208 s hy : y \u2208 {x} + r \u2022 closedBall z \u03b5 r\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 rpos : 0 < r a : E az : a \u2208 closedBall z \u03b5 ya : y = x + r \u2022 a norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 I : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) \u22a2 \u2016\u2191(f' x - A) (z - a)\u2016 \u2264 \u2016f' x - A\u2016 * \u2016z - a\u2016 ** apply ContinuousLinearMap.le_op_norm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.isHaarMeasure_map ** \ud835\udd5c : Type u_1 G : Type u_2 H\u271d : Type u_3 inst\u271d\u00b9\u00b2 : MeasurableSpace G inst\u271d\u00b9\u00b9 : MeasurableSpace H\u271d inst\u271d\u00b9\u2070 : Group G inst\u271d\u2079 : TopologicalSpace G \u03bc : Measure G inst\u271d\u2078 : IsHaarMeasure \u03bc inst\u271d\u2077 : BorelSpace G inst\u271d\u2076 : TopologicalGroup G H : Type u_4 inst\u271d\u2075 : Group H inst\u271d\u2074 : TopologicalSpace H inst\u271d\u00b3 : MeasurableSpace H inst\u271d\u00b2 : BorelSpace H inst\u271d\u00b9 : T2Space H inst\u271d : TopologicalGroup H f : G \u2192* H hf : Continuous \u2191f h_surj : Surjective \u2191f h_prop : Tendsto (\u2191f) (cocompact G) (cocompact H) \u22a2 \u2200 \u2983K : Set H\u2984, IsCompact K \u2192 \u2191\u2191(map (\u2191f) \u03bc) K < \u22a4 ** intro K hK ** \ud835\udd5c : Type u_1 G : Type u_2 H\u271d : Type u_3 inst\u271d\u00b9\u00b2 : MeasurableSpace G inst\u271d\u00b9\u00b9 : MeasurableSpace H\u271d inst\u271d\u00b9\u2070 : Group G inst\u271d\u2079 : TopologicalSpace G \u03bc : Measure G inst\u271d\u2078 : IsHaarMeasure \u03bc inst\u271d\u2077 : BorelSpace G inst\u271d\u2076 : TopologicalGroup G H : Type u_4 inst\u271d\u2075 : Group H inst\u271d\u2074 : TopologicalSpace H inst\u271d\u00b3 : MeasurableSpace H inst\u271d\u00b2 : BorelSpace H inst\u271d\u00b9 : T2Space H inst\u271d : TopologicalGroup H f : G \u2192* H hf : Continuous \u2191f h_surj : Surjective \u2191f h_prop : Tendsto (\u2191f) (cocompact G) (cocompact H) K : Set H hK : IsCompact K \u22a2 \u2191\u2191(map (\u2191f) \u03bc) K < \u22a4 ** rw [map_apply hf.measurable hK.measurableSet] ** \ud835\udd5c : Type u_1 G : Type u_2 H\u271d : Type u_3 inst\u271d\u00b9\u00b2 : MeasurableSpace G inst\u271d\u00b9\u00b9 : MeasurableSpace H\u271d inst\u271d\u00b9\u2070 : Group G inst\u271d\u2079 : TopologicalSpace G \u03bc : Measure G inst\u271d\u2078 : IsHaarMeasure \u03bc inst\u271d\u2077 : BorelSpace G inst\u271d\u2076 : TopologicalGroup G H : Type u_4 inst\u271d\u2075 : Group H inst\u271d\u2074 : TopologicalSpace H inst\u271d\u00b3 : MeasurableSpace H inst\u271d\u00b2 : BorelSpace H inst\u271d\u00b9 : T2Space H inst\u271d : TopologicalGroup H f : G \u2192* H hf : Continuous \u2191f h_surj : Surjective \u2191f h_prop : Tendsto (\u2191f) (cocompact G) (cocompact H) K : Set H hK : IsCompact K \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' K) < \u22a4 ** exact IsCompact.measure_lt_top ((\u27e8\u27e8f, hf\u27e9, h_prop\u27e9 : CocompactMap G H).isCompact_preimage hK) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mem_lpMeasSubgroup_toLp_of_trim ** \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 Lp F p } \u22a2 Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p) \u2208 lpMeasSubgroup F m p \u03bc ** let hf_mem_\u2112p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f) ** \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 Lp F p } hf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f) \u22a2 Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p) \u2208 lpMeasSubgroup F m p \u03bc ** rw [mem_lpMeasSubgroup_iff_aeStronglyMeasurable'] ** \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 Lp F p } hf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f) \u22a2 AEStronglyMeasurable' m (\u2191\u2191(Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p))) \u03bc ** refine' AEStronglyMeasurable'.congr _ (Mem\u2112p.coeFn_toLp hf_mem_\u2112p).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\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 Lp F p } hf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f) \u22a2 AEStronglyMeasurable' m (\u2191\u2191f) \u03bc ** refine' aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim hm _ ** \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 Lp F p } hf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f) \u22a2 AEStronglyMeasurable' m (\u2191\u2191f) (Measure.trim \u03bc hm) ** exact Lp.aestronglyMeasurable f ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_sum_indicator_atTop_iff ** \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop ** have h\u2081 := (martingale_martingalePart hf hint).ae_not_tendsto_atTop_atTop\n  (martingalePart_bdd_difference \u2131 hbdd) ** \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop ** have h\u2082 := (martingale_martingalePart hf hint).ae_not_tendsto_atTop_atBot\n  (martingalePart_bdd_difference \u2131 hbdd) ** \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop ** have h\u2083 : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, 0 \u2264 (\u03bc[f (n + 1) - f n|\u2131 n]) \u03c9 := by\n  refine' ae_all_iff.2 fun n => condexp_nonneg _\n  filter_upwards [ae_all_iff.1 hfmono n] with \u03c9 h\u03c9 using sub_nonneg.2 h\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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop ** filter_upwards [h\u2081, h\u2082, h\u2083, hfmono] with \u03c9 h\u03c9\u2081 h\u03c9\u2082 h\u03c9\u2083 h\u03c9\u2084 ** 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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 \u22a2 Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop ** constructor <;> intro ht ** \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 ** refine' ae_all_iff.2 fun n => condexp_nonneg _ ** \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot n : \u2115 \u22a2 0 \u2264\u1d50[\u03bc] f (n + 1) - f n ** filter_upwards [ae_all_iff.1 hfmono n] with \u03c9 h\u03c9 using sub_nonneg.2 h\u03c9 ** case h.mp \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun n => f n \u03c9) atTop atTop \u22a2 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop ** refine' tendsto_atTop_atTop_of_monotone' _ _ ** case h.mp.refine'_2 \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun n => f n \u03c9) atTop atTop \u22a2 \u00acBddAbove (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9) ** rintro \u27e8b, hbdd\u27e9 ** case h.mp.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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun n => f n \u03c9) atTop atTop b : \u211d hbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9) \u22a2 False ** rw [\u2190 tendsto_neg_atBot_iff] at ht ** case h.mp.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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun x => -f x \u03c9) atTop atBot b : \u211d hbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9) \u22a2 False ** simp only [martingalePart, sub_eq_add_neg] at h\u03c9\u2081 ** case h.mp.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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun x => -f x \u03c9) atTop atBot b : \u211d hbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9) h\u03c9\u2081 : \u00acTendsto (fun n => (f n + -predictablePart f \u2131 \u03bc n) \u03c9) atTop atTop \u22a2 False ** exact h\u03c9\u2081 (tendsto_atTop_add_right_of_le _ (-b) (tendsto_neg_atBot_iff.1 ht) fun n =>\n  neg_le_neg (hbdd \u27e8n, rfl\u27e9)) ** case h.mp.refine'_1 \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun n => f n \u03c9) atTop atTop \u22a2 Monotone fun n => predictablePart f \u2131 \u03bc n \u03c9 ** intro n m hnm ** case h.mp.refine'_1 \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun n => f n \u03c9) atTop atTop n m : \u2115 hnm : n \u2264 m \u22a2 (fun n => predictablePart f \u2131 \u03bc n \u03c9) n \u2264 (fun n => predictablePart f \u2131 \u03bc n \u03c9) m ** simp only [predictablePart, Finset.sum_apply] ** case h.mp.refine'_1 \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun n => f n \u03c9) atTop atTop n m : \u2115 hnm : n \u2264 m \u22a2 \u2211 c in Finset.range n, (\u03bc[f (c + 1) - f c|\u2191\u2131 c]) \u03c9 \u2264 \u2211 c in Finset.range m, (\u03bc[f (c + 1) - f c|\u2191\u2131 c]) \u03c9 ** refine' Finset.sum_mono_set_of_nonneg h\u03c9\u2083 (Finset.range_mono hnm) ** case h.mpr \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop \u22a2 Tendsto (fun n => f n \u03c9) atTop atTop ** refine' tendsto_atTop_atTop_of_monotone' (monotone_nat_of_le_succ h\u03c9\u2084) _ ** case h.mpr \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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop \u22a2 \u00acBddAbove (Set.range fun n => f n \u03c9) ** rintro \u27e8b, hbdd\u27e9 ** case h.mpr.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 hfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 hf : Adapted \u2131 f hint : \u2200 (n : \u2115), Integrable (f n) hbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R h\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 \u03c9 : \u03a9 h\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop h\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot h\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9 h\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9 ht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop b : \u211d hbdd : b \u2208 upperBounds (Set.range fun n => f n \u03c9) \u22a2 False ** exact h\u03c9\u2082 ((tendsto_atBot_add_left_of_ge _ b fun n =>\n  hbdd \u27e8n, rfl\u27e9) <| tendsto_neg_atBot_iff.2 ht) ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.eval\u2082Hom_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 : \u03c4 \u2192 S h : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 \u2191(eval\u2082Hom f g) (\u2191(bind\u2081 h) \u03c6) = \u2191(eval\u2082Hom f fun i => \u2191(eval\u2082Hom f g) (h i)) \u03c6 ** rw [hom_bind\u2081, eval\u2082Hom_comp_C] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PartrecToTM2.trStmts\u2081_supports ** S : Finset \u039b' q : \u039b' H\u2081 : \u039b'.Supports S q HS\u2081 : trStmts\u2081 q \u2286 S \u22a2 Supports (trStmts\u2081 q) S ** have W := fun {q} => trStmts\u2081_self q ** S : Finset \u039b' q : \u039b' H\u2081 : \u039b'.Supports S q HS\u2081 : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q \u22a2 Supports (trStmts\u2081 q) S ** induction' q with _ _ _ q q_ih _ _ q q_ih q q_ih _ _ q q_ih q q_ih q q_ih q\u2081 q\u2082 q\u2081_ih q\u2082_ih _ <;>\n  simp [trStmts\u2081, -Finset.singleton_subset_iff] at HS\u2081 \u22a2 ** case move S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q p\u271d : \u0393' \u2192 Bool k\u2081\u271d k\u2082\u271d : K' q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) HS\u2081 : insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q) \u2286 S \u22a2 Supports (insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q)) S  case clear S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q p\u271d : \u0393' \u2192 Bool k\u271d : K' q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.clear p\u271d k\u271d q) HS\u2081 : insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q) \u2286 S \u22a2 Supports (insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q)) S  case copy S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.copy q) HS\u2081 : insert (\u039b'.copy q) (trStmts\u2081 q) \u2286 S \u22a2 Supports (insert (\u039b'.copy q) (trStmts\u2081 q)) S  case push S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q k\u271d : K' s\u271d : Option \u0393' \u2192 Option \u0393' q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.push k\u271d s\u271d q) HS\u2081 : insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q) \u2286 S \u22a2 Supports (insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)) S  case read S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q : Option \u0393' \u2192 \u039b' q_ih : \u2200 (a : Option \u0393'), \u039b'.Supports S (q a) \u2192 trStmts\u2081 (q a) \u2286 S \u2192 Supports (trStmts\u2081 (q a)) S H\u2081 : \u039b'.Supports S (\u039b'.read q) HS\u2081 : insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2286 S \u22a2 Supports (insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))) S  case succ S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.succ q) HS\u2081 : insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)) \u2286 S \u22a2 Supports (insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) S  case pred S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S \u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S  case ret S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q k\u271d : Cont' H\u2081 : \u039b'.Supports S (\u039b'.ret k\u271d) HS\u2081 : {\u039b'.ret k\u271d} \u2286 S \u22a2 Supports {\u039b'.ret k\u271d} S ** any_goals\n  cases' Finset.insert_subset_iff.1 HS\u2081 with h\u2081 h\u2082\n  first | have h\u2083 := h\u2082 W | try simp [Finset.subset_iff] at h\u2082 ** case pred S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S \u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S ** cases' Finset.insert_subset_iff.1 HS\u2081 with h\u2081 h\u2082 ** case pred.intro S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S h\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S h\u2082 : insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) \u2286 S \u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S ** first | have h\u2083 := h\u2082 W | try simp [Finset.subset_iff] at h\u2082 ** case push.intro S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q k\u271d : K' s\u271d : Option \u0393' \u2192 Option \u0393' q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.push k\u271d s\u271d q) HS\u2081 : insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q) \u2286 S h\u2081 : \u039b'.push k\u271d s\u271d q \u2208 S h\u2082 : trStmts\u2081 q \u2286 S \u22a2 Supports (insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)) S ** have h\u2083 := h\u2082 W ** case pred.intro S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S h\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S h\u2082 : insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) \u2286 S \u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S ** try simp [Finset.subset_iff] at h\u2082 ** case pred.intro S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S h\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S h\u2082 : insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) \u2286 S \u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S ** simp [Finset.subset_iff] at h\u2082 ** case move.intro S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q p\u271d : \u0393' \u2192 Bool k\u2081\u271d k\u2082\u271d : K' q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) HS\u2081 : insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q) \u2286 S h\u2081 : \u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q \u2208 S h\u2082 : trStmts\u2081 q \u2286 S h\u2083 : q \u2208 S \u22a2 Supports (insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q)) S ** exact supports_insert.2 \u27e8\u27e8fun _ => h\u2083, fun _ => h\u2081\u27e9, q_ih H\u2081 h\u2082\u27e9 ** case clear.intro S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q p\u271d : \u0393' \u2192 Bool k\u271d : K' q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.clear p\u271d k\u271d q) HS\u2081 : insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q) \u2286 S h\u2081 : \u039b'.clear p\u271d k\u271d q \u2208 S h\u2082 : trStmts\u2081 q \u2286 S h\u2083 : q \u2208 S \u22a2 Supports (insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q)) S ** exact supports_insert.2 \u27e8\u27e8fun _ => h\u2083, fun _ => h\u2081\u27e9, q_ih H\u2081 h\u2082\u27e9 ** case copy.intro S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.copy q) HS\u2081 : insert (\u039b'.copy q) (trStmts\u2081 q) \u2286 S h\u2081 : \u039b'.copy q \u2208 S h\u2082 : trStmts\u2081 q \u2286 S h\u2083 : q \u2208 S \u22a2 Supports (insert (\u039b'.copy q) (trStmts\u2081 q)) S ** exact supports_insert.2 \u27e8\u27e8fun _ => h\u2081, fun _ => h\u2083\u27e9, q_ih H\u2081 h\u2082\u27e9 ** case push.intro S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q k\u271d : K' s\u271d : Option \u0393' \u2192 Option \u0393' q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.push k\u271d s\u271d q) HS\u2081 : insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q) \u2286 S h\u2081 : \u039b'.push k\u271d s\u271d q \u2208 S h\u2082 : trStmts\u2081 q \u2286 S h\u2083 : q \u2208 S \u22a2 Supports (insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)) S ** exact supports_insert.2 \u27e8\u27e8fun _ => h\u2083, fun _ => h\u2083\u27e9, q_ih H\u2081 h\u2082\u27e9 ** case read.intro S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q : Option \u0393' \u2192 \u039b' q_ih : \u2200 (a : Option \u0393'), \u039b'.Supports S (q a) \u2192 trStmts\u2081 (q a) \u2286 S \u2192 Supports (trStmts\u2081 (q a)) S H\u2081 : \u039b'.Supports S (\u039b'.read q) HS\u2081 : insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2286 S h\u2081 : \u039b'.read q \u2208 S h\u2082 : \u2200 \u2983x : \u039b'\u2984 (x_1 : Option \u0393'), x \u2208 trStmts\u2081 (q x_1) \u2192 x \u2208 S \u22a2 Supports (insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))) S ** refine' supports_insert.2 \u27e8fun _ => h\u2082 _ W, _\u27e9 ** case read.intro S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q : Option \u0393' \u2192 \u039b' q_ih : \u2200 (a : Option \u0393'), \u039b'.Supports S (q a) \u2192 trStmts\u2081 (q a) \u2286 S \u2192 Supports (trStmts\u2081 (q a)) S H\u2081 : \u039b'.Supports S (\u039b'.read q) HS\u2081 : insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2286 S h\u2081 : \u039b'.read q \u2208 S h\u2082 : \u2200 \u2983x : \u039b'\u2984 (x_1 : Option \u0393'), x \u2208 trStmts\u2081 (q x_1) \u2192 x \u2208 S \u22a2 Supports (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) S ** exact supports_biUnion.2 fun _ => q_ih _ (H\u2081 _) fun _ h => h\u2082 _ h ** case succ.intro S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.succ q) HS\u2081 : insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)) \u2286 S h\u2081 : \u039b'.succ q \u2208 S h\u2082 : unrev q \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q \u2192 a \u2208 S \u22a2 Supports (insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) S ** refine' supports_insert.2 \u27e8\u27e8fun _ => h\u2081, fun _ => h\u2082.1, fun _ => h\u2082.1\u27e9, _\u27e9 ** case succ.intro S : Finset \u039b' q\u271d : \u039b' H\u2081\u271d : \u039b'.Supports S q\u271d HS\u2081\u271d : trStmts\u2081 q\u271d \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q : \u039b' q_ih : \u039b'.Supports S q \u2192 trStmts\u2081 q \u2286 S \u2192 Supports (trStmts\u2081 q) S H\u2081 : \u039b'.Supports S (\u039b'.succ q) HS\u2081 : insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)) \u2286 S h\u2081 : \u039b'.succ q \u2208 S h\u2082 : unrev q \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q \u2192 a \u2208 S \u22a2 Supports (insert (unrev q) (trStmts\u2081 q)) S ** exact supports_insert.2 \u27e8\u27e8fun _ => h\u2082.2 _ W, fun _ => h\u2082.1\u27e9, q_ih H\u2081 h\u2082.2\u27e9 ** case pred.intro S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S h\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S h\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S \u22a2 Supports (insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082))) S ** refine' supports_insert.2 \u27e8\u27e8fun _ => h\u2081, fun _ => h\u2082.2 _ (Or.inl W), fun _ => h\u2082.1, fun _ => h\u2082.1\u27e9, _\u27e9 ** case pred.intro S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S h\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S h\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S \u22a2 Supports (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) S ** refine' supports_insert.2 \u27e8\u27e8fun _ => h\u2082.2 _ (Or.inr W), fun _ => h\u2082.1\u27e9, _\u27e9 ** case pred.intro S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S h\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S h\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S \u22a2 Supports (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082) S ** refine' supports_union.2 \u27e8_, _\u27e9 ** case pred.intro.refine'_1 S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S h\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S h\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S \u22a2 Supports (trStmts\u2081 q\u2081) S ** exact q\u2081_ih H\u2081.1 fun _ h => h\u2082.2 _ (Or.inl h) ** case pred.intro.refine'_2 S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q q\u2081 q\u2082 : \u039b' q\u2081_ih : \u039b'.Supports S q\u2081 \u2192 trStmts\u2081 q\u2081 \u2286 S \u2192 Supports (trStmts\u2081 q\u2081) S q\u2082_ih : \u039b'.Supports S q\u2082 \u2192 trStmts\u2081 q\u2082 \u2286 S \u2192 Supports (trStmts\u2081 q\u2082) S H\u2081 : \u039b'.Supports S (\u039b'.pred q\u2081 q\u2082) HS\u2081 : insert (\u039b'.pred q\u2081 q\u2082) (insert (unrev q\u2082) (trStmts\u2081 q\u2081 \u222a trStmts\u2081 q\u2082)) \u2286 S h\u2081 : \u039b'.pred q\u2081 q\u2082 \u2208 S h\u2082 : unrev q\u2082 \u2208 S \u2227 \u2200 (a : \u039b'), a \u2208 trStmts\u2081 q\u2081 \u2228 a \u2208 trStmts\u2081 q\u2082 \u2192 a \u2208 S \u22a2 Supports (trStmts\u2081 q\u2082) S ** exact q\u2082_ih H\u2081.2 fun _ h => h\u2082.2 _ (Or.inr h) ** case ret S : Finset \u039b' q : \u039b' H\u2081\u271d : \u039b'.Supports S q HS\u2081\u271d : trStmts\u2081 q \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q k\u271d : Cont' H\u2081 : \u039b'.Supports S (\u039b'.ret k\u271d) HS\u2081 : {\u039b'.ret k\u271d} \u2286 S \u22a2 Supports {\u039b'.ret k\u271d} S ** exact supports_singleton.2 (ret_supports H\u2081) ** Qed",
        "informal": ""
    },
    {
        "formal": "ComputablePred.to_re ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 p : \u03b1 \u2192 Prop hp : ComputablePred p \u22a2 RePred p ** obtain \u27e8f, hf, rfl\u27e9 := computable_iff.1 hp ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 Bool hf : Computable f hp : ComputablePred fun a => f a = true \u22a2 RePred fun a => f a = true ** unfold RePred ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 Bool hf : Computable f hp : ComputablePred fun a => f a = true \u22a2 Partrec fun a => Part.assert ((fun a => f a = true) a) fun x => Part.some () ** dsimp only [] ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 Bool hf : Computable f hp : ComputablePred fun a => f a = true \u22a2 Partrec fun a => Part.assert (f a = true) fun x => Part.some () ** refine'\n  (Partrec.cond hf (Decidable.Partrec.const' (Part.some ())) Partrec.none).of_eq fun n =>\n    Part.ext fun a => _ ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 Bool hf : Computable f hp : ComputablePred fun a => f a = true n : \u03b1 a : Unit \u22a2 (a \u2208 bif f n then Part.some () else Part.none) \u2194 a \u2208 Part.assert (f n = true) fun x => Part.some () ** cases a ** case intro.intro.unit \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 Bool hf : Computable f hp : ComputablePred fun a => f a = true n : \u03b1 \u22a2 (PUnit.unit \u2208 bif f n then Part.some () else Part.none) \u2194 PUnit.unit \u2208 Part.assert (f n = true) fun x => Part.some () ** cases f n <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image\u2082_inter_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 hf : Injective2 f \u22a2 \u2191(image\u2082 f s (t \u2229 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 \u03b2 hf : Injective2 f \u22a2 image2 f (\u2191s) (\u2191t \u2229 \u2191t') = image2 f \u2191s \u2191t \u2229 image2 f \u2191s \u2191t' ** exact image2_inter_right hf ** Qed",
        "informal": ""
    },
    {
        "formal": "blimsup_cthickening_ae_le_of_eventually_mul_le ** \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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) 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 ** let R\u2081 i := max 0 (r\u2081 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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 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 ** let R\u2082 i := max 0 (r\u2082 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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (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 ** have hRp : 0 \u2264 R\u2081 := fun i => le_max_left 0 (r\u2081 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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 \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 ** replace hMr : \u2200\u1da0 i in atTop, M * R\u2081 i \u2264 R\u2082 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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 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 ** simp_rw [\u2190 cthickening_max_zero (r\u2081 _), \u2190 cthickening_max_zero (r\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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i \u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\u1d50[\u03bc]     blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p ** cases' le_or_lt 1 M with hM' hM' ** case hMr \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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 \u22a2 \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i ** refine' hMr.mono fun i hi => _ ** case hMr \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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 i : \u2115 hi : M * r\u2081 i \u2264 r\u2082 i \u22a2 M * R\u2081 i \u2264 R\u2082 i ** rw [mul_max_of_nonneg _ _ hM.le, mul_zero] ** case hMr \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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 i : \u2115 hi : M * r\u2081 i \u2264 r\u2082 i \u22a2 max 0 (M * r\u2081 i) \u2264 R\u2082 i ** exact max_le_max (le_refl 0) hi ** case 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 M : \u211d hM : 0 < M r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i hM' : 1 \u2264 M \u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\u1d50[\u03bc]     blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p ** apply HasSubset.Subset.eventuallyLE ** case inl.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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i hM' : 1 \u2264 M \u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2286     blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p ** change _ \u2264 _ ** case inl.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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i hM' : 1 \u2264 M \u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264     blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p ** refine' mono_blimsup' (hMr.mono fun i hi _ => cthickening_mono _ (s i)) ** case inl.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\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i hM' : 1 \u2264 M i : \u2115 hi : M * R\u2081 i \u2264 R\u2082 i x\u271d : p i \u22a2 max 0 (r\u2081 i) \u2264 max 0 (r\u2082 i) ** exact (le_mul_of_one_le_left (hRp i) hM').trans hi ** case 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 M : \u211d hM : 0 < M r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i hM' : M < 1 \u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (s i)) atTop p \u2264\u1d50[\u03bc]     blimsup (fun i => cthickening (max 0 (r\u2082 i)) (s i)) atTop p ** simp only [\u2190 @cthickening_closure _ _ _ (s _)] ** case 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 M : \u211d hM : 0 < M r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i hM' : M < 1 \u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (closure (s i))) atTop p \u2264\u1d50[\u03bc]     blimsup (fun i => cthickening (max 0 (r\u2082 i)) (closure (s i))) atTop p ** have hs : \u2200 i, IsClosed (closure (s i)) := fun i => isClosed_closure ** case 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 M : \u211d hM : 0 < M r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) R\u2081 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2081 i) R\u2082 : \u2115 \u2192 \u211d := fun i => max 0 (r\u2082 i) hRp : 0 \u2264 R\u2081 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * R\u2081 i \u2264 R\u2082 i hM' : M < 1 hs : \u2200 (i : \u2115), IsClosed (closure (s i)) \u22a2 blimsup (fun i => cthickening (max 0 (r\u2081 i)) (closure (s i))) atTop p \u2264\u1d50[\u03bc]     blimsup (fun i => cthickening (max 0 (r\u2082 i)) (closure (s i))) atTop p ** exact blimsup_cthickening_ae_le_of_eventually_mul_le_aux \u03bc p hs\n  (tendsto_nhds_max_right hr) hRp hM hM' hMr ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.QuasiMeasurePreserving.prod_of_left ** \u03b1\u271d : Type u_1 \u03b1' : Type u_2 \u03b2\u271d : Type u_3 \u03b2' : Type u_4 \u03b3\u271d : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2\u271d inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3\u271d \u03bc\u271d \u03bc' : Measure \u03b1\u271d \u03bd\u271d \u03bd' : Measure \u03b2\u271d \u03c4\u271d : Measure \u03b3\u271d inst\u271d\u2075 : NormedAddCommGroup E \u03b1 : Type u_7 \u03b2 : Type u_8 \u03b3 : Type u_9 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 \u03c4 : Measure \u03b3 hf : Measurable f inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : SigmaFinite \u03bd h2f : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, QuasiMeasurePreserving fun x => f (x, y) \u22a2 QuasiMeasurePreserving f ** convert (QuasiMeasurePreserving.prod_of_right (hf.comp measurable_swap) h2f).comp\n    ((measurable_swap.measurePreserving (\u03bd.prod \u03bc)).symm\n        MeasurableEquiv.prodComm).quasiMeasurePreserving ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos' ** \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 hn :   \u00ac\u2200 (n : \u2115),       \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264           restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** by_cases s \u2264[i] 0 ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 hn :   \u00ac\u2200 (n : \u2115),       \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264           restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h : \u00acrestrict s i \u2264 restrict 0 i \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** push_neg at hn ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** set k := Nat.find hn ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** have hk\u2082 : s \u2264[i \\ \u22c3 l < k, restrictNonposSeq s i l] 0 := Nat.find_spec hn ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** have hmeas : MeasurableSet (\u22c3 (l : \u2115) (_ : l < k), restrictNonposSeq s i l) :=\n  MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _ ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** refine' \u27e8i \\ \u22c3 l < k, restrictNonposSeq s i l, hi\u2081.diff hmeas, Set.diff_subset _ _, hk\u2082, _\u27e9 ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2191s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) < 0 ** rw [of_diff hmeas hi\u2081, s.of_disjoint_iUnion_nat] ** case pos \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 hn :   \u00ac\u2200 (n : \u2115),       \u00acrestrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264           restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h : restrict s i \u2264 restrict 0 i \u22a2 \u2203 j, MeasurableSet j \u2227 j \u2286 i \u2227 restrict s j \u2264 restrict 0 j \u2227 \u2191s j < 0 ** exact \u27e8i, hi\u2081, Set.Subset.refl _, h, hi\u2082\u27e9 ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2191s i - \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0 ** have h\u2081 : \u2200 l < k, 0 \u2264 s (restrictNonposSeq s i l) := by\n  intro l hl\n  refine' le_of_lt (measure_of_restrictNonposSeq h _ _)\n  refine' mt (restrict_le_zero_subset _ (hi\u2081.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl)\n  exact\n    MeasurableSet.iUnion fun _ =>\n      MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _ ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2191s i - \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0 ** suffices 0 \u2264 \u2211' l : \u2115, s (\u22c3 _ : l < k, restrictNonposSeq s i l) by\n  rw [sub_neg]\n  exact lt_of_lt_of_le hi\u2082 this ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 0 \u2264 \u2211' (l : \u2115), \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** refine' tsum_nonneg _ ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2200 (i_1 : \u2115), 0 \u2264 \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) ** intro l ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 \u22a2 0 \u2264 \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** by_cases l < k ** \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** intro l hl ** \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 hl : l < k \u22a2 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** refine' le_of_lt (measure_of_restrictNonposSeq h _ _) ** \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 hl : l < k \u22a2 \u00acrestrict s (i \\ \u22c3 k, \u22c3 (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u2264       restrict 0 (i \\ \u22c3 k, \u22c3 (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ** refine' mt (restrict_le_zero_subset _ (hi\u2081.diff _) (Set.Subset.refl _)) (Nat.find_min hn hl) ** \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 hl : l < k \u22a2 MeasurableSet (\u22c3 k, \u22c3 (_ : k < l), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ** exact\n  MeasurableSet.iUnion fun _ =>\n    MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _ ** \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) this : 0 \u2264 \u2211' (l : \u2115), \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2191s i - \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) < 0 ** rw [sub_neg] ** \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) this : 0 \u2264 \u2211' (l : \u2115), \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2191s i < \u2211' (i_1 : \u2115), \u2191s (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) ** exact lt_of_lt_of_le hi\u2082 this ** case pos \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h\u271d : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 h : l < k \u22a2 0 \u2264 \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** convert h\u2081 _ h ** case h.e'_4.h.e'_7 \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h\u271d : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 h : l < k \u22a2 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l =     MeasureTheory.SignedMeasure.restrictNonposSeq s i l ** ext x ** case h.e'_4.h.e'_7.h \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h\u271d : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 h : l < k x : \u03b1 \u22a2 x \u2208 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2194     x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l ** rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp] ** case h.e'_4.h.e'_7.h \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h\u271d : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 h : l < k x : \u03b1 \u22a2 x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2192 l < k ** exact fun _ => h ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h\u271d : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 h : \u00acl < k \u22a2 0 \u2264 \u2191s (\u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** convert le_of_eq s.empty.symm ** case h.e'_4.h.e'_7 \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h\u271d : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 h : \u00acl < k \u22a2 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l = \u2205 ** ext ** case h.e'_4.h.e'_7.h \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h\u271d : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 h : \u00acl < k x\u271d : \u03b1 \u22a2 x\u271d \u2208 \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2194 x\u271d \u2208 \u2205 ** simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff] ** case h.e'_4.h.e'_7.h \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h\u271d : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) h\u2081 : \u2200 (l : \u2115), l < k \u2192 0 \u2264 \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l) l : \u2115 h : \u00acl < k x\u271d : \u03b1 \u22a2 l < Nat.find hn \u2192 \u00acx\u271d \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i l ** exact fun h' => False.elim (h h') ** case neg.hf\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 : SignedMeasure \u03b1 i j : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2200 (i_1 : \u2115), MeasurableSet (\u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1) ** intro ** case neg.hf\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 : SignedMeasure \u03b1 i j : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) i\u271d : \u2115 \u22a2 MeasurableSet (\u22c3 (_ : i\u271d < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i\u271d) ** exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _ ** case neg.hf\u2082 \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 Pairwise (Disjoint on fun l => \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ** intro a b hab ** case neg.hf\u2082 \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a b : \u2115 hab : a \u2260 b \u22a2 (Disjoint on fun l => \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a b ** refine' Set.disjoint_iUnion_left.mpr fun _ => _ ** case neg.hf\u2082 \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a b : \u2115 hab : a \u2260 b x\u271d : a < k \u22a2 Disjoint (MeasureTheory.SignedMeasure.restrictNonposSeq s i a)     ((fun l => \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) b) ** refine' Set.disjoint_iUnion_right.mpr fun _ => _ ** case neg.hf\u2082 \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a b : \u2115 hab : a \u2260 b x\u271d\u00b9 : a < k x\u271d : b < k \u22a2 Disjoint (MeasureTheory.SignedMeasure.restrictNonposSeq s i a) (MeasureTheory.SignedMeasure.restrictNonposSeq s i b) ** exact restrictNonposSeq_disjoint hab ** case neg \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l \u2286 i ** apply Set.iUnion_subset ** case neg.h \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u22a2 \u2200 (i_1 : \u2115), \u22c3 (_ : i_1 < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i i_1 \u2286 i ** intro a x ** case neg.h \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a : \u2115 x : \u03b1 \u22a2 x \u2208 \u22c3 (_ : a < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i a \u2192 x \u2208 i ** simp only [and_imp, exists_prop, Set.mem_iUnion] ** case neg.h \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a : \u2115 x : \u03b1 \u22a2 a < Nat.find hn \u2192 x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i a \u2192 x \u2208 i ** intro _ hx ** case neg.h \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\u2081 : MeasurableSet i hi\u2082 : \u2191s i < 0 h : \u00acrestrict s i \u2264 restrict 0 i hn :   \u2203 n,     restrict s (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264       restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) k : \u2115 := Nat.find hn hk\u2082 :   restrict s (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) \u2264     restrict 0 (i \\ \u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) hmeas : MeasurableSet (\u22c3 l, \u22c3 (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) a : \u2115 x : \u03b1 a\u271d : a < Nat.find hn hx : x \u2208 MeasureTheory.SignedMeasure.restrictNonposSeq s i a \u22a2 x \u2208 i ** exact restrictNonposSeq_subset _ hx ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image_add_left_Icc ** \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) (Icc a b) = Icc (c + a) (c + b) ** rw [\u2190 map_add_left_Icc, map_eq_image, addLeftEmbedding, Embedding.coeFn_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lpMeas.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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 \u22a2 \u2191\u2191\u2191f =\u1d50[\u03bc] 0 ** obtain \u27e8g, hg_sm, hfg\u27e9 := lpMeas.ae_fin_strongly_measurable' hm f hp_ne_zero hp_ne_top ** case intro.intro \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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2191\u2191\u2191f =\u1d50[\u03bc] g \u22a2 \u2191\u2191\u2191f =\u1d50[\u03bc] 0 ** refine' hfg.trans _ ** case intro.intro \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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2191\u2191\u2191f =\u1d50[\u03bc] g \u22a2 g =\u1d50[\u03bc] 0 ** unfold Filter.EventuallyEq at hfg ** case intro.intro \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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x \u22a2 g =\u1d50[\u03bc] 0 ** refine' ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim hm _ _ hg_sm ** case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s ** intro s hs h\u03bcs ** case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 IntegrableOn g s ** have hfg_restrict : f =\u1d50[\u03bc.restrict s] g := ae_restrict_of_ae hfg ** case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 hfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g \u22a2 IntegrableOn g s ** rw [IntegrableOn, integrable_congr hfg_restrict.symm] ** case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 hfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g \u22a2 Integrable \u2191\u2191\u2191f ** exact hf_int_finite s hs h\u03bcs ** case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, g x \u2202\u03bc = 0 ** intro s hs h\u03bcs ** case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, g x \u2202\u03bc = 0 ** have hfg_restrict : f =\u1d50[\u03bc.restrict s] g := ae_restrict_of_ae hfg ** case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 hfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g \u22a2 \u222b (x : \u03b1) in s, g x \u2202\u03bc = 0 ** rw [integral_congr_ae hfg_restrict.symm] ** case intro.intro.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 lpMeas E' \ud835\udd5c m p \u03bc } 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\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f x \u2202\u03bc = 0 g : \u03b1 \u2192 E' hg_sm : FinStronglyMeasurable g (Measure.trim \u03bc hm) hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191f x = g x s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 hfg_restrict : \u2191\u2191\u2191f =\u1d50[Measure.restrict \u03bc s] g \u22a2 \u222b (a : \u03b1) in s, \u2191\u2191\u2191f a \u2202\u03bc = 0 ** exact hf_zero s hs h\u03bcs ** Qed",
        "informal": ""
    },
    {
        "formal": "VitaliFamily.ae_tendsto_average ** \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => \u2a0d (y : \u03b1) in a, f y \u2202\u03bc) (filterAt v x) (\ud835\udcdd (f x)) ** filter_upwards [v.ae_tendsto_average_norm_sub hf, v.ae_eventually_measure_pos] with x hx h'x ** case h \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f x : \u03b1 hx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a \u22a2 Tendsto (fun a => \u2a0d (y : \u03b1) in a, f y \u2202\u03bc) (filterAt v x) (\ud835\udcdd (f x)) ** rw [tendsto_iff_norm_sub_tendsto_zero] ** case h \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f x : \u03b1 hx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a \u22a2 Tendsto (fun e => \u2016\u2a0d (y : \u03b1) in e, f y \u2202\u03bc - f x\u2016) (filterAt v x) (\ud835\udcdd 0) ** refine' squeeze_zero' (eventually_of_forall fun a => norm_nonneg _) _ hx ** case h \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f x : \u03b1 hx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a \u22a2 \u2200\u1da0 (t : Set \u03b1) in filterAt v x, \u2016\u2a0d (y : \u03b1) in t, f y \u2202\u03bc - f x\u2016 \u2264 \u2a0d (y : \u03b1) in t, \u2016f y - f x\u2016 \u2202\u03bc ** filter_upwards [h'x, v.eventually_measure_lt_top x, v.eventually_filterAt_integrableOn x hf]\n  with a ha h'a h''a ** case h \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f x : \u03b1 hx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a a : Set \u03b1 ha : 0 < \u2191\u2191\u03bc a h'a : \u2191\u2191\u03bc a < \u22a4 h''a : IntegrableOn f a \u22a2 \u2016\u2a0d (y : \u03b1) in a, f y \u2202\u03bc - f x\u2016 \u2264 \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc ** nth_rw 1 [\u2190 setAverage_const ha.ne' h'a.ne (f x)] ** case h \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f x : \u03b1 hx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a a : Set \u03b1 ha : 0 < \u2191\u2191\u03bc a h'a : \u2191\u2191\u03bc a < \u22a4 h''a : IntegrableOn f a \u22a2 \u2016\u2a0d (y : \u03b1) in a, f y \u2202\u03bc - \u2a0d (x_1 : \u03b1) in a, f x \u2202\u03bc\u2016 \u2264 \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc ** simp_rw [setAverage_eq'] ** case h \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f x : \u03b1 hx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a a : Set \u03b1 ha : 0 < \u2191\u2191\u03bc a h'a : \u2191\u2191\u03bc a < \u22a4 h''a : IntegrableOn f a \u22a2 \u2016\u222b (y : \u03b1), f y \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a - \u222b (x_1 : \u03b1), f x \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a\u2016 \u2264     \u222b (y : \u03b1), \u2016f y - f x\u2016 \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a ** rw [\u2190 integral_sub] ** case h \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f x : \u03b1 hx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a a : Set \u03b1 ha : 0 < \u2191\u2191\u03bc a h'a : \u2191\u2191\u03bc a < \u22a4 h''a : IntegrableOn f a \u22a2 \u2016\u222b (a : \u03b1), f a - f x \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a\u2016 \u2264 \u222b (y : \u03b1), \u2016f y - f x\u2016 \u2202(\u2191\u2191\u03bc a)\u207b\u00b9 \u2022 Measure.restrict \u03bc a ** exact norm_integral_le_integral_norm _ ** case h.hf \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f x : \u03b1 hx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a a : Set \u03b1 ha : 0 < \u2191\u2191\u03bc a h'a : \u2191\u2191\u03bc a < \u22a4 h''a : IntegrableOn f a \u22a2 Integrable fun y => f y ** exact (integrable_inv_smul_measure ha.ne' h'a.ne).2 h''a ** case h.hg \u03b1 : Type u_1 inst\u271d\u2077 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d\u00b2 : IsLocallyFiniteMeasure \u03c1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f x : \u03b1 hx : Tendsto (fun a => \u2a0d (y : \u03b1) in a, \u2016f y - f x\u2016 \u2202\u03bc) (filterAt v x) (\ud835\udcdd 0) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a a : Set \u03b1 ha : 0 < \u2191\u2191\u03bc a h'a : \u2191\u2191\u03bc a < \u22a4 h''a : IntegrableOn f a \u22a2 Integrable fun x_1 => f x ** exact (integrable_inv_smul_measure ha.ne' h'a.ne).2 (integrableOn_const.2 (Or.inr h'a)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.mul_eq_one_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 s * t = 1 \u2194 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1 ** refine' \u27e8fun h => _, _\u27e9 ** case refine'_1 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 h : s * t = 1 \u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1 ** have hst : (s * t).Nonempty := h.symm.subst one_nonempty ** case refine'_1 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 h : s * t = 1 hst : Set.Nonempty (s * t) \u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1 ** obtain \u27e8a, ha\u27e9 := hst.of_image2_left ** case refine'_1.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 h : s * t = 1 hst : Set.Nonempty (s * t) a : \u03b1 ha : a \u2208 s \u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1 ** obtain \u27e8b, hb\u27e9 := hst.of_image2_right ** case refine'_1.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 h : s * t = 1 hst : Set.Nonempty (s * t) a : \u03b1 ha : a \u2208 s b : \u03b1 hb : b \u2208 t \u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1 ** have H : \u2200 {a b}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = (1 : \u03b1) := fun {a b} ha hb =>\n  h.subset <| mem_image2_of_mem ha hb ** case refine'_1.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 h : s * t = 1 hst : Set.Nonempty (s * t) a : \u03b1 ha : a \u2208 s b : \u03b1 hb : b \u2208 t H : \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = 1 \u22a2 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1 ** refine' \u27e8a, b, _, _, H ha hb\u27e9 <;> refine' eq_singleton_iff_unique_mem.2 \u27e8\u2039_\u203a, fun x hx => _\u27e9 ** case refine'_1.intro.intro.refine'_1 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 h : s * t = 1 hst : Set.Nonempty (s * t) a : \u03b1 ha : a \u2208 s b : \u03b1 hb : b \u2208 t H : \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = 1 x : \u03b1 hx : x \u2208 s \u22a2 x = a ** exact (eq_inv_of_mul_eq_one_left <| H hx hb).trans (inv_eq_of_mul_eq_one_left <| H ha hb) ** case refine'_1.intro.intro.refine'_2 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 h : s * t = 1 hst : Set.Nonempty (s * t) a : \u03b1 ha : a \u2208 s b : \u03b1 hb : b \u2208 t H : \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 t \u2192 a * b = 1 x : \u03b1 hx : x \u2208 t \u22a2 x = b ** exact (eq_inv_of_mul_eq_one_right <| H ha hx).trans (inv_eq_of_mul_eq_one_right <| H ha hb) ** case refine'_2 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 b, s = {a} \u2227 t = {b} \u2227 a * b = 1) \u2192 s * t = 1 ** rintro \u27e8b, c, rfl, rfl, h\u27e9 ** case refine'_2.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 b c : \u03b1 h : b * c = 1 \u22a2 {b} * {c} = 1 ** rw [singleton_mul_singleton, h, singleton_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "IsUnifLocDoublingMeasure.exists_eventually_forall_measure_closedBall_le_mul ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d \u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5) ** let C := doublingConstant \u03bc ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) \u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5) ** rcases lt_or_le K 1 with (hK | hK) ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc \u22a2 \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) ** intro n ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc n : \u2115 \u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) ** induction' n with n ih ** case succ \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc n : \u2115 ih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) \u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0,     \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ Nat.succ n * \u03b5)) \u2264 \u2191(C ^ Nat.succ n) * \u2191\u2191\u03bc (closedBall x \u03b5) ** replace ih := eventually_nhdsWithin_pos_mul_left (two_pos : 0 < (2 : \u211d)) ih ** case succ \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc n : \u2115 ih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0,     \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ Nat.succ n * \u03b5)) \u2264 \u2191(C ^ Nat.succ n) * \u2191\u2191\u03bc (closedBall x \u03b5) ** refine' (ih.and (exists_measure_closedBall_le_mul' \u03bc)).mono fun \u03b5 h\u03b5 x => _ ** case succ \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc n : \u2115 ih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227     \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5) x : \u03b1 \u22a2 \u2191\u2191\u03bc (closedBall x (2 ^ Nat.succ n * \u03b5)) \u2264 \u2191(C ^ Nat.succ n) * \u2191\u2191\u03bc (closedBall x \u03b5) ** calc\n  \u03bc (closedBall x ((2 : \u211d) ^ (n + 1) * \u03b5)) = \u03bc (closedBall x ((2 : \u211d) ^ n * (2 * \u03b5))) := by\n    rw [pow_succ', mul_assoc]\n  _ \u2264 \u2191(C ^ n) * \u03bc (closedBall x (2 * \u03b5)) := (h\u03b5.1 x)\n  _ \u2264 \u2191(C ^ n) * (C * \u03bc (closedBall x \u03b5)) := by gcongr; exact h\u03b5.2 x\n  _ = \u2191(C ^ (n + 1)) * \u03bc (closedBall x \u03b5) := by rw [\u2190 mul_assoc, pow_succ', ENNReal.coe_mul] ** case zero \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc \u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ Nat.zero * \u03b5)) \u2264 \u2191(C ^ Nat.zero) * \u2191\u2191\u03bc (closedBall x \u03b5) ** simp ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc n : \u2115 ih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227     \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5) x : \u03b1 \u22a2 \u2191\u2191\u03bc (closedBall x (2 ^ (n + 1) * \u03b5)) = \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) ** rw [pow_succ', mul_assoc] ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc n : \u2115 ih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227     \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5) x : \u03b1 \u22a2 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(C ^ n) * (\u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)) ** gcongr ** case bc \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc n : \u2115 ih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227     \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5) x : \u03b1 \u22a2 \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5) ** exact h\u03b5.2 x ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc n : \u2115 ih : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * (2 * \u03b5))) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x (2 * \u03b5))) \u2227     \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(doublingConstant \u03bc) * \u2191\u2191\u03bc (closedBall x \u03b5) x : \u03b1 \u22a2 \u2191(C ^ n) * (\u2191C * \u2191\u2191\u03bc (closedBall x \u03b5)) = \u2191(C ^ (n + 1)) * \u2191\u2191\u03bc (closedBall x \u03b5) ** rw [\u2190 mul_assoc, pow_succ', ENNReal.coe_mul] ** case inl \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : K < 1 \u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5) ** refine' \u27e81, _\u27e9 ** case inl \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : K < 1 \u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u21911 * \u2191\u2191\u03bc (closedBall x \u03b5) ** simp only [ENNReal.coe_one, one_mul] ** case inl \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : K < 1 \u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191\u2191\u03bc (closedBall x \u03b5) ** exact\n  eventually_mem_nhdsWithin.mono fun \u03b5 h\u03b5 x t ht =>\n    measure_mono <| closedBall_subset_closedBall (by nlinarith [mem_Ioi.mp h\u03b5]) ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : K < 1 \u03b5 : \u211d h\u03b5 : \u03b5 \u2208 Ioi 0 x : \u03b1 t : \u211d ht : t \u2264 K \u22a2 t * \u03b5 \u2264 \u03b5 ** nlinarith [mem_Ioi.mp h\u03b5] ** case inr \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : 1 \u2264 K \u22a2 \u2203 C, \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1) (t : \u211d), t \u2264 K \u2192 \u2191\u2191\u03bc (closedBall x (t * \u03b5)) \u2264 \u2191C * \u2191\u2191\u03bc (closedBall x \u03b5) ** refine'\n  \u27e8C ^ \u2308Real.logb 2 K\u2309\u208a,\n    ((h\u03bc \u2308Real.logb 2 K\u2309\u208a).and eventually_mem_nhdsWithin).mono fun \u03b5 h\u03b5 x t ht =>\n      le_trans (measure_mono <| closedBall_subset_closedBall _) (h\u03b5.1 x)\u27e9 ** case inr \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : 1 \u2264 K \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227     \u03b5 \u2208 Ioi 0 x : \u03b1 t : \u211d ht : t \u2264 K \u22a2 t * \u03b5 \u2264 2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5 ** refine' mul_le_mul_of_nonneg_right (ht.trans _) (mem_Ioi.mp h\u03b5.2).le ** case inr \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : 1 \u2264 K \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227     \u03b5 \u2208 Ioi 0 x : \u03b1 t : \u211d ht : t \u2264 K \u22a2 K \u2264 2 ^ \u2308Real.logb 2 K\u2309\u208a ** conv_lhs => rw [\u2190 Real.rpow_logb two_pos (by norm_num) (by linarith : 0 < K)] ** case inr \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : 1 \u2264 K \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227     \u03b5 \u2208 Ioi 0 x : \u03b1 t : \u211d ht : t \u2264 K \u22a2 2 ^ Real.logb 2 K \u2264 2 ^ \u2308Real.logb 2 K\u2309\u208a ** rw [\u2190 Real.rpow_nat_cast] ** case inr \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : 1 \u2264 K \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227     \u03b5 \u2208 Ioi 0 x : \u03b1 t : \u211d ht : t \u2264 K \u22a2 2 ^ Real.logb 2 K \u2264 2 ^ \u2191\u2308Real.logb 2 K\u2309\u208a ** exact Real.rpow_le_rpow_of_exponent_le one_le_two (Nat.le_ceil (Real.logb 2 K)) ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : 1 \u2264 K \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227     \u03b5 \u2208 Ioi 0 x : \u03b1 t : \u211d ht : t \u2264 K \u22a2 2 \u2260 1 ** norm_num ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d C : \u211d\u22650 := doublingConstant \u03bc h\u03bc : \u2200 (n : \u2115), \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ n * \u03b5)) \u2264 \u2191(C ^ n) * \u2191\u2191\u03bc (closedBall x \u03b5) hK : 1 \u2264 K \u03b5 : \u211d h\u03b5 :   (\u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x (2 ^ \u2308Real.logb 2 K\u2309\u208a * \u03b5)) \u2264 \u2191(C ^ \u2308Real.logb 2 K\u2309\u208a) * \u2191\u2191\u03bc (closedBall x \u03b5)) \u2227     \u03b5 \u2208 Ioi 0 x : \u03b1 t : \u211d ht : t \u2264 K \u22a2 0 < K ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.ofMultiset_apply_of_not_mem ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : Multiset \u03b1 hs : s \u2260 0 a : \u03b1 ha : \u00aca \u2208 s \u22a2 \u2191(ofMultiset s hs) a = 0 ** simpa only [ofMultiset_apply, ENNReal.div_eq_zero_iff, Nat.cast_eq_zero, Multiset.count_eq_zero,\n  ENNReal.nat_ne_top, or_false_iff] using ha ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.fold_hom ** \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 op' : \u03b3 \u2192 \u03b3 \u2192 \u03b3 inst\u271d\u00b9 : IsCommutative \u03b3 op' inst\u271d : IsAssociative \u03b3 op' m : \u03b2 \u2192 \u03b3 hm : \u2200 (x y : \u03b2), m (op x y) = op' (m x) (m y) \u22a2 fold op' (m b) (fun x => m (f x)) s = m (fold op b f s) ** rw [fold, fold, \u2190 Multiset.fold_hom op hm, Multiset.map_map] ** \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 op' : \u03b3 \u2192 \u03b3 \u2192 \u03b3 inst\u271d\u00b9 : IsCommutative \u03b3 op' inst\u271d : IsAssociative \u03b3 op' m : \u03b2 \u2192 \u03b3 hm : \u2200 (x y : \u03b2), m (op x y) = op' (m x) (m y) \u22a2 Multiset.fold op' (m b) (Multiset.map (fun x => m (f x)) s.val) = Multiset.fold op' (m b) (Multiset.map (m \u2218 f) s.val) ** simp only [Function.comp_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.replicate_succ_to_snoc ** \u03b1 : Type u_1 n : \u2115 xs : Vector \u03b1 n val : \u03b1 \u22a2 replicate (n + 1) val = snoc (replicate n val) val ** clear xs ** \u03b1 : Type u_1 n : \u2115 val : \u03b1 \u22a2 replicate (n + 1) val = snoc (replicate n val) val ** induction n ** case zero \u03b1 : Type u_1 n : \u2115 val : \u03b1 \u22a2 replicate (Nat.zero + 1) val = snoc (replicate Nat.zero val) val  case succ \u03b1 : Type u_1 n : \u2115 val : \u03b1 n\u271d : \u2115 n_ih\u271d : replicate (n\u271d + 1) val = snoc (replicate n\u271d val) val \u22a2 replicate (Nat.succ n\u271d + 1) val = snoc (replicate (Nat.succ n\u271d) val) val ** case zero => rfl ** case succ \u03b1 : Type u_1 n : \u2115 val : \u03b1 n\u271d : \u2115 n_ih\u271d : replicate (n\u271d + 1) val = snoc (replicate n\u271d val) val \u22a2 replicate (Nat.succ n\u271d + 1) val = snoc (replicate (Nat.succ n\u271d) val) val ** case succ n ih =>\n  rw [replicate_succ]\n  conv => {\n    rhs; rw [replicate_succ]\n  }\n  rw[snoc_cons, ih] ** \u03b1 : Type u_1 n : \u2115 val : \u03b1 \u22a2 replicate (Nat.zero + 1) val = snoc (replicate Nat.zero val) val ** rfl ** \u03b1 : Type u_1 n\u271d : \u2115 val : \u03b1 n : \u2115 ih : replicate (n + 1) val = snoc (replicate n val) val \u22a2 replicate (Nat.succ n + 1) val = snoc (replicate (Nat.succ n) val) val ** rw [replicate_succ] ** \u03b1 : Type u_1 n\u271d : \u2115 val : \u03b1 n : \u2115 ih : replicate (n + 1) val = snoc (replicate n val) val \u22a2 val ::\u1d65 replicate (n + 1) val = snoc (replicate (Nat.succ n) val) val ** conv => {\n  rhs; rw [replicate_succ]\n} ** \u03b1 : Type u_1 n\u271d : \u2115 val : \u03b1 n : \u2115 ih : replicate (n + 1) val = snoc (replicate n val) val \u22a2 val ::\u1d65 replicate (n + 1) val = snoc (val ::\u1d65 replicate n val) val ** rw[snoc_cons, ih] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mem\u2112p_norm_rpow_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\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G q : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc q_zero : q \u2260 0 q_top : q \u2260 \u22a4 \u22a2 Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q) \u2194 Mem\u2112p f p ** refine' \u27e8fun h => _, fun h => h.norm_rpow_div q\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\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G q : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc q_zero : q \u2260 0 q_top : q \u2260 \u22a4 h : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q) \u22a2 Mem\u2112p f p ** apply (mem\u2112p_norm_iff 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\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G q : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc q_zero : q \u2260 0 q_top : q \u2260 \u22a4 h : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q) \u22a2 Mem\u2112p (fun x => \u2016f x\u2016) p ** convert h.norm_rpow_div q\u207b\u00b9 using 1 ** case h.e'_5 \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\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G q : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc q_zero : q \u2260 0 q_top : q \u2260 \u22a4 h : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q) \u22a2 (fun x => \u2016f x\u2016) = fun x => \u2016\u2016f x\u2016 ^ ENNReal.toReal q\u2016 ^ ENNReal.toReal q\u207b\u00b9 ** ext x ** case h.e'_5.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\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G q : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc q_zero : q \u2260 0 q_top : q \u2260 \u22a4 h : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q) x : \u03b1 \u22a2 \u2016f x\u2016 = \u2016\u2016f x\u2016 ^ ENNReal.toReal q\u2016 ^ ENNReal.toReal q\u207b\u00b9 ** rw [Real.norm_eq_abs, Real.abs_rpow_of_nonneg (norm_nonneg _), \u2190 Real.rpow_mul (abs_nonneg _),\n  ENNReal.toReal_inv, mul_inv_cancel, abs_of_nonneg (norm_nonneg _), Real.rpow_one] ** case h.e'_5.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\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G q : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc q_zero : q \u2260 0 q_top : q \u2260 \u22a4 h : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q) x : \u03b1 \u22a2 ENNReal.toReal q \u2260 0 ** simp [ENNReal.toReal_eq_zero_iff, not_or, q_zero, q_top] ** case h.e'_6 \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\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G q : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc q_zero : q \u2260 0 q_top : q \u2260 \u22a4 h : Mem\u2112p (fun x => \u2016f x\u2016 ^ ENNReal.toReal q) (p / q) \u22a2 p = p / q / q\u207b\u00b9 ** rw [div_eq_mul_inv, inv_inv, div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel q_zero q_top,\n  mul_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.of_measure_le_smul ** \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 \u03bc' : Measure \u03b1 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u22a2 Mem\u2112p f p ** refine' \u27e8hf.1.mono' (Measure.absolutelyContinuous_of_le_smul h\u03bc'_le), _\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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03bc' : Measure \u03b1 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u22a2 snorm f p \u03bc' < \u22a4 ** refine' (snorm_mono_measure f h\u03bc'_le).trans_lt _ ** \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 \u03bc' : Measure \u03b1 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u22a2 snorm f p (c \u2022 \u03bc) < \u22a4 ** by_cases hc0 : c = 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 \u03bc' : Measure \u03b1 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p hc0 : \u00acc = 0 \u22a2 snorm f p (c \u2022 \u03bc) < \u22a4 ** rw [snorm_smul_measure_of_ne_zero hc0, smul_eq_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 \u03bc' : Measure \u03b1 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p hc0 : \u00acc = 0 \u22a2 c ^ ENNReal.toReal (1 / p) * snorm f p \u03bc < \u22a4 ** refine' ENNReal.mul_lt_top _ hf.2.ne ** 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 \u03bc' : Measure \u03b1 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p hc0 : \u00acc = 0 \u22a2 c ^ ENNReal.toReal (1 / p) \u2260 \u22a4 ** simp [hc, hc0] ** 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 \u03bc' : Measure \u03b1 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p hc0 : c = 0 \u22a2 snorm f p (c \u2022 \u03bc) < \u22a4 ** simp [hc0] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mem\u2112p_top_of_bound ** \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 C : \u211d hfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 C \u22a2 snorm f \u22a4 \u03bc < \u22a4 ** rw [snorm_exponent_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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc C : \u211d hfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 C \u22a2 snormEssSup f \u03bc < \u22a4 ** exact snormEssSup_lt_top_of_ae_bound hfC ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_condexpL2_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 snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc \u2264 snorm (\u2191\u2191f) 2 \u03bc ** rw [lpMeas_coe, \u2190 ENNReal.toReal_le_toReal (Lp.snorm_ne_top _) (Lp.snorm_ne_top _), \u2190\n  Lp.norm_def, \u2190 Lp.norm_def, Submodule.norm_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 \u2016\u2191(condexpL2 E \ud835\udd5c hm) f\u2016 \u2264 \u2016f\u2016 ** exact norm_condexpL2_le hm f ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.card_le_of_interleaved ** 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 t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y \u22a2 card s \u2264 card t + 1 ** replace h : \u2200 (x) (_ : x \u2208 s) (y) (_ : y \u2208 s), x < y \u2192 \u2203 z \u2208 t, x < z \u2227 z < y ** 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 t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y \u22a2 card s \u2264 card t + 1 ** set f : \u03b1 \u2192 WithTop \u03b1 := fun x => (t.filter fun y => x < y).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 t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y f : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t) \u22a2 card s \u2264 card t + 1 ** have f_mono : StrictMonoOn f s := by\n  intro x hx y hy hxy\n  rcases h x hx y hy hxy with \u27e8a, hat, hxa, hay\u27e9\n  calc\n    f x \u2264 a := min_le (mem_filter.2 \u27e8hat, by simpa\u27e9)\n    _ < f y :=\n      (Finset.lt_inf_iff <| WithTop.coe_lt_top a).2 fun b hb =>\n        WithTop.coe_lt_coe.2 <| hay.trans (by simpa using (mem_filter.1 hb).2) ** 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 t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y f : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t) f_mono : StrictMonoOn f \u2191s \u22a2 card s \u2264 card t + 1 ** calc\n  s.card = (s.image f).card := (card_image_of_injOn f_mono.injOn).symm\n  _ \u2264 (insert \u22a4 (t.image (\u2191)) : Finset (WithTop \u03b1)).card :=\n    card_mono <| image_subset_iff.2 fun x _ =>\n        insert_subset_insert _ (image_subset_image <| filter_subset _ _)\n          (min_mem_insert_top_image_coe _)\n  _ \u2264 t.card + 1 := (card_insert_le _ _).trans (add_le_add_right card_image_le _) ** case 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\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y \u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y ** intro x hx y hy hxy ** case 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\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x\u271d : \u03b1 s t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x < y \u22a2 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y ** rcases exists_next_right \u27e8y, hy, hxy\u27e9 with \u27e8a, has, hxa, ha\u27e9 ** case h.intro.intro.intro 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\u271d : \u03b1 s t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x < y a : \u03b1 has : a \u2208 s hxa : x < a ha : \u2200 (z : \u03b1), z \u2208 s \u2192 x < z \u2192 a \u2264 z \u22a2 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y ** rcases h x hx a has hxa fun z hzs hz => hz.2.not_le <| ha _ hzs hz.1 with \u27e8b, hbt, hxb, hba\u27e9 ** case h.intro.intro.intro.intro.intro.intro 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\u271d : \u03b1 s t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 (\u2200 (z : \u03b1), z \u2208 s \u2192 \u00acz \u2208 Set.Ioo x y) \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x < y a : \u03b1 has : a \u2208 s hxa : x < a ha : \u2200 (z : \u03b1), z \u2208 s \u2192 x < z \u2192 a \u2264 z b : \u03b1 hbt : b \u2208 t hxb : x < b hba : b < a \u22a2 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y ** exact \u27e8b, hbt, hxb, hba.trans_le <| ha _ hy hxy\u27e9 ** 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 t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y f : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t) \u22a2 StrictMonoOn f \u2191s ** intro x hx y hy hxy ** 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\u271d : \u03b1 s t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y f : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t) x : \u03b1 hx : x \u2208 \u2191s y : \u03b1 hy : y \u2208 \u2191s hxy : x < y \u22a2 f x < f y ** rcases h x hx y hy hxy with \u27e8a, hat, hxa, hay\u27e9 ** case intro.intro.intro 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\u271d : \u03b1 s t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y f : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t) x : \u03b1 hx : x \u2208 \u2191s y : \u03b1 hy : y \u2208 \u2191s hxy : x < y a : \u03b1 hat : a \u2208 t hxa : x < a hay : a < y \u22a2 f x < f y ** calc\n  f x \u2264 a := min_le (mem_filter.2 \u27e8hat, by simpa\u27e9)\n  _ < f y :=\n    (Finset.lt_inf_iff <| WithTop.coe_lt_top a).2 fun b hb =>\n      WithTop.coe_lt_coe.2 <| hay.trans (by simpa using (mem_filter.1 hb).2) ** 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\u271d : \u03b1 s t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y f : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t) x : \u03b1 hx : x \u2208 \u2191s y : \u03b1 hy : y \u2208 \u2191s hxy : x < y a : \u03b1 hat : a \u2208 t hxa : x < a hay : a < y \u22a2 x < a ** simpa ** 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\u271d : \u03b1 s t : Finset \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 \u2203 z, z \u2208 t \u2227 x < z \u2227 z < y f : \u03b1 \u2192 WithTop \u03b1 := fun x => Finset.min (filter (fun y => x < y) t) x : \u03b1 hx : x \u2208 \u2191s y : \u03b1 hy : y \u2208 \u2191s hxy : x < y a : \u03b1 hat : a \u2208 t hxa : x < a hay : a < y b : \u03b1 hb : b \u2208 filter (fun y_1 => y < y_1) t \u22a2 y < b ** simpa using (mem_filter.1 hb).2 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.ext_iff_of_sUnion_eq_univ ** \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 : Set (Set \u03b1) hc : Set.Countable S hs : \u22c3\u2080 S = univ \u22a2 \u22c3 i \u2208 S, i = univ ** rwa [\u2190 sUnion_eq_biUnion] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.sum_eapproxDiff ** \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 n : \u2115 a : \u03b1 \u22a2 \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a ** induction' n with n IH ** case zero \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 a : \u03b1 \u22a2 \u2211 k in Finset.range (Nat.zero + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f Nat.zero) a ** simp only [Nat.zero_eq, Nat.zero_add, Finset.sum_singleton, Finset.range_one] ** case zero \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 a : \u03b1 \u22a2 \u2191(\u2191(eapproxDiff f 0) a) = \u2191(eapprox f 0) a ** rfl ** case succ \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 a : \u03b1 n : \u2115 IH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a \u22a2 \u2211 k in Finset.range (Nat.succ n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f (Nat.succ n)) a ** erw [Finset.sum_range_succ, Nat.succ_eq_add_one, IH, eapproxDiff, coe_map, Function.comp_apply,\n  coe_sub, Pi.sub_apply, ENNReal.coe_toNNReal,\n  add_tsub_cancel_of_le (monotone_eapprox f (Nat.le_succ _) _)] ** case succ \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 a : \u03b1 n : \u2115 IH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a \u22a2 \u2191(eapprox f (Nat.add n 0 + 1)) a - \u2191(eapprox f (Nat.add n 0)) a \u2260 \u22a4 ** apply (lt_of_le_of_lt _ (eapprox_lt_top f (n + 1) a)).ne ** \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 a : \u03b1 n : \u2115 IH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a \u22a2 \u2191(eapprox f (Nat.add n 0 + 1)) a - \u2191(eapprox f (Nat.add n 0)) a \u2264 \u2191(eapprox f (n + 1)) a ** rw [tsub_le_iff_right] ** \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 a : \u03b1 n : \u2115 IH : \u2211 k in Finset.range (n + 1), \u2191(\u2191(eapproxDiff f k) a) = \u2191(eapprox f n) a \u22a2 \u2191(eapprox f (Nat.add n 0 + 1)) a \u2264 \u2191(eapprox f (n + 1)) a + \u2191(eapprox f (Nat.add n 0)) a ** exact le_self_add ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.generateMeasurable_eq_rec ** \u03b1 : Type u s : Set (Set \u03b1) \u22a2 {t | GenerateMeasurable s t} = \u22c3 i, generateMeasurableRec s i ** ext t ** case h \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 \u22a2 t \u2208 {t | GenerateMeasurable s t} \u2194 t \u2208 \u22c3 i, generateMeasurableRec s i ** refine' \u27e8fun ht => _, fun ht => _\u27e9 ** case h.refine'_1 \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 ht : t \u2208 {t | GenerateMeasurable s t} \u22a2 t \u2208 \u22c3 i, generateMeasurableRec s i ** inhabit \u03c9\u2081 ** case h.refine'_1 \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 ht : t \u2208 {t | GenerateMeasurable s t} inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 \u22a2 t \u2208 \u22c3 i, generateMeasurableRec s i ** induction' ht with u hu u _ IH f _ IH ** case h.refine'_1.basic \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 u : Set \u03b1 hu : u \u2208 s \u22a2 u \u2208 \u22c3 i, generateMeasurableRec s i ** exact mem_iUnion.2 \u27e8default, self_subset_generateMeasurableRec s _ hu\u27e9 ** case h.refine'_1.empty \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 \u22a2 \u2205 \u2208 \u22c3 i, generateMeasurableRec s i ** exact mem_iUnion.2 \u27e8default, empty_mem_generateMeasurableRec s _\u27e9 ** case h.refine'_1.compl \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 u : Set \u03b1 a\u271d : GenerateMeasurable s u IH : u \u2208 \u22c3 i, generateMeasurableRec s i \u22a2 u\u1d9c \u2208 \u22c3 i, generateMeasurableRec s i ** rcases mem_iUnion.1 IH with \u27e8i, hi\u27e9 ** case h.refine'_1.compl.intro \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 u : Set \u03b1 a\u271d : GenerateMeasurable s u IH : u \u2208 \u22c3 i, generateMeasurableRec s i i : (Quotient.out (ord (aleph 1))).\u03b1 hi : u \u2208 generateMeasurableRec s i \u22a2 u\u1d9c \u2208 \u22c3 i, generateMeasurableRec s i ** obtain \u27e8j, hj\u27e9 := exists_gt i ** case h.refine'_1.compl.intro.intro \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 u : Set \u03b1 a\u271d : GenerateMeasurable s u IH : u \u2208 \u22c3 i, generateMeasurableRec s i i : (Quotient.out (ord (aleph 1))).\u03b1 hi : u \u2208 generateMeasurableRec s i j : (Quotient.out (ord (aleph 1))).\u03b1 hj : i < j \u22a2 u\u1d9c \u2208 \u22c3 i, generateMeasurableRec s i ** exact mem_iUnion.2 \u27e8j, compl_mem_generateMeasurableRec hj hi\u27e9 ** case h.refine'_1.iUnion \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 f : \u2115 \u2192 Set \u03b1 a\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n) IH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i \u22a2 \u22c3 i, f i \u2208 \u22c3 i, generateMeasurableRec s i ** have : \u2200 n, \u2203 i, f n \u2208 generateMeasurableRec s i := fun n => by simpa using IH n ** case h.refine'_1.iUnion \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 f : \u2115 \u2192 Set \u03b1 a\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n) IH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i this : \u2200 (n : \u2115), \u2203 i, f n \u2208 generateMeasurableRec s i \u22a2 \u22c3 i, f i \u2208 \u22c3 i, generateMeasurableRec s i ** choose I hI using this ** \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 f : \u2115 \u2192 Set \u03b1 a\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n) IH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i n : \u2115 \u22a2 \u2203 i, f n \u2208 generateMeasurableRec s i ** simpa using IH n ** case h.refine'_1.iUnion.refine'_1 \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 f : \u2115 \u2192 Set \u03b1 a\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n) IH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i I : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1 hI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n) this : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1 \u22a2 (Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)) < Ordinal.type fun x x_1 => x < x_1 ** rw [Ordinal.type_lt] ** case h.refine'_1.iUnion.refine'_1 \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 f : \u2115 \u2192 Set \u03b1 a\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n) IH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i I : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1 hI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n) this : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1 \u22a2 (Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)) < ord (aleph 1) ** refine' Ordinal.lsub_lt_ord_lift _ fun i => Ordinal.typein_lt_self _ ** case h.refine'_1.iUnion.refine'_1 \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 f : \u2115 \u2192 Set \u03b1 a\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n) IH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i I : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1 hI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n) this : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1 \u22a2 lift.{u, 0} #\u2115 < Ordinal.cof (ord (aleph 1)) ** rw [mk_denumerable, lift_aleph0, isRegular_aleph_one.cof_eq] ** case h.refine'_1.iUnion.refine'_1 \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 f : \u2115 \u2192 Set \u03b1 a\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n) IH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i I : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1 hI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n) this : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1 \u22a2 \u2135\u2080 < aleph 1 ** exact aleph0_lt_aleph_one ** case h.refine'_1.iUnion.refine'_2 \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).\u03b1 f : \u2115 \u2192 Set \u03b1 a\u271d : \u2200 (n : \u2115), GenerateMeasurable s (f n) IH : \u2200 (n : \u2115), f n \u2208 \u22c3 i, generateMeasurableRec s i I : \u2115 \u2192 (Quotient.out (ord (aleph 1))).\u03b1 hI : \u2200 (n : \u2115), f n \u2208 generateMeasurableRec s (I n) this : IsWellOrder (Quotient.out (ord (aleph 1))).\u03b1 fun x x_1 => x < x_1 n : \u2115 \u22a2 Ordinal.typein (fun x x_1 => x < x_1) (I n) < Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n) ** apply Ordinal.lt_lsub fun n : \u2115 => _ ** case h.refine'_2 \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 ht : t \u2208 \u22c3 i, generateMeasurableRec s i \u22a2 t \u2208 {t | GenerateMeasurable s t} ** rcases ht with \u27e8t, \u27e8i, rfl\u27e9, hx\u27e9 ** case h.refine'_2.intro.intro.intro \u03b1 : Type u s : Set (Set \u03b1) t : Set \u03b1 i : (Quotient.out (ord (aleph 1))).\u03b1 hx : t \u2208 (fun i => generateMeasurableRec s i) i \u22a2 t \u2208 {t | GenerateMeasurable s t} ** revert t ** case h.refine'_2.intro.intro.intro \u03b1 : Type u s : Set (Set \u03b1) i : (Quotient.out (ord (aleph 1))).\u03b1 \u22a2 \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) i \u2192 t \u2208 {t | GenerateMeasurable s t} ** apply (aleph 1).ord.out.wo.wf.induction i ** case h.refine'_2.intro.intro.intro \u03b1 : Type u s : Set (Set \u03b1) i : (Quotient.out (ord (aleph 1))).\u03b1 \u22a2 \u2200 (x : (Quotient.out (ord (aleph 1))).\u03b1),     (\u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),         WellOrder.r (Quotient.out (ord (aleph 1))) y x \u2192           \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t}) \u2192       \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) x \u2192 t \u2208 {t | GenerateMeasurable s t} ** intro j H t ht ** case h.refine'_2.intro.intro.intro \u03b1 : Type u s : Set (Set \u03b1) i j : (Quotient.out (ord (aleph 1))).\u03b1 H :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192       \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t} t : Set \u03b1 ht : t \u2208 (fun i => generateMeasurableRec s i) j \u22a2 t \u2208 {t | GenerateMeasurable s t} ** unfold generateMeasurableRec at ht ** case h.refine'_2.intro.intro.intro \u03b1 : Type u s : Set (Set \u03b1) i j : (Quotient.out (ord (aleph 1))).\u03b1 H :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192       \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t} t : Set \u03b1 ht :   t \u2208     let i := j;     let S := \u22c3 j, generateMeasurableRec s \u2191j;     s \u222a {\u2205} \u222a compl '' S \u222a range fun f => \u22c3 n, \u2191(f n) \u22a2 t \u2208 {t | GenerateMeasurable s t} ** rcases ht with (((h | (rfl : t = \u2205)) | \u27e8u, \u27e8-, \u27e8\u27e8k, hk\u27e9, rfl\u27e9, hu\u27e9, rfl\u27e9) | \u27e8f, rfl\u27e9) ** case h.refine'_2.intro.intro.intro.inl.inl.inl \u03b1 : Type u s : Set (Set \u03b1) i j : (Quotient.out (ord (aleph 1))).\u03b1 H :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192       \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t} t : Set \u03b1 h : t \u2208 s \u22a2 t \u2208 {t | GenerateMeasurable s t} ** exact .basic t h ** case h.refine'_2.intro.intro.intro.inl.inl.inr \u03b1 : Type u s : Set (Set \u03b1) i j : (Quotient.out (ord (aleph 1))).\u03b1 H :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192       \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t} \u22a2 \u2205 \u2208 {t | GenerateMeasurable s t} ** exact .empty ** case h.refine'_2.intro.intro.intro.inl.inr.intro.intro.intro.intro.intro.mk \u03b1 : Type u s : Set (Set \u03b1) i j : (Quotient.out (ord (aleph 1))).\u03b1 H :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192       \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t} u : Set \u03b1 k : (Quotient.out (ord (aleph 1))).\u03b1 hk : k \u2208 Iio j hu : u \u2208 (fun j_1 => generateMeasurableRec s \u2191j_1) { val := k, property := hk } \u22a2 u\u1d9c \u2208 {t | GenerateMeasurable s t} ** exact .compl u (H k hk u hu) ** case h.refine'_2.intro.intro.intro.inr.intro \u03b1 : Type u s : Set (Set \u03b1) i j : (Quotient.out (ord (aleph 1))).\u03b1 H :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192       \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t} f : \u2115 \u2192 \u2191(\u22c3 j_1, generateMeasurableRec s \u2191j_1) \u22a2 (fun f => \u22c3 n, \u2191(f n)) f \u2208 {t | GenerateMeasurable s t} ** refine .iUnion _ @fun n => ?_ ** case h.refine'_2.intro.intro.intro.inr.intro \u03b1 : Type u s : Set (Set \u03b1) i j : (Quotient.out (ord (aleph 1))).\u03b1 H :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192       \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t} f : \u2115 \u2192 \u2191(\u22c3 j_1, generateMeasurableRec s \u2191j_1) n : \u2115 \u22a2 GenerateMeasurable s \u2191(f n) ** obtain \u27e8-, \u27e8\u27e8k, hk\u27e9, rfl\u27e9, hf\u27e9 := (f n).prop ** case h.refine'_2.intro.intro.intro.inr.intro.intro.intro.intro.mk \u03b1 : Type u s : Set (Set \u03b1) i j : (Quotient.out (ord (aleph 1))).\u03b1 H :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y j \u2192       \u2200 (t : Set \u03b1), t \u2208 (fun i => generateMeasurableRec s i) y \u2192 t \u2208 {t | GenerateMeasurable s t} f : \u2115 \u2192 \u2191(\u22c3 j_1, generateMeasurableRec s \u2191j_1) n : \u2115 k : (Quotient.out (ord (aleph 1))).\u03b1 hk : k \u2208 Iio j hf : \u2191(f n) \u2208 (fun j_1 => generateMeasurableRec s \u2191j_1) { val := k, property := hk } \u22a2 GenerateMeasurable s \u2191(f n) ** exact H k hk _ hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.range_IicExtend ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : LinearOrder \u03b1 a b : \u03b1 h : a \u2264 b x : \u03b1 f : \u2191(Iic b) \u2192 \u03b2 \u22a2 range (IicExtend f) = range f ** simp only [IicExtend, range_comp f, range_projIic, range_id', image_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_mul_Icc_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' Icc a b = Icc (b / c) (a / c) ** simpa only [mul_comm] using preimage_mul_const_Icc_of_neg a b h ** Qed",
        "informal": ""
    },
    {
        "formal": "surjOn_Ici_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 (Ici a) (Ici (f a)) ** rw [\u2190 Ioi_union_left, \u2190 Ioi_union_left] ** \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 \u222a {a}) (Ioi (f a) \u222a {f a}) ** exact\n  (surjOn_Ioi_of_monotone_surjective h_mono h_surj a).union_union\n    (@image_singleton _ _ f a \u25b8 surjOn_image _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.predictablePart_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), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R ** simp_rw [predictablePart, Finset.sum_apply, Finset.sum_range_succ_sub_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\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), |(\u03bc[f (i + 1) - f i|\u2191\u2131 i]) \u03c9| \u2264 \u2191R ** exact ae_all_iff.2 fun i => ae_bdd_condexp_of_ae_bdd <| ae_all_iff.1 hbdd i ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.integral_truncation_eq_intervalIntegral ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc A : \u211d hA : 0 \u2264 A \u22a2 \u222b (x : \u03b1), truncation f A x \u2202\u03bc = \u222b (y : \u211d) in -A..A, y \u2202Measure.map f \u03bc ** simpa using moment_truncation_eq_intervalIntegral hf hA one_ne_zero ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.integrable_kernel_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) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4 \u22a2 Integrable fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) ** constructor ** case 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) hs : MeasurableSet s h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4 \u22a2 AEStronglyMeasurable (fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))) (\u2191\u03ba a) ** exact (measurable_kernel_prod_mk_left' hs a).ennreal_toReal.aestronglyMeasurable ** case right \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) hs : MeasurableSet s 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)) ** exact hasFiniteIntegral_prod_mk_left a h2s ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.negOfNat_mul_negSucc ** m n : Nat \u22a2 negOfNat n * -[m+1] = ofNat (n * succ m) ** rw [Int.mul_comm, negSucc_mul_negOfNat, Nat.mul_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.strong_law_aux4 ** \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 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n =>       \u2191\u230ac ^ n\u230b\u208a ** filter_upwards [strong_law_aux2 X hint hindep hident hnonneg c_one] 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 \u03c9 : \u03a9 h\u03c9 :   (fun n =>       \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) =o[atTop]     fun n => \u2191\u230ac ^ n\u230b\u208a \u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a ** have A : Tendsto (fun n : \u2115 => \u230ac ^ n\u230b\u208a) atTop atTop :=\n  tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt c_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 \u03c9 : \u03a9 h\u03c9 :   (fun n =>       \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) =o[atTop]     fun n => \u2191\u230ac ^ n\u230b\u208a A : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop \u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a ** convert h\u03c9.add ((strong_law_aux3 X hint hident).comp_tendsto 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 c : \u211d c_one : 1 < c \u03c9 : \u03a9 h\u03c9 :   (fun n =>       \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) =o[atTop]     fun n => \u2191\u230ac ^ n\u230b\u208a A : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop \u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) = fun x =>     (\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -         \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a) +       ((fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) \u2218 fun n =>           \u230ac ^ n\u230b\u208a)         x ** 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 c : \u211d c_one : 1 < c \u03c9 : \u03a9 h\u03c9 :   (fun n =>       \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) =o[atTop]     fun n => \u2191\u230ac ^ n\u230b\u208a A : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop n : \u2115 \u22a2 \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a =     (\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) +       ((fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) \u2218 fun n =>           \u230ac ^ n\u230b\u208a)         n ** simp ** 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 hp1 : 1 \u2264 p \u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g 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 g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hp1 : 1 \u2264 p hp0 : \u00acp = 0 \u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \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 : \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 hp1 : 1 \u2264 p hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc ** have hp1_real : 1 \u2264 p.toReal := by\n  rwa [\u2190 ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top 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 g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hp1 : 1 \u2264 p hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 hp1_real : 1 \u2264 ENNReal.toReal p \u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc ** repeat 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 g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hp1 : 1 \u2264 p hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 hp1_real : 1 \u2264 ENNReal.toReal p \u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal p) \u03bc + snorm' g (ENNReal.toReal p) \u03bc ** exact snorm'_add_le hf hg hp1_real ** 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 g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hp1 : 1 \u2264 p hp0 : p = 0 \u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g 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 g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hp1 : 1 \u2264 p hp0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc ** simp [hp_top, snormEssSup_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 hp1 : 1 \u2264 p hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 1 \u2264 ENNReal.toReal p ** rwa [\u2190 ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top 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 g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hp1 : 1 \u2264 p hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 hp1_real : 1 \u2264 ENNReal.toReal p \u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal p) \u03bc + snorm g p \u03bc ** rw [snorm_eq_snorm' hp0 hp_top] ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.natSize_to_nat ** \u03b1 : Type u_1 n : Num \u22a2 natSize n = Nat.size \u2191n ** rw [\u2190 size_eq_natSize, size_to_nat] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableEmbedding.measurable_extend ** \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\u03b1 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 g\u271d : \u03b2 \u2192 \u03b3 hf : MeasurableEmbedding f g : \u03b1 \u2192 \u03b3 g' : \u03b2 \u2192 \u03b3 hg : Measurable g hg' : Measurable g' \u22a2 Measurable (extend f g g') ** refine' measurable_of_restrict_of_restrict_compl hf.measurableSet_range _ _ ** case refine'_1 \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\u03b1 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 g\u271d : \u03b2 \u2192 \u03b3 hf : MeasurableEmbedding f g : \u03b1 \u2192 \u03b3 g' : \u03b2 \u2192 \u03b3 hg : Measurable g hg' : Measurable g' \u22a2 Measurable (restrict (range f) (extend f g g')) ** rw [restrict_extend_range] ** case refine'_1 \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\u03b1 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 g\u271d : \u03b2 \u2192 \u03b3 hf : MeasurableEmbedding f g : \u03b1 \u2192 \u03b3 g' : \u03b2 \u2192 \u03b3 hg : Measurable g hg' : Measurable g' \u22a2 Measurable fun x => g (Exists.choose (_ : \u2191x \u2208 range f)) ** simpa only [rangeSplitting] using hg.comp hf.measurable_rangeSplitting ** case refine'_2 \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\u03b1 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 g\u271d : \u03b2 \u2192 \u03b3 hf : MeasurableEmbedding f g : \u03b1 \u2192 \u03b3 g' : \u03b2 \u2192 \u03b3 hg : Measurable g hg' : Measurable g' \u22a2 Measurable (restrict (range f)\u1d9c (extend f g g')) ** rw [restrict_extend_compl_range] ** case refine'_2 \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\u03b1 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 g\u271d : \u03b2 \u2192 \u03b3 hf : MeasurableEmbedding f g : \u03b1 \u2192 \u03b3 g' : \u03b2 \u2192 \u03b3 hg : Measurable g hg' : Measurable g' \u22a2 Measurable (g' \u2218 Subtype.val) ** exact hg'.comp measurable_subtype_coe ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.coe_eq_pair ** \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 : \u03b1 s : Finset \u03b1 a b : \u03b1 \u22a2 \u2191s = {a, b} \u2194 s = {a, b} ** rw [\u2190 coe_pair, coe_inj] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.borel_eq_generateFrom_Iic_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, {Iic \u2191a}) ** rw [borel_eq_generateFrom_Ioi_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 => Ioi \u2191a) = MeasurableSpace.generateFrom (range fun a => Iic \u2191a) ** refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>\nrintro _ \u27e8q, rfl\u27e9 <;>\ndsimp only <;>\n[rw [\u2190 compl_Iic]; rw [\u2190 compl_Ioi]] <;>\nexact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.insert_size ** \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 : m.size = Buckets.size m.buckets \u22a2 (insert m k v).size = Buckets.size (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 : 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,           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))).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,             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 ** 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 v : \u03b2 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 (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))).size =     Buckets.size       (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 ** 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 v : \u03b2 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,         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) }.size =     Buckets.size       { 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 ** unfold Buckets.size ** 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 v : \u03b2 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,         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) }.size =     Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         { 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.val.data) ** refine have \u27e8_, _, h\u2081, _, eq\u27e9 := Buckets.exists_of_update ..; eq \u25b8 ?_ ** 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 v : \u03b2 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 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.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)).val.data =     w\u271d\u00b9 ++       AssocList.replace k v           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::         w\u271d \u22a2 { 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) }.size =     Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         (w\u271d\u00b9 ++           AssocList.replace k v               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] ** case h_2.inl \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 : 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 h\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val \u22a2 { 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) }.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.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 ** unfold Buckets.size ** case h_2.inl \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 : 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 h\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val \u22a2 { 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) }.size =     Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         { 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.val.data) ** refine have \u27e8_, _, h\u2081, _, eq\u27e9 := Buckets.exists_of_update ..; eq \u25b8 ?_ ** case h_2.inl \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 : 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 h\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val 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.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)).val.data =     w\u271d\u00b9 ++       AssocList.cons k v           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::         w\u271d \u22a2 { 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) }.size =     Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         (w\u271d\u00b9 ++           AssocList.cons k v               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, Nat.succ_add] ** case h_2.inl \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 : 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 h\u271d : numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val 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.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)).val.data =     w\u271d\u00b9 ++       AssocList.cons k v           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) +       Nat.succ         (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)) ** rfl ** case h_2.inr \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 : 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 h\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val \u22a2 (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))).size =     Buckets.size       (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 [expand_size] ** case h_2.inr \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 : 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 h\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val \u22a2 (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))).size =     Buckets.size       (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)) ** simp [h, expand, Buckets.size] ** case h_2.inr \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 : 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 h\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val \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.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)).val.data) ** refine have \u27e8_, _, h\u2081, _, eq\u27e9 := Buckets.exists_of_update ..; eq \u25b8 ?_ ** case h_2.inr \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 : 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 h\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val 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.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)).val.data =     w\u271d\u00b9 ++       AssocList.cons k v           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.cons k v               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::             w\u271d)) ** simp [h\u2081, Buckets.size_eq, Nat.succ_add] ** case h_2.inr \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 : 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 h\u271d : \u00acnumBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val 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.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)).val.data =     w\u271d\u00b9 ++       AssocList.cons k v           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) +       Nat.succ         (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)) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_measure_Ioc_atBot ** \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\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 \u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a))) ** haveI : Nonempty \u03b1 := \u27e8a\u27e9 ** \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\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 \u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a))) ** have h_mono : Antitone fun x => \u03bc (Ioc x a) := fun i j hij =>\n  measure_mono (Ioc_subset_Ioc_left 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 m0 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) \u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a))) ** convert tendsto_atBot_iSup h_mono ** case h.e'_5.h.e'_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\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) \u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 i, \u2191\u2191\u03bc (Ioc i a) ** obtain \u27e8xs, hxs_mono, hxs_tendsto\u27e9 := exists_seq_antitone_tendsto_atTop_atBot \u03b1 ** case h.e'_5.h.e'_3.intro.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 m0 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) xs : \u2115 \u2192 \u03b1 hxs_mono : Antitone xs hxs_tendsto : Tendsto xs atTop atBot \u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 i, \u2191\u2191\u03bc (Ioc i a) ** have h_Iic : Iic a = \u22c3 n, Ioc (xs n) a := by\n  ext1 x\n  simp only [mem_Iic, mem_iUnion, mem_Ioc, exists_and_right, iff_and_self]\n  intro\n  obtain \u27e8y, hxy\u27e9 := NoMinOrder.exists_lt x\n  obtain \u27e8n, hn\u27e9 := tendsto_atTop_atBot.mp hxs_tendsto y\n  exact \u27e8n, (hn n le_rfl).trans_lt hxy\u27e9 ** case h.e'_5.h.e'_3.intro.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 m0 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) xs : \u2115 \u2192 \u03b1 hxs_mono : Antitone xs hxs_tendsto : Tendsto xs atTop atBot h_Iic : Iic a = \u22c3 n, Ioc (xs n) a \u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 i, \u2191\u2191\u03bc (Ioc i a) ** rw [h_Iic, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_antitone h_mono hxs_tendsto] ** case h.e'_5.h.e'_3.intro.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 m0 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) xs : \u2115 \u2192 \u03b1 hxs_mono : Antitone xs hxs_tendsto : Tendsto xs atTop atBot h_Iic : Iic a = \u22c3 n, Ioc (xs n) a \u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => Ioc (xs n) a ** exact Monotone.directed_le fun i j hij => Ioc_subset_Ioc_left (hxs_mono 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 m0 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) xs : \u2115 \u2192 \u03b1 hxs_mono : Antitone xs hxs_tendsto : Tendsto xs atTop atBot \u22a2 Iic a = \u22c3 n, Ioc (xs n) a ** ext1 x ** 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\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) xs : \u2115 \u2192 \u03b1 hxs_mono : Antitone xs hxs_tendsto : Tendsto xs atTop atBot x : \u03b1 \u22a2 x \u2208 Iic a \u2194 x \u2208 \u22c3 n, Ioc (xs n) a ** simp only [mem_Iic, mem_iUnion, mem_Ioc, exists_and_right, iff_and_self] ** 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\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) xs : \u2115 \u2192 \u03b1 hxs_mono : Antitone xs hxs_tendsto : Tendsto xs atTop atBot x : \u03b1 \u22a2 x \u2264 a \u2192 \u2203 x_1, xs x_1 < x ** intro ** 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\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) xs : \u2115 \u2192 \u03b1 hxs_mono : Antitone xs hxs_tendsto : Tendsto xs atTop atBot x : \u03b1 a\u271d : x \u2264 a \u22a2 \u2203 x_1, xs x_1 < x ** obtain \u27e8y, hxy\u27e9 := NoMinOrder.exists_lt x ** case h.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 m0 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) xs : \u2115 \u2192 \u03b1 hxs_mono : Antitone xs hxs_tendsto : Tendsto xs atTop atBot x : \u03b1 a\u271d : x \u2264 a y : \u03b1 hxy : y < x \u22a2 \u2203 x_1, xs x_1 < x ** obtain \u27e8n, hn\u27e9 := tendsto_atTop_atBot.mp hxs_tendsto y ** case h.intro.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 m0 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : NoMinOrder \u03b1 inst\u271d : IsCountablyGenerated atBot \u03bc : Measure \u03b1 a : \u03b1 this : Nonempty \u03b1 h_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a) xs : \u2115 \u2192 \u03b1 hxs_mono : Antitone xs hxs_tendsto : Tendsto xs atTop atBot x : \u03b1 a\u271d : x \u2264 a y : \u03b1 hxy : y < x n : \u2115 hn : \u2200 (a : \u2115), n \u2264 a \u2192 xs a \u2264 y \u22a2 \u2203 x_1, xs x_1 < x ** exact \u27e8n, (hn n le_rfl).trans_lt hxy\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.pi_eval_preimage_null ** \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) i : \u03b9 s : Set (\u03b1 i) hs : \u2191\u2191(\u03bc i) s = 0 \u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' s) = 0 ** rcases exists_measurable_superset_of_null hs with \u27e8t, hst, _, h\u03bct\u27e9 ** case intro.intro.intro \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) i : \u03b9 s : Set (\u03b1 i) hs : \u2191\u2191(\u03bc i) s = 0 t : Set (\u03b1 i) hst : s \u2286 t left\u271d : MeasurableSet t h\u03bct : \u2191\u2191(\u03bc i) t = 0 \u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' s) = 0 ** suffices : Measure.pi \u03bc (eval i \u207b\u00b9' t) = 0 ** case intro.intro.intro \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) i : \u03b9 s : Set (\u03b1 i) hs : \u2191\u2191(\u03bc i) s = 0 t : Set (\u03b1 i) hst : s \u2286 t left\u271d : MeasurableSet t h\u03bct : \u2191\u2191(\u03bc i) t = 0 this : \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0 \u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' s) = 0  case this \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) i : \u03b9 s : Set (\u03b1 i) hs : \u2191\u2191(\u03bc i) s = 0 t : Set (\u03b1 i) hst : s \u2286 t left\u271d : MeasurableSet t h\u03bct : \u2191\u2191(\u03bc i) t = 0 \u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0 ** exact measure_mono_null (preimage_mono hst) this ** case this \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) i : \u03b9 s : Set (\u03b1 i) hs : \u2191\u2191(\u03bc i) s = 0 t : Set (\u03b1 i) hst : s \u2286 t left\u271d : MeasurableSet t h\u03bct : \u2191\u2191(\u03bc i) t = 0 \u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0 ** clear! s ** case this \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) i : \u03b9 t : Set (\u03b1 i) left\u271d : MeasurableSet t h\u03bct : \u2191\u2191(\u03bc i) t = 0 \u22a2 \u2191\u2191(Measure.pi \u03bc) (eval i \u207b\u00b9' t) = 0 ** rw [\u2190 univ_pi_update_univ, pi_pi] ** case this \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) i : \u03b9 t : Set (\u03b1 i) left\u271d : MeasurableSet t h\u03bct : \u2191\u2191(\u03bc i) t = 0 \u22a2 \u220f i_1 : \u03b9, \u2191\u2191(\u03bc i_1) (update (fun j => univ) i t i_1) = 0 ** apply Finset.prod_eq_zero (Finset.mem_univ i) ** case this \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) i : \u03b9 t : Set (\u03b1 i) left\u271d : MeasurableSet t h\u03bct : \u2191\u2191(\u03bc i) t = 0 \u22a2 \u2191\u2191(\u03bc i) (update (fun j => univ) i t i) = 0 ** simp [h\u03bct] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.replicate_sublist_replicate ** \u03b1 : Type u_1 m n : Nat a : \u03b1 \u22a2 replicate m a <+ replicate n a \u2194 m \u2264 n ** refine \u27e8fun h => ?_, fun h => ?_\u27e9 ** case refine_1 \u03b1 : Type u_1 m n : Nat a : \u03b1 h : replicate m a <+ replicate n a \u22a2 m \u2264 n ** have := h.length_le ** case refine_1 \u03b1 : Type u_1 m n : Nat a : \u03b1 h : replicate m a <+ replicate n a this : length (replicate m a) \u2264 length (replicate n a) \u22a2 m \u2264 n ** simp only [length_replicate] at this \u22a2 ** case refine_1 \u03b1 : Type u_1 m n : Nat a : \u03b1 h : replicate m a <+ replicate n a this : m \u2264 n \u22a2 m \u2264 n ** exact this ** case refine_2 \u03b1 : Type u_1 m n : Nat a : \u03b1 h : m \u2264 n \u22a2 replicate m a <+ replicate n a ** induction h with\n| refl => apply Sublist.refl\n| step => simp [*, replicate, Sublist.cons] ** case refine_2.refl \u03b1 : Type u_1 m n : Nat a : \u03b1 \u22a2 replicate m a <+ replicate m a ** apply Sublist.refl ** case refine_2.step \u03b1 : Type u_1 m n : Nat a : \u03b1 m\u271d : Nat a\u271d : Nat.le m m\u271d a_ih\u271d : replicate m a <+ replicate m\u271d a \u22a2 replicate m a <+ replicate (succ m\u271d) a ** simp [*, replicate, Sublist.cons] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image_uncurry_product ** \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\u2077 : DecidableEq \u03b1' inst\u271d\u2076 : DecidableEq \u03b2' inst\u271d\u2075 : DecidableEq \u03b3 inst\u271d\u2074 : DecidableEq \u03b3' inst\u271d\u00b3 : DecidableEq \u03b4 inst\u271d\u00b2 : DecidableEq \u03b4' inst\u271d\u00b9 : DecidableEq \u03b5 inst\u271d : DecidableEq \u03b5' f\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s\u271d s' : Finset \u03b1 t\u271d t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 s : Finset \u03b1 t : Finset \u03b2 \u22a2 image (uncurry f) (s \u00d7\u02e2 t) = image\u2082 f s t ** rw [\u2190 image\u2082_curry, curry_uncurry] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBSet.ModifyWF.of_eq ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering f : \u03b1 \u2192 \u03b1 t : RBSet \u03b1 cmp H : \u2200 {x : \u03b1}, RBNode.find? cut t.val = some x \u2192 cmpEq cmp (f x) x \u22a2 ModifyWF t cut f ** refine \u27e8.modify ?_ t.2\u27e9 ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering f : \u03b1 \u2192 \u03b1 t : RBSet \u03b1 cmp H : \u2200 {x : \u03b1}, RBNode.find? cut t.val = some x \u2192 cmpEq cmp (f x) x \u22a2 OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst ** revert H ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering f : \u03b1 \u2192 \u03b1 t : RBSet \u03b1 cmp \u22a2 (\u2200 {x : \u03b1}, RBNode.find? cut t.val = some x \u2192 cmpEq cmp (f x) x) \u2192     OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst ** rw [find?_eq_zoom] ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering f : \u03b1 \u2192 \u03b1 t : RBSet \u03b1 cmp \u22a2 (\u2200 {x : \u03b1}, root? (zoom cut t.val Path.root).fst = some x \u2192 cmpEq cmp (f x) x) \u2192     OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst ** cases (t.1.zoom cut).1 <;> intro H <;> [trivial; exact H rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.castNum_or ** \u03b1 : Type u_1 \u22a2 \u2200 (m n : Num), \u2191(m ||| n) = \u2191m ||| \u2191n ** apply castNum_eq_bitwise fun x y => pos (PosNum.lor x y) <;>\n intros <;> (try cases_type* Bool) <;> rfl ** case pbb \u03b1 : Type u_1 a\u271d b\u271d : Bool m\u271d n\u271d : PosNum \u22a2 pos (lor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d)) = bit (a\u271d || b\u271d) (pos (lor 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 pos (lor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d)) = bit (a\u271d || b\u271d) (pos (lor m\u271d n\u271d)) ** cases_type* Bool ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_set_lintegral_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 l : Filter \u03b9 s : \u03b9 \u2192 Set \u03b1 hl : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0) \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1) in s i, f x \u2202\u03bc) l (\ud835\udcdd 0) ** simp only [ENNReal.nhds_zero, tendsto_iInf, tendsto_principal, mem_Iio,\n  \u2190 pos_iff_ne_zero] at hl \u22a2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 l : Filter \u03b9 s : \u03b9 \u2192 Set \u03b1 hl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i \u22a2 \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < i ** intro \u03b5 \u03b50 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 l : Filter \u03b9 s : \u03b9 \u2192 Set \u03b1 hl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 \u22a2 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < \u03b5 ** rcases exists_pos_set_lintegral_lt_of_measure_lt h \u03b50.ne' with \u27e8\u03b4, \u03b40, h\u03b4\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 l : Filter \u03b9 s : \u03b9 \u2192 Set \u03b1 hl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 \u03b4 : \u211d\u22650\u221e \u03b40 : \u03b4 > 0 h\u03b4 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5 \u22a2 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < \u03b5 ** exact (hl \u03b4 \u03b40).mono fun i => h\u03b4 _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.nat_div ** \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 \u22a2 Primrec\u2082 fun x x_1 => x / x_1 ** refine of_graph \u27e8_, fst, fun p => Nat.div_le_self _ _\u27e9 ?_ ** \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 \u22a2 PrimrecRel fun a b => (fun x x_1 => x / x_1) a.1 a.2 = b ** have : PrimrecRel fun (a : \u2115 \u00d7 \u2115) (b : \u2115) => (a.2 = 0 \u2227 b = 0) \u2228\n    (0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2) :=\n  PrimrecPred.or\n    (.and (const 0 |> Primrec.eq.comp (fst |> snd.comp)) (const 0 |> Primrec.eq.comp snd))\n    (.and (nat_lt.comp (const 0) (fst |> snd.comp)) <|\n        .and (nat_le.comp (nat_mul.comp snd (fst |> snd.comp)) (fst |> fst.comp))\n        (nat_lt.comp (fst.comp fst) (nat_mul.comp (Primrec.succ.comp snd) (snd.comp fst)))) ** \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 this : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 \u22a2 PrimrecRel fun a b => (fun x x_1 => x / x_1) a.1 a.2 = b ** refine this.of_eq ?_ ** \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 this : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 \u22a2 \u2200 (a : \u2115 \u00d7 \u2115) (b : \u2115),     a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 \u2194 (fun x x_1 => x / x_1) a.1 a.2 = b ** rintro \u27e8a, k\u27e9 q ** case mk \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 this : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 a k q : \u2115 \u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194     (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q ** if H : k = 0 then simp [H, eq_comm]\nelse\n  have : q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k := by\n    rw [le_antisymm_iff, \u2190 (@Nat.lt_succ _ q), Nat.le_div_iff_mul_le' (Nat.pos_of_ne_zero H),\n        Nat.div_lt_iff_lt_mul' (Nat.pos_of_ne_zero H)]\n  simpa [H, zero_lt_iff, eq_comm (b := q)] ** \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 this : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 a k q : \u2115 H : k = 0 \u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194     (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q ** simp [H, eq_comm] ** \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 this : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 a k q : \u2115 H : \u00ack = 0 \u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194     (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q ** have : q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k := by\n  rw [le_antisymm_iff, \u2190 (@Nat.lt_succ _ q), Nat.le_div_iff_mul_le' (Nat.pos_of_ne_zero H),\n      Nat.div_lt_iff_lt_mul' (Nat.pos_of_ne_zero 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 this\u271d : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 a k q : \u2115 H : \u00ack = 0 this : q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k \u22a2 (a, k).2 = 0 \u2227 q = 0 \u2228 0 < (a, k).2 \u2227 q * (a, k).2 \u2264 (a, k).1 \u2227 (a, k).1 < (q + 1) * (a, k).2 \u2194     (fun x x_1 => x / x_1) (a, k).1 (a, k).2 = q ** simpa [H, zero_lt_iff, eq_comm (b := q)] ** \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 this : PrimrecRel fun a b => a.2 = 0 \u2227 b = 0 \u2228 0 < a.2 \u2227 b * a.2 \u2264 a.1 \u2227 a.1 < (b + 1) * a.2 a k q : \u2115 H : \u00ack = 0 \u22a2 q * k \u2264 a \u2227 a < (q + 1) * k \u2194 q = a / k ** rw [le_antisymm_iff, \u2190 (@Nat.lt_succ _ q), Nat.le_div_iff_mul_le' (Nat.pos_of_ne_zero H),\n    Nat.div_lt_iff_lt_mul' (Nat.pos_of_ne_zero H)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.isOrdered_iff' ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering L R : Option \u03b1 inst\u271d : TransCmp cmp t : RBNode \u03b1 \u22a2 isOrdered cmp t L R = true \u2194     (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) t) \u2227       (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) t) \u2227         (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp t ** induction t generalizing L R with\n| nil =>\n  simp [isOrdered]; split <;> simp [cmpLT_iff]\n  next h => intro _ ha _ hb; cases h _ _ ha hb\n| node _ l v r =>\n  simp [isOrdered, *]\n  exact \u27e8\n    fun \u27e8\u27e8Ll, lv, Lv, ol\u27e9, \u27e8vr, rR, vR, or\u27e9\u27e9 => \u27e8\n      fun _ h => \u27e8Lv _ h, Ll _ h, (Lv _ h).trans_l vr\u27e9,\n      fun _ h => \u27e8vR _ h, (vR _ h).trans_r lv, rR _ h\u27e9,\n      fun _ hL _ hR => (Lv _ hL).trans (vR _ hR),\n      lv, vr, ol, or\u27e9,\n    fun \u27e8hL, hR, _, lv, vr, ol, or\u27e9 => \u27e8\n      \u27e8fun _ h => (hL _ h).2.1, lv, fun _ h => (hL _ h).1, ol\u27e9,\n      \u27e8vr, fun _ h => (hR _ h).2.2, fun _ h => (hR _ h).1, or\u27e9\u27e9\u27e9 ** case nil \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering inst\u271d : TransCmp cmp L R : Option \u03b1 \u22a2 isOrdered cmp nil L R = true \u2194     (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) nil) \u2227       (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) nil) \u2227         (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp nil ** simp [isOrdered] ** case nil \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering inst\u271d : TransCmp cmp L R : Option \u03b1 \u22a2 (match L, R with       | some l, some r => decide (cmp l r = Ordering.lt)       | x, x_1 => true) =       true \u2194     \u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmpLT cmp a b ** split <;> simp [cmpLT_iff] ** case nil.h_2 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering inst\u271d : TransCmp cmp L R : Option \u03b1 l\u271d r\u271d : optParam (Option \u03b1) none x\u271d : \u2200 (l r : \u03b1), L = some l \u2192 R = some r \u2192 False \u22a2 \u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmp a b = Ordering.lt ** next h => intro _ ha _ hb; cases h _ _ ha hb ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering inst\u271d : TransCmp cmp L R : Option \u03b1 l\u271d r\u271d : optParam (Option \u03b1) none h : \u2200 (l r : \u03b1), L = some l \u2192 R = some r \u2192 False \u22a2 \u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmp a b = Ordering.lt ** intro _ ha _ hb ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering inst\u271d : TransCmp cmp L R : Option \u03b1 l\u271d r\u271d : optParam (Option \u03b1) none h : \u2200 (l r : \u03b1), L = some l \u2192 R = some r \u2192 False a\u271d : \u03b1 ha : L = some a\u271d b\u271d : \u03b1 hb : R = some b\u271d \u22a2 cmp a\u271d b\u271d = Ordering.lt ** cases h _ _ ha hb ** case node \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering inst\u271d : TransCmp cmp c\u271d : RBColor l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 l_ih\u271d :   \u2200 {L R : Option \u03b1},     isOrdered cmp l L R = true \u2194       (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) l) \u2227         (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) l) \u2227           (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp l r_ih\u271d :   \u2200 {L R : Option \u03b1},     isOrdered cmp r L R = true \u2194       (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) r) \u2227         (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) r) \u2227           (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp r L R : Option \u03b1 \u22a2 isOrdered cmp (node c\u271d l v r) L R = true \u2194     (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) (node c\u271d l v r)) \u2227       (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) (node c\u271d l v r)) \u2227         (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp (node c\u271d l v r) ** simp [isOrdered, *] ** case node \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering inst\u271d : TransCmp cmp c\u271d : RBColor l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 l_ih\u271d :   \u2200 {L R : Option \u03b1},     isOrdered cmp l L R = true \u2194       (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) l) \u2227         (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) l) \u2227           (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp l r_ih\u271d :   \u2200 {L R : Option \u03b1},     isOrdered cmp r L R = true \u2194       (\u2200 (a : \u03b1), a \u2208 L \u2192 All (fun x => cmpLT cmp a x) r) \u2227         (\u2200 (a : \u03b1), a \u2208 R \u2192 All (fun x => cmpLT cmp x a) r) \u2227           (\u2200 (a : \u03b1), a \u2208 L \u2192 \u2200 (b : \u03b1), b \u2208 R \u2192 cmpLT cmp a b) \u2227 Ordered cmp r L R : Option \u03b1 \u22a2 ((\u2200 (a : \u03b1), L = some a \u2192 All (fun x => cmpLT cmp a x) l) \u2227         All (fun x => cmpLT cmp x v) l \u2227 (\u2200 (a : \u03b1), L = some a \u2192 cmpLT cmp a v) \u2227 Ordered cmp l) \u2227       All (fun x => cmpLT cmp v x) r \u2227         (\u2200 (a : \u03b1), R = some a \u2192 All (fun x => cmpLT cmp x a) r) \u2227           (\u2200 (b : \u03b1), R = some b \u2192 cmpLT cmp v b) \u2227 Ordered cmp r \u2194     (\u2200 (a : \u03b1), L = some a \u2192 cmpLT cmp a v \u2227 All (fun x => cmpLT cmp a x) l \u2227 All (fun x => cmpLT cmp a x) r) \u2227       (\u2200 (a : \u03b1), R = some a \u2192 cmpLT cmp v a \u2227 All (fun x => cmpLT cmp x a) l \u2227 All (fun x => cmpLT cmp x a) r) \u2227         (\u2200 (a : \u03b1), L = some a \u2192 \u2200 (b : \u03b1), R = some b \u2192 cmpLT cmp a b) \u2227           All (fun x => cmpLT cmp x v) l \u2227 All (fun x => cmpLT cmp v x) r \u2227 Ordered cmp l \u2227 Ordered cmp r ** exact \u27e8\n  fun \u27e8\u27e8Ll, lv, Lv, ol\u27e9, \u27e8vr, rR, vR, or\u27e9\u27e9 => \u27e8\n    fun _ h => \u27e8Lv _ h, Ll _ h, (Lv _ h).trans_l vr\u27e9,\n    fun _ h => \u27e8vR _ h, (vR _ h).trans_r lv, rR _ h\u27e9,\n    fun _ hL _ hR => (Lv _ hL).trans (vR _ hR),\n    lv, vr, ol, or\u27e9,\n  fun \u27e8hL, hR, _, lv, vr, ol, or\u27e9 => \u27e8\n    \u27e8fun _ h => (hL _ h).2.1, lv, fun _ h => (hL _ h).1, ol\u27e9,\n    \u27e8vr, fun _ h => (hR _ h).2.2, fun _ h => (hR _ h).1, or\u27e9\u27e9\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.stoppedValue_leastGE_snorm_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 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc 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 snorm (stoppedValue f (leastGE f r i)) 1 \u03bc \u2264 2 * \u2191\u2191\u03bc Set.univ * ENNReal.ofReal (r + \u2191R) ** refine' snorm_one_le_of_le' ((hf.stoppedValue_leastGE r).integrable _) _\n  (norm_stoppedValue_leastGE_le hr hf0 hbdd i) ** \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 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 0 \u2264 \u222b (x : \u03a9), stoppedValue f (leastGE f r i) x \u2202\u03bc ** rw [\u2190 integral_univ] ** \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 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 0 \u2264 \u222b (x : \u03a9) in Set.univ, stoppedValue f (leastGE f r i) x \u2202\u03bc ** refine' le_trans _ ((hf.stoppedValue_leastGE r).set_integral_le (zero_le _) MeasurableSet.univ) ** \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 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 0 \u2264 \u222b (\u03c9 : \u03a9) in Set.univ, stoppedValue f (leastGE f r 0) \u03c9 \u2202\u03bc ** simp_rw [stoppedValue, leastGE, hitting_of_le le_rfl, hf0, integral_zero', le_rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.one_lt_ncard_iff ** \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 1 < ncard s \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b ** rw [one_lt_ncard hs] ** \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 s \u2227 a \u2260 b) \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b ** simp only [exists_prop, exists_and_left] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator ** \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_meas_f : Measurable f \u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** revert f ** \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} \u22a2 \u2200 {f : \u03a9 \u2192 \u211d\u22650\u221e},     Measurable f \u2192       \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** have h_mul_indicator : \u2200 g, Measurable g \u2192 Measurable fun a => g a * T.indicator (fun _ => c) a :=\n  fun g h_mg => h_mg.mul (measurable_const.indicator h_meas_T) ** \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a \u22a2 \u2200 {f : \u03a9 \u2192 \u211d\u22650\u221e},     Measurable f \u2192       \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** apply @Measurable.ennreal_induction _ Mf ** case h_ind \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a \u22a2 \u2200 (c_1 : \u211d\u22650\u221e) \u2983s : Set \u03a9\u2984,     MeasurableSet s \u2192       \u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c_1) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =         (\u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c_1) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** intro c' s' h_meas_s' ** case h_ind \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a c' : \u211d\u22650\u221e s' : Set \u03a9 h_meas_s' : MeasurableSet s' \u22a2 \u222b\u207b (\u03c9 : \u03a9), indicator s' (fun x => c') \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =     (\u222b\u207b (\u03c9 : \u03a9), indicator s' (fun x => c') \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** simp_rw [\u2190 inter_indicator_mul] ** case h_ind \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a c' : \u211d\u22650\u221e s' : Set \u03a9 h_meas_s' : MeasurableSet s' \u22a2 \u222b\u207b (\u03c9 : \u03a9), indicator (s' \u2229 T) (fun x => c' * c) \u03c9 \u2202\u03bc =     (\u222b\u207b (\u03c9 : \u03a9), indicator s' (fun x => c') \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** rw [lintegral_indicator _ (MeasurableSet.inter (hMf _ h_meas_s') h_meas_T),\n  lintegral_indicator _ (hMf _ h_meas_s'), lintegral_indicator _ h_meas_T] ** case h_ind \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a c' : \u211d\u22650\u221e s' : Set \u03a9 h_meas_s' : MeasurableSet s' \u22a2 \u222b\u207b (a : \u03a9) in s' \u2229 T, c' * c \u2202\u03bc = (\u222b\u207b (a : \u03a9) in s', c' \u2202\u03bc) * \u222b\u207b (a : \u03a9) in T, c \u2202\u03bc ** simp only [measurable_const, lintegral_const, univ_inter, lintegral_const_mul,\n  MeasurableSet.univ, Measure.restrict_apply] ** case h_ind \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a c' : \u211d\u22650\u221e s' : Set \u03a9 h_meas_s' : MeasurableSet s' \u22a2 c' * c * \u2191\u2191\u03bc (s' \u2229 T) = c' * \u2191\u2191\u03bc s' * (c * \u2191\u2191\u03bc T) ** rw [IndepSets_iff] at h_ind ** case h_ind \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : \u2200 (t1 t2 : Set \u03a9), t1 \u2208 {s | MeasurableSet s} \u2192 t2 \u2208 {T} \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2 h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a c' : \u211d\u22650\u221e s' : Set \u03a9 h_meas_s' : MeasurableSet s' \u22a2 c' * c * \u2191\u2191\u03bc (s' \u2229 T) = c' * \u2191\u2191\u03bc s' * (c * \u2191\u2191\u03bc T) ** rw [mul_mul_mul_comm, h_ind s' T h_meas_s' (Set.mem_singleton _)] ** case h_add \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a \u22a2 \u2200 \u2983f g : \u03a9 \u2192 \u211d\u22650\u221e\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Measurable f \u2192         Measurable g \u2192           \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =               (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc \u2192             \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =                 (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc \u2192               \u222b\u207b (\u03c9 : \u03a9), (f + g) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =                 (\u222b\u207b (\u03c9 : \u03a9), (f + g) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** intro f' g _ h_meas_f' _ h_ind_f' h_ind_g ** case h_add \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g\u271d : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a f' g : \u03a9 \u2192 \u211d\u22650\u221e a\u271d\u00b9 : Disjoint (Function.support f') (Function.support g) h_meas_f' : Measurable f' a\u271d : Measurable g h_ind_f' :   \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_ind_g :   \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc \u22a2 \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =     (\u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** have h_measM_f' : Measurable f' := h_meas_f'.mono hMf le_rfl ** case h_add \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g\u271d : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a f' g : \u03a9 \u2192 \u211d\u22650\u221e a\u271d\u00b9 : Disjoint (Function.support f') (Function.support g) h_meas_f' : Measurable f' a\u271d : Measurable g h_ind_f' :   \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_ind_g :   \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_measM_f' : Measurable f' \u22a2 \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =     (\u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** simp_rw [Pi.add_apply, right_distrib] ** case h_add \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g\u271d : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a f' g : \u03a9 \u2192 \u211d\u22650\u221e a\u271d\u00b9 : Disjoint (Function.support f') (Function.support g) h_meas_f' : Measurable f' a\u271d : Measurable g h_ind_f' :   \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_ind_g :   \u222b\u207b (\u03c9 : \u03a9), g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_measM_f' : Measurable f' \u22a2 \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 * indicator T (fun x => c) \u03c9 + g \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =     (\u222b\u207b (\u03c9 : \u03a9), f' \u03c9 + g \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** rw [lintegral_add_left (h_mul_indicator _ h_measM_f'), lintegral_add_left h_measM_f',\n  right_distrib, h_ind_f', h_ind_g] ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a \u22a2 \u2200 \u2983f : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\u2984,     (\u2200 (n : \u2115), Measurable (f n)) \u2192       Monotone f \u2192         (\u2200 (n : \u2115),             \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =               (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc) \u2192           \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =             (\u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** intro f h_meas_f h_mono_f h_ind_f ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a f : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f : \u2200 (n : \u2115), Measurable (f n) h_mono_f : Monotone f h_ind_f :   \u2200 (n : \u2115),     \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =       (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc \u22a2 \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =     (\u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** have h_measM_f : \u2200 n, Measurable (f n) := fun n => (h_meas_f n).mono hMf le_rfl ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a f : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f : \u2200 (n : \u2115), Measurable (f n) h_mono_f : Monotone f h_ind_f :   \u2200 (n : \u2115),     \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =       (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_measM_f : \u2200 (n : \u2115), Measurable (f n) \u22a2 \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =     (\u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f n x) \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** simp_rw [ENNReal.iSup_mul] ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a f : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f : \u2200 (n : \u2115), Measurable (f n) h_mono_f : Monotone f h_ind_f :   \u2200 (n : \u2115),     \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =       (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_measM_f : \u2200 (n : \u2115), Measurable (f n) \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2a06 i, f i \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =     (\u222b\u207b (\u03c9 : \u03a9), \u2a06 n, f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** rw [lintegral_iSup h_measM_f h_mono_f, lintegral_iSup, ENNReal.iSup_mul] ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a f : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f : \u2200 (n : \u2115), Measurable (f n) h_mono_f : Monotone f h_ind_f :   \u2200 (n : \u2115),     \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =       (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_measM_f : \u2200 (n : \u2115), Measurable (f n) \u22a2 \u2a06 n, \u222b\u207b (a : \u03a9), f n a * indicator T (fun x => c) a \u2202\u03bc =     \u2a06 i, (\u222b\u207b (a : \u03a9), f i a \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc ** simp_rw [\u2190 h_ind_f] ** case h_iSup.hf \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a f : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f : \u2200 (n : \u2115), Measurable (f n) h_mono_f : Monotone f h_ind_f :   \u2200 (n : \u2115),     \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =       (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_measM_f : \u2200 (n : \u2115), Measurable (f n) \u22a2 \u2200 (n : \u2115), Measurable fun \u03c9 => f n \u03c9 * indicator T (fun x => c) \u03c9 ** exact fun n => h_mul_indicator _ (h_measM_f n) ** case h_iSup.h_mono \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 c : \u211d\u22650\u221e T : Set \u03a9 h_meas_T : MeasurableSet T h_ind : IndepSets {s | MeasurableSet s} {T} h_mul_indicator : \u2200 (g : \u03a9 \u2192 \u211d\u22650\u221e), Measurable g \u2192 Measurable fun a => g a * indicator T (fun x => c) a f : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f : \u2200 (n : \u2115), Measurable (f n) h_mono_f : Monotone f h_ind_f :   \u2200 (n : \u2115),     \u222b\u207b (\u03c9 : \u03a9), f n \u03c9 * indicator T (fun x => c) \u03c9 \u2202\u03bc =       (\u222b\u207b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator T (fun x => c) \u03c9 \u2202\u03bc h_measM_f : \u2200 (n : \u2115), Measurable (f n) \u22a2 Monotone fun i \u03c9 => f i \u03c9 * indicator T (fun x => c) \u03c9 ** exact fun m n h_le a => mul_le_mul_right' (h_mono_f h_le a) _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite' ** \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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc \u22a2 f =\u1d50[\u03bc] g ** rw [\u2190 ae_eq_trim_iff_of_aeStronglyMeasurable' hm hfm hgm] ** \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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc \u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm ** have hf_mk_int_finite :\n  \u2200 s, MeasurableSet[m] s \u2192 \u03bc.trim hm s < \u221e \u2192 @IntegrableOn _ _ m _ (hfm.mk f) s (\u03bc.trim hm) := by\n  intro s hs h\u03bcs\n  rw [trim_measurableSet_eq hm hs] at h\u03bcs\n  unfold IntegrableOn\n  rw [restrict_trim hm _ hs]\n  refine' Integrable.trim hm _ hfm.stronglyMeasurable_mk\n  exact Integrable.congr (hf_int_finite s hs h\u03bcs) (ae_restrict_of_ae hfm.ae_eq_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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s \u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm ** have hg_mk_int_finite :\n  \u2200 s, MeasurableSet[m] s \u2192 \u03bc.trim hm s < \u221e \u2192 @IntegrableOn _ _ m _ (hgm.mk g) s (\u03bc.trim hm) := by\n  intro s hs h\u03bcs\n  rw [trim_measurableSet_eq hm hs] at h\u03bcs\n  unfold IntegrableOn\n  rw [restrict_trim hm _ hs]\n  refine' Integrable.trim hm _ hgm.stronglyMeasurable_mk\n  exact Integrable.congr (hg_int_finite s hs h\u03bcs) (ae_restrict_of_ae hgm.ae_eq_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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s hg_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s \u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm ** have hfg_mk_eq :\n  \u2200 s : Set \u03b1,\n    MeasurableSet[m] s \u2192\n      \u03bc.trim hm s < \u221e \u2192 \u222b x in s, hfm.mk f x \u2202\u03bc.trim hm = \u222b x in s, hgm.mk g x \u2202\u03bc.trim hm := by\n  intro s hs h\u03bcs\n  rw [trim_measurableSet_eq hm hs] at h\u03bcs\n  rw [restrict_trim hm _ hs, \u2190 integral_trim hm hfm.stronglyMeasurable_mk, \u2190\n    integral_trim hm hgm.stronglyMeasurable_mk,\n    integral_congr_ae (ae_restrict_of_ae hfm.ae_eq_mk.symm),\n    integral_congr_ae (ae_restrict_of_ae hgm.ae_eq_mk.symm)]\n  exact hfg_eq s 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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s hg_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s hfg_mk_eq :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192         \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =           \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm \u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk g hgm ** exact ae_eq_of_forall_set_integral_eq_of_sigmaFinite hf_mk_int_finite hg_mk_int_finite hfg_mk_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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s ** intro s 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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u22a2 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s ** rw [trim_measurableSet_eq hm hs] at 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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s ** unfold IntegrableOn ** \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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 Integrable (AEStronglyMeasurable'.mk f hfm) ** exact Integrable.congr (hf_int_finite s hs h\u03bcs) (ae_restrict_of_ae hfm.ae_eq_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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s ** intro s 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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u22a2 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s ** rw [trim_measurableSet_eq hm hs] at 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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s ** unfold IntegrableOn ** \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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 Integrable (AEStronglyMeasurable'.mk g hgm) ** exact Integrable.congr (hg_int_finite s hs h\u03bcs) (ae_restrict_of_ae hgm.ae_eq_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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s hg_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s \u22a2 \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192         \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =           \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm ** intro s 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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s hg_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =     \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm ** rw [trim_measurableSet_eq hm hs] at 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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s hg_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk f hfm x \u2202Measure.trim \u03bc hm =     \u222b (x : \u03b1) in s, AEStronglyMeasurable'.mk g hgm x \u2202Measure.trim \u03bc hm ** rw [restrict_trim hm _ hs, \u2190 integral_trim hm hfm.stronglyMeasurable_mk, \u2190\n  integral_trim hm hgm.stronglyMeasurable_mk,\n  integral_congr_ae (ae_restrict_of_ae hfm.ae_eq_mk.symm),\n  integral_congr_ae (ae_restrict_of_ae hgm.ae_eq_mk.symm)] ** \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\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 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s hfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc hf_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk f hfm) s hg_mk_int_finite :   \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u2192 IntegrableOn (AEStronglyMeasurable'.mk g hgm) s s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = \u222b (a : \u03b1) in s, g a \u2202\u03bc ** exact hfg_eq s hs h\u03bcs ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toMeasure_bindOnSupport_apply ** \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 s : Set \u03b2 inst\u271d : MeasurableSpace \u03b2 hs : MeasurableSet s \u22a2 \u2191\u2191(toMeasure (bindOnSupport p f)) s = \u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191\u2191(toMeasure (f a h)) s ** simp only [toMeasure_apply_eq_toOuterMeasure_apply _ _ hs, toOuterMeasure_bindOnSupport_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.list_append ** \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 l\u2081 l\u2082 : List \u03b1 \u22a2 List.foldr (fun b s => ((l\u2081, l\u2082), b, s).2.1 :: ((l\u2081, l\u2082), b, s).2.2) (l\u2081, l\u2082).2 (l\u2081, l\u2082).1 = l\u2081 ++ l\u2082 ** induction l\u2081 <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image ** 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 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) ** 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 hf : InjOn f s u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n \u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) ** 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 hf : InjOn f s u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) \u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) ** 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 hf : InjOn f s 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 hf : InjOn f s 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 hf : InjOn f s 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 hf : InjOn f s 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 hf : InjOn f s 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 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =     \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** conv_lhs => 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 hf : InjOn f s 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 \u222b\u207b (x : E) in \u22c3 n, s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =     \u2211' (n : \u2115), \u222b\u207b (x : E) in 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 hf : InjOn f s 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 hf : InjOn f s 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 hf : InjOn f s 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 _ _ ** 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 : \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 \u2264     \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s 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 \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' (s \u2229 u n)) ** apply\n  lintegral_abs_det_fderiv_le_addHaar_image_aux2 \u03bc (hs.inter (u_meas n)) _\n    (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _)) (hf.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 hf : InjOn f s 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 hf : InjOn f s 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 hf : InjOn f s 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)) = \u2191\u2191\u03bc (f '' s) ** conv_rhs => 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 hf : InjOn f s 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)) = \u2191\u2191\u03bc (\u22c3 i, f '' (s \u2229 u i)) ** 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s 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 i => f '' (s \u2229 u i)) ** intro i j hij ** 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 hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s 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 i j : \u2115 hij : i \u2260 j \u22a2 (Disjoint on fun i => f '' (s \u2229 u i)) i j ** apply Disjoint.image _ hf (inter_subset_left _ _) (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 hf : InjOn f s 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 i j : \u2115 hij : i \u2260 j \u22a2 Disjoint (s \u2229 u i) (s \u2229 u j) ** exact\n  Disjoint.mono (inter_subset_right _ _) (inter_subset_right _ _)\n    (disjoint_disjointed _ hij) ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s 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 (f '' (s \u2229 u i)) ** intro i ** 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 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s 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 i : \u2115 \u22a2 MeasurableSet (f '' (s \u2229 u i)) ** exact\n  measurable_image_of_fderivWithin (hs.inter (u_meas i))\n    (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _))\n    (hf.mono (inter_subset_left _ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.sizeOf_lt_sizeOf_of_mem ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : SizeOf \u03b1 x : \u03b1 s : Finset \u03b1 hx : x \u2208 s \u22a2 sizeOf x < sizeOf s ** cases s ** case mk \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : SizeOf \u03b1 x : \u03b1 val\u271d : Multiset \u03b1 nodup\u271d : Nodup val\u271d hx : x \u2208 { val := val\u271d, nodup := nodup\u271d } \u22a2 sizeOf x < sizeOf { val := val\u271d, nodup := nodup\u271d } ** dsimp [SizeOf.sizeOf, SizeOf.sizeOf, Multiset.sizeOf] ** case mk \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : SizeOf \u03b1 x : \u03b1 val\u271d : Multiset \u03b1 nodup\u271d : Nodup val\u271d hx : x \u2208 { val := val\u271d, nodup := nodup\u271d } \u22a2 sizeOf x < 1 + Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1) ** rw [add_comm] ** case mk \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : SizeOf \u03b1 x : \u03b1 val\u271d : Multiset \u03b1 nodup\u271d : Nodup val\u271d hx : x \u2208 { val := val\u271d, nodup := nodup\u271d } \u22a2 sizeOf x < Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1) + 1 ** refine' lt_trans _ (Nat.lt_succ_self _) ** case mk \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : SizeOf \u03b1 x : \u03b1 val\u271d : Multiset \u03b1 nodup\u271d : Nodup val\u271d hx : x \u2208 { val := val\u271d, nodup := nodup\u271d } \u22a2 sizeOf x < Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1) ** exact Multiset.sizeOf_lt_sizeOf_of_mem hx ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.sigmaFiniteTrim_mono ** \u03b1 : Type u_1 m\u271d m0\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 s : Set \u03b1 m m\u2082 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 hm\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) \u22a2 SigmaFinite (Measure.trim \u03bc hm) ** have _ := Measure.FiniteSpanningSetsIn (\u03bc.trim (hm\u2082.trans hm)) Set.univ ** \u03b1 : Type u_1 m\u271d m0\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 s : Set \u03b1 m m\u2082 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 hm\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) x\u271d : Type u_1 \u22a2 SigmaFinite (Measure.trim \u03bc hm) ** refine' Measure.FiniteSpanningSetsIn.sigmaFinite _ ** case refine'_1 \u03b1 : Type u_1 m\u271d m0\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 s : Set \u03b1 m m\u2082 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 hm\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) x\u271d : Type u_1 \u22a2 Set (Set \u03b1) ** exact Set.univ ** case refine'_2 \u03b1 : Type u_1 m\u271d m0\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 s : Set \u03b1 m m\u2082 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 hm\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) x\u271d : Type u_1 \u22a2 Measure.FiniteSpanningSetsIn (Measure.trim \u03bc hm) Set.univ ** refine'\n  { set := spanningSets (\u03bc.trim (hm\u2082.trans hm))\n    set_mem := fun _ => Set.mem_univ _\n    finite := fun i => _ spanning := iUnion_spanningSets _ } ** case refine'_2 \u03b1 : Type u_1 m\u271d m0\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 s : Set \u03b1 m m\u2082 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 hm\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) x\u271d : Type u_1 i : \u2115 \u22a2 \u2191\u2191(Measure.trim \u03bc hm) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i) < \u22a4 ** calc\n  (\u03bc.trim hm) (spanningSets (\u03bc.trim (hm\u2082.trans hm)) i) =\n      ((\u03bc.trim hm).trim hm\u2082) (spanningSets (\u03bc.trim (hm\u2082.trans hm)) i) :=\n    by rw [@trim_measurableSet_eq \u03b1 m\u2082 m (\u03bc.trim hm) _ hm\u2082 (measurable_spanningSets _ _)]\n  _ = (\u03bc.trim (hm\u2082.trans hm)) (spanningSets (\u03bc.trim (hm\u2082.trans hm)) i) := by\n    rw [@trim_trim _ _ \u03bc _ _ hm\u2082 hm]\n  _ < \u221e := measure_spanningSets_lt_top _ _ ** \u03b1 : Type u_1 m\u271d m0\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 s : Set \u03b1 m m\u2082 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 hm\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) x\u271d : Type u_1 i : \u2115 \u22a2 \u2191\u2191(Measure.trim \u03bc hm) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i) =     \u2191\u2191(Measure.trim (Measure.trim \u03bc hm) hm\u2082) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i) ** rw [@trim_measurableSet_eq \u03b1 m\u2082 m (\u03bc.trim hm) _ hm\u2082 (measurable_spanningSets _ _)] ** \u03b1 : Type u_1 m\u271d m0\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 s : Set \u03b1 m m\u2082 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 hm\u2082 : m\u2082 \u2264 m inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) x\u271d : Type u_1 i : \u2115 \u22a2 \u2191\u2191(Measure.trim (Measure.trim \u03bc hm) hm\u2082) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i) =     \u2191\u2191(Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) (spanningSets (Measure.trim \u03bc (_ : m\u2082 \u2264 m0)) i) ** rw [@trim_trim _ _ \u03bc _ _ hm\u2082 hm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.setToL1S_sub ** \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 f g : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 setToL1S T (f - g) = setToL1S T f - setToL1S T g ** rw [sub_eq_add_neg, setToL1S_add T h_zero h_add, setToL1S_neg h_zero h_add, sub_eq_add_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.trim_eq_trim_iff ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m m\u2081 m\u2082 : OuterMeasure \u03b1 \u22a2 trim m\u2081 = trim m\u2082 \u2194 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191m\u2081 s = \u2191m\u2082 s ** simp only [le_antisymm_iff, trim_le_trim_iff, forall_and] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.WF.out ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 h : WF m \u22a2 m.size = Buckets.size m.buckets \u2227 Buckets.WF m.buckets ** induction h with\n| mk h\u2081 h\u2082 => exact \u27e8h\u2081, h\u2082\u27e9\n| @empty' _ h => exact \u27e8(Buckets.mk_size h).symm, .mk' h\u27e9\n| insert _ ih => exact \u27e8insert_size ih.1, insert_WF ih.2\u27e9\n| erase _ ih => exact \u27e8erase_size ih.1, erase_WF ih.2\u27e9\n| modify _ ih => exact \u27e8modify_size ih.1, modify_WF ih.2\u27e9 ** case mk \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m m\u271d : Imp \u03b1 \u03b2 h\u2081 : m\u271d.size = Buckets.size m\u271d.buckets h\u2082 : Buckets.WF m\u271d.buckets \u22a2 m\u271d.size = Buckets.size m\u271d.buckets \u2227 Buckets.WF m\u271d.buckets ** exact \u27e8h\u2081, h\u2082\u27e9 ** case empty' \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 n\u271d : Nat h : 0 < n\u271d \u22a2 (Imp.empty' n\u271d).size = Buckets.size (Imp.empty' n\u271d).buckets \u2227 Buckets.WF (Imp.empty' n\u271d).buckets ** exact \u27e8(Buckets.mk_size h).symm, .mk' h\u27e9 ** case insert \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m m\u271d : Imp \u03b1 \u03b2 a\u271d\u00b9 : \u03b1 b\u271d : \u03b2 a\u271d : WF m\u271d ih : m\u271d.size = Buckets.size m\u271d.buckets \u2227 Buckets.WF m\u271d.buckets \u22a2 (Imp.insert m\u271d a\u271d\u00b9 b\u271d).size = Buckets.size (Imp.insert m\u271d a\u271d\u00b9 b\u271d).buckets \u2227 Buckets.WF (Imp.insert m\u271d a\u271d\u00b9 b\u271d).buckets ** exact \u27e8insert_size ih.1, insert_WF ih.2\u27e9 ** case erase \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m m\u271d : Imp \u03b1 \u03b2 a\u271d\u00b9 : \u03b1 a\u271d : WF m\u271d ih : m\u271d.size = Buckets.size m\u271d.buckets \u2227 Buckets.WF m\u271d.buckets \u22a2 (Imp.erase m\u271d a\u271d\u00b9).size = Buckets.size (Imp.erase m\u271d a\u271d\u00b9).buckets \u2227 Buckets.WF (Imp.erase m\u271d a\u271d\u00b9).buckets ** exact \u27e8erase_size ih.1, erase_WF ih.2\u27e9 ** case modify \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m m\u271d : Imp \u03b1 \u03b2 a\u271d\u00b9 : \u03b1 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 a\u271d : WF m\u271d ih : m\u271d.size = Buckets.size m\u271d.buckets \u2227 Buckets.WF m\u271d.buckets \u22a2 (Imp.modify m\u271d a\u271d\u00b9 f\u271d).size = Buckets.size (Imp.modify m\u271d a\u271d\u00b9 f\u271d).buckets \u2227 Buckets.WF (Imp.modify m\u271d a\u271d\u00b9 f\u271d).buckets ** exact \u27e8modify_size ih.1, modify_WF ih.2\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.mul_div_assoc' ** b a c : Int h : c \u2223 a \u22a2 div (a * b) c = div a c * b ** rw [Int.mul_comm, Int.mul_div_assoc _ h, Int.mul_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.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\u2098[\u03bc] \u03b2 \u22a2 Integrable f \u2192 Integrable g \u2192 Integrable (f + g) ** refine' induction_on\u2082 f g fun f hf g hg hfi hgi => _ ** \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 g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2 f : \u03b1 \u2192 \u03b2 hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 \u03b2 hg : AEStronglyMeasurable g \u03bc hfi : Integrable (mk f hf) hgi : Integrable (mk g hg) \u22a2 Integrable (mk f hf + mk g hg) ** simp only [integrable_mk, mk_add_mk] at hfi hgi \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\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f\u271d g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2 f : \u03b1 \u2192 \u03b2 hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 \u03b2 hg : AEStronglyMeasurable g \u03bc hfi : MeasureTheory.Integrable f hgi : MeasureTheory.Integrable g \u22a2 MeasureTheory.Integrable (f + g) ** exact hfi.add hgi ** Qed",
        "informal": ""
    },
    {
        "formal": "tendsto_measure_cthickening ** \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 \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) ** have A : Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n  rw [closure_eq_iInter_cthickening]\n  exact\n    tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.measurableSet)\n      (fun i j _ ij => cthickening_mono ij _) 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 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 A : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) ** have B : Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n  apply Tendsto.congr' _ tendsto_const_nhds\n  filter_upwards [self_mem_nhdsWithin (\u03b1 := \u211d)] with _ hr\n  rw [cthickening_of_nonpos hr] ** \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 A : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) B : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) ** convert B.sup A ** case h.e'_4 \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 A : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) B : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) \u22a2 \ud835\udcdd 0 = \ud835\udcdd[Iic 0] 0 \u2294 \ud835\udcdd[Ioi 0] 0 ** exact (nhds_left_sup_nhds_right' 0).symm ** \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 \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) ** rw [closure_eq_iInter_cthickening] ** \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 \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 \u03b4, \u22c2 (_ : 0 < \u03b4), cthickening \u03b4 s))) ** exact\n  tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.measurableSet)\n    (fun i j _ ij => cthickening_mono ij _) 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 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 A : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) ** apply Tendsto.congr' _ tendsto_const_nhds ** \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 A : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) \u22a2 (fun x => \u2191\u2191\u03bc (closure s)) =\u1da0[\ud835\udcdd[Iic 0] 0] fun r => \u2191\u2191\u03bc (cthickening r s) ** filter_upwards [self_mem_nhdsWithin (\u03b1 := \u211d)] with _ hr ** 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 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 A : Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closure s))) a\u271d : \u211d hr : a\u271d \u2208 Iic 0 \u22a2 \u2191\u2191\u03bc (closure s) = \u2191\u2191\u03bc (cthickening a\u271d s) ** rw [cthickening_of_nonpos hr] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image_add_left_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_1 => x + x_1) c) (Ioc a b) = Ioc (c + a) (c + b) ** rw [\u2190 map_add_left_Ioc, map_eq_image, addLeftEmbedding, Embedding.coeFn_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "torusIntegral_sub ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f g : (Fin n \u2192 \u2102) \u2192 E c : Fin n \u2192 \u2102 R : Fin n \u2192 \u211d hf : TorusIntegrable f c R hg : TorusIntegrable g c R \u22a2 (\u222f (x : Fin n \u2192 \u2102) in T(c, R), f x - g x) = (\u222f (x : Fin n \u2192 \u2102) in T(c, R), f x) - \u222f (x : Fin n \u2192 \u2102) in T(c, R), g x ** simpa only [sub_eq_add_neg, \u2190 torusIntegral_neg] using torusIntegral_add hf hg.neg ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.liftpPreservation_iff_uniform ** n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F \u22a2 LiftPPreservation \u2194 IsUniform ** rw [\u2190 suppPreservation_iff_liftpPreservation, suppPreservation_iff_isUniform] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.extract_zero_endPos ** c : Char cs : List Char \u22a2 extract { data := c :: cs } 0 (endPos { data := c :: cs }) = { data := c :: cs } ** simp [extract, Nat.ne_of_gt add_csize_pos] ** c : Char cs : List Char \u22a2 { data := extract.go\u2081 (c :: cs) 0 0 { byteIdx := utf8Len cs + csize c } } = { data := c :: cs } ** congr ** case e_data c : Char cs : List Char \u22a2 extract.go\u2081 (c :: cs) 0 0 { byteIdx := utf8Len cs + csize c } = c :: cs ** apply extract.go\u2081_zero_utf8Len ** Qed",
        "informal": ""
    },
    {
        "formal": "SNum.bit_one ** b : Bool \u22a2 (b::zero (decide \u00acb = true)) = nz (msb b) ** cases b <;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.cons_subset_cons ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s t : Finset \u03b1 a b : \u03b1 hs : \u00aca \u2208 s ht : \u00aca \u2208 t \u22a2 cons a s hs \u2286 cons a t ht \u2194 s \u2286 t ** rwa [\u2190 coe_subset, coe_cons, coe_cons, Set.insert_subset_insert_iff, coe_subset] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.vars_add_of_disjoint ** 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 : DecidableEq \u03c3 h : Disjoint (vars p) (vars q) \u22a2 vars (p + q) = vars p \u222a vars q ** apply Finset.Subset.antisymm (vars_add_subset p q) ** 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 : DecidableEq \u03c3 h : Disjoint (vars p) (vars q) \u22a2 vars p \u222a vars q \u2286 vars (p + q) ** intro x hx ** 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 : DecidableEq \u03c3 h : Disjoint (vars p) (vars q) x : \u03c3 hx : x \u2208 vars p \u222a vars q \u22a2 x \u2208 vars (p + q) ** simp only [vars_def, Multiset.disjoint_toFinset] at h hx \u22a2 ** 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 : DecidableEq \u03c3 x : \u03c3 h : Multiset.Disjoint (degrees p) (degrees q) hx : x \u2208 Multiset.toFinset (degrees p) \u222a Multiset.toFinset (degrees q) \u22a2 x \u2208 Multiset.toFinset (degrees (p + q)) ** rw [degrees_add_of_disjoint h, Multiset.toFinset_union] ** 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 : DecidableEq \u03c3 x : \u03c3 h : Multiset.Disjoint (degrees p) (degrees q) hx : x \u2208 Multiset.toFinset (degrees p) \u222a Multiset.toFinset (degrees q) \u22a2 x \u2208 Multiset.toFinset (degrees p) \u222a Multiset.toFinset (degrees q) ** exact hx ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.le_total_of_le_of_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx : x \u2264 z hy : y \u2264 z \u22a2 x \u2264 y \u2228 y \u2264 x ** rcases Part.eq_none_or_eq_some x with (h | \u27e8b, h\u2080\u27e9) ** case inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx : x \u2264 z hy : y \u2264 z b : \u03b1 h\u2080 : x = some b \u22a2 x \u2264 y \u2228 y \u2264 x ** right ** case inr.intro.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx : x \u2264 z hy : y \u2264 z b : \u03b1 h\u2080 : x = some b \u22a2 y \u2264 x ** intro b' h\u2081 ** case inr.intro.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx : x \u2264 z hy : y \u2264 z b : \u03b1 h\u2080 : x = some b b' : \u03b1 h\u2081 : b' \u2208 y \u22a2 b' \u2208 x ** rw [Part.eq_some_iff] at h\u2080 ** case inr.intro.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx : x \u2264 z hy : y \u2264 z b : \u03b1 h\u2080 : b \u2208 x b' : \u03b1 h\u2081 : b' \u2208 y \u22a2 b' \u2208 x ** have hx := hx _ h\u2080 ** case inr.intro.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx\u271d : x \u2264 z hy : y \u2264 z b : \u03b1 h\u2080 : b \u2208 x b' : \u03b1 h\u2081 : b' \u2208 y hx : b \u2208 z \u22a2 b' \u2208 x ** have hy := hy _ h\u2081 ** case inr.intro.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx\u271d : x \u2264 z hy\u271d : y \u2264 z b : \u03b1 h\u2080 : b \u2208 x b' : \u03b1 h\u2081 : b' \u2208 y hx : b \u2208 z hy : b' \u2208 z \u22a2 b' \u2208 x ** have hx := Part.mem_unique hx hy ** case inr.intro.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx\u271d\u00b9 : x \u2264 z hy\u271d : y \u2264 z b : \u03b1 h\u2080 : b \u2208 x b' : \u03b1 h\u2081 : b' \u2208 y hx\u271d : b \u2208 z hy : b' \u2208 z hx : b = b' \u22a2 b' \u2208 x ** subst hx ** case inr.intro.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx\u271d : x \u2264 z hy\u271d : y \u2264 z b : \u03b1 h\u2080 : b \u2208 x hx : b \u2208 z h\u2081 : b \u2208 y hy : b \u2208 z \u22a2 b \u2208 x ** exact h\u2080 ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx : x \u2264 z hy : y \u2264 z h : x = none \u22a2 x \u2264 y \u2228 y \u2264 x ** rw [h] ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx : x \u2264 z hy : y \u2264 z h : x = none \u22a2 none \u2264 y \u2228 y \u2264 none ** left ** case inl.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 x y z : Part \u03b1 hx : x \u2264 z hy : y \u2264 z h : x = none \u22a2 none \u2264 y ** apply OrderBot.bot_le _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.pdf.integrable_iff_integrable_mul_pdf ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f \u22a2 (Integrable fun x => f (X x)) \u2194 Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) ** erw [\u2190 integrable_map_measure hf.aestronglyMeasurable (HasPDF.measurable X \u2119 \u03bc).aemeasurable,\n  map_eq_withDensity_pdf X \u2119 \u03bc, integrable_withDensity_iff (measurable_pdf _ _ _) ae_lt_top] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.AssocList.mapKey_toList ** \u03b1 : Type u_1 \u03b4 : Type u_2 \u03b2 : Type u_3 f : \u03b1 \u2192 \u03b4 l : AssocList \u03b1 \u03b2 \u22a2 toList (mapKey f l) =     List.map       (fun x =>         match x with         | (a, b) => (f a, b))       (toList l) ** induction l <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toOuterMeasure_apply_inter_support ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 \u22a2 \u2191(toOuterMeasure p) (s \u2229 support p) = \u2191(toOuterMeasure p) s ** simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.mk_preimage_prod_right_fn_eq_if ** \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 inst\u271d : DecidablePred fun x => x \u2208 s g : \u03b4 \u2192 \u03b2 \u22a2 (fun b => (a, g b)) \u207b\u00b9' s \u00d7\u02e2 t = if a \u2208 s then g \u207b\u00b9' t else \u2205 ** rw [\u2190 mk_preimage_prod_right_eq_if, prod_preimage_right, preimage_preimage] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.stoppedValue_sub_eq_sum' ** \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 : AddCommGroup \u03b2 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u22a2 stoppedValue u \u03c0 - stoppedValue u \u03c4 = fun \u03c9 =>     Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (u (i + 1) - u i)) \u03c9 ** rw [stoppedValue_sub_eq_sum hle] ** \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 : AddCommGroup \u03b2 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u22a2 (fun \u03c9 => Finset.sum (Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9)) (fun i => u (i + 1) - u i) \u03c9) = fun \u03c9 =>     Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (u (i + 1) - u i)) \u03c9 ** ext \u03c9 ** 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 : AddCommGroup \u03b2 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u03c9 : \u03a9 \u22a2 Finset.sum (Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9)) (fun i => u (i + 1) - u i) \u03c9 =     Finset.sum (Finset.range (N + 1)) (fun i => Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9} (u (i + 1) - u i)) \u03c9 ** simp only [Finset.sum_apply, Finset.sum_indicator_eq_sum_filter] ** 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 : AddCommGroup \u03b2 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u03c9 : \u03a9 \u22a2 \u2211 c in Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9), (u (c + 1) - u c) \u03c9 =     \u2211 c in Finset.filter (fun i => \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}) (Finset.range (N + 1)), (u (c + 1) - u c) \u03c9 ** refine' Finset.sum_congr _ fun _ _ => rfl ** 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 : AddCommGroup \u03b2 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u03c9 : \u03a9 \u22a2 Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9) = Finset.filter (fun i => \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}) (Finset.range (N + 1)) ** ext i ** case h.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 : AddCommGroup \u03b2 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u03c9 : \u03a9 i : \u2115 \u22a2 i \u2208 Finset.Ico (\u03c4 \u03c9) (\u03c0 \u03c9) \u2194 i \u2208 Finset.filter (fun i => \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9}) (Finset.range (N + 1)) ** simp only [Finset.mem_filter, Set.mem_setOf_eq, Finset.mem_range, Finset.mem_Ico] ** case h.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 : AddCommGroup \u03b2 hle : \u03c4 \u2264 \u03c0 N : \u2115 hbdd : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N \u03c9 : \u03a9 i : \u2115 \u22a2 \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9 \u2194 i < N + 1 \u2227 \u03c4 \u03c9 \u2264 i \u2227 i < \u03c0 \u03c9 ** exact \u27e8fun h => \u27e8lt_trans h.2 (Nat.lt_succ_iff.2 <| hbdd _), h\u27e9, fun h => h.2\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.StronglyMeasurable.integral_prod_right ** \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd  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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : \u00acCompleteSpace E \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd ** borelize 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd ** haveI : SeparableSpace (range (uncurry f) \u222a {0} : Set E) :=\n  hf.separableSpace_range_union_singleton ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : SeparableSpace \u2191(range (uncurry f) \u222a {0}) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd ** let s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn _ hf.measurable (range (uncurry f) \u222a {0}) 0 (by simp) ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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\u03bd ** let s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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\u03bd ** let f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => {x | Integrable (f x) \u03bd}.indicator fun x => (s' n x).integral \u03bd ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (s' n x) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd ** have hf' : \u2200 n, StronglyMeasurable (f' n) := by\n  intro n; refine' StronglyMeasurable.indicator _ (measurableSet_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 measurable_measure_prod_mk_left\n  exact (s n).measurableSet_fiber x ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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\u03bd) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd ** exact stronglyMeasurable_of_tendsto _ hf' h2f' ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : \u00acCompleteSpace E \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd ** simp [integral, hE, stronglyMeasurable_const] ** \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : SeparableSpace \u2191(range (uncurry f) \u222a {0}) \u22a2 0 \u2208 range (uncurry f) \u222a {0} ** simp ** \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (s' n x) \u22a2 \u2200 (n : \u2115), StronglyMeasurable (f' n) ** intro n ** \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (s' n x) n : \u2115 \u22a2 StronglyMeasurable (f' n) ** refine' StronglyMeasurable.indicator _ (measurableSet_integrable 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (s' n x) n : \u2115 \u22a2 StronglyMeasurable fun x => SimpleFunc.integral \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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 \u03bd (s' n x) ** simp only [SimpleFunc.integral_eq_sum_of_subset (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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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\u03bd             (\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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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\u03bd           (\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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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\u03bd       (\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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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\u03bd       (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 measurable_measure_prod_mk_left ** case 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (s' n x) n : \u2115 x : \u03b1 \u22a2 SimpleFunc.range (s' n x) \u2286 SimpleFunc.range (s n) ** intro y ** \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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\u03bd) ** rw [tendsto_pi_nhds] ** \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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\u03bd)) ** intro x ** \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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\u03bd)) ** by_cases hfx : Integrable (f x) \u03bd ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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\u03bd)) ** have : \u2200 n, Integrable (s' n x) \u03bd := 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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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\u03bd)) ** simp only [hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,\n  mem_setOf_eq] ** 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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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\u03bd)     atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u03bd)) ** 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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) n : \u2115 \u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u03bd, \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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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\u03bd,       \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 ** refine' fun n => eventually_of_forall fun y => SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n ** case pos.refine'_1.refine'_1 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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 Measurable (uncurry f) ** exact hf.measurable ** case pos.refine'_1.refine'_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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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\u03bd,     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 _ _ _ ** case pos.refine'_2.refine'_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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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.refine'_3.a \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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] ** case pos.refine'_2.refine'_1 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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 Measurable fun x_1 => uncurry f (x, x_1) ** exact hf.measurable.of_uncurry_left ** case pos.refine'_2.refine'_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\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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : 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 \u03bd (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 \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 : NormedSpace \u211d E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) hE : CompleteSpace E this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : 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 \u03bd (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\u03bd)) ** simp [hfx, integral_undef] ** Qed",
        "informal": ""
    },
    {
        "formal": "UnionFind.lt_rankMax ** \u03b1 : Type u_1 self : UnionFind \u03b1 i : \u2115 \u22a2 rank self i < rankMax self ** simp [rank] ** \u03b1 : Type u_1 self : UnionFind \u03b1 i : \u2115 \u22a2 (if h : i < size self then self.arr[i].rank else 0) < rankMax self ** split ** case inl \u03b1 : Type u_1 self : UnionFind \u03b1 i : \u2115 h\u271d : i < size self \u22a2 self.arr[i].rank < rankMax self  case inr \u03b1 : Type u_1 self : UnionFind \u03b1 i : \u2115 h\u271d : \u00aci < size self \u22a2 0 < rankMax self ** {apply lt_rankMax'} ** case inr \u03b1 : Type u_1 self : UnionFind \u03b1 i : \u2115 h\u271d : \u00aci < size self \u22a2 0 < rankMax self ** apply Nat.succ_pos ** Qed",
        "informal": ""
    },
    {
        "formal": "WType.cardinal_mk_eq_sum ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u \u22a2 #(WType \u03b2) = sum fun a => #(WType \u03b2) ^ #(\u03b2 a) ** simp only [Cardinal.power_def, \u2190 Cardinal.mk_sigma] ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u \u22a2 #(WType \u03b2) = #((i : \u03b1) \u00d7 (\u03b2 i \u2192 WType \u03b2)) ** exact mk_congr (equivSigma \u03b2) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_normalize_iff_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 nonzero : \u03bc \u2260 0 \u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) \u2194     Tendsto \u03bcs F (\ud835\udcdd \u03bc) ** constructor ** case mp \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 nonzero : \u03bc \u2260 0 \u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) \u2192     Tendsto \u03bcs F (\ud835\udcdd \u03bc) ** rintro \u27e8normalized_lim, mass_lim\u27e9 ** case mp.intro \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 nonzero : \u03bc \u2260 0 normalized_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)) \u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) ** exact tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass normalized_lim mass_lim ** case mpr \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 nonzero : \u03bc \u2260 0 \u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2192     Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) ** intro \u03bcs_lim ** case mpr \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 nonzero : \u03bc \u2260 0 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) ** refine' \u27e8tendsto_normalize_of_tendsto \u03bcs_lim nonzero, \u03bcs_lim.mass\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.mem_vars_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 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R j : \u03c4 h : j \u2208 vars (\u2191(bind\u2081 f) \u03c6) \u22a2 \u2203 i, i \u2208 vars \u03c6 \u2227 j \u2208 vars (f i) ** classical\nsimpa only [exists_prop, Finset.mem_biUnion, mem_support_iff, Ne.def] using vars_bind\u2081 f \u03c6 h ** \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 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R j : \u03c4 h : j \u2208 vars (\u2191(bind\u2081 f) \u03c6) \u22a2 \u2203 i, i \u2208 vars \u03c6 \u2227 j \u2208 vars (f i) ** simpa only [exists_prop, Finset.mem_biUnion, mem_support_iff, Ne.def] using vars_bind\u2081 f \u03c6 h ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Icc_subset_Ioo_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 h\u2081 : a\u2081 \u2264 b\u2081 \u22a2 Icc a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082 \u2194 a\u2082 < a\u2081 \u2227 b\u2081 < b\u2082 ** rw [\u2190 coe_subset, coe_Icc, coe_Ioo, Set.Icc_subset_Ioo_iff h\u2081] ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.natAdd_subNat_cast ** n m : Nat i : Fin (n + m) h : n \u2264 \u2191i \u22a2 natAdd n (subNat n (cast (_ : n + m = m + n) i) h) = i ** simp [\u2190 cast_addNat] ** n m : Nat i : Fin (n + m) h : n \u2264 \u2191i \u22a2 cast (_ : n + m = n + m) i = i ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u22a2 0 \u2264\u1d50[\u03bc] f ** simp_rw [EventuallyLE, Pi.zero_apply] ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x ** rw [ae_const_le_iff_forall_lt_measure_zero] ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u22a2 \u2200 (b : \u211d), b < 0 \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0 ** intro b hb_neg ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 \u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0 ** let s := {x | f x \u2264 b} ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} \u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0 ** have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s \u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0 ** have mus : \u03bc s < \u221e := by\n  let c : \u211d\u22650 := \u27e8|b|, abs_nonneg _\u27e9\n  have c_pos : (c : \u211d\u22650\u221e) \u2260 0 := by simpa [\u2190 NNReal.coe_eq_zero] using hb_neg.ne\n  calc\n    \u03bc s \u2264 \u03bc {x | (c : \u211d\u22650\u221e) \u2264 \u2016f x\u2016\u208a} := by\n      apply measure_mono\n      intro x hx\n      simp only [Set.mem_setOf_eq] at hx\n      simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,\n        Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx\n    _ \u2264 (\u222b\u207b x, \u2016f x\u2016\u208a \u2202\u03bc) / c :=\n      (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)\n    _ < \u221e := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0 ** have h_int_gt : (\u222b x in s, f x \u2202\u03bc) \u2264 b * (\u03bc s).toReal := by\n  have h_const_le : (\u222b x in s, f x \u2202\u03bc) \u2264 \u222b _ in s, b \u2202\u03bc := by\n    refine'\n      set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _\n    rw [EventuallyLE, ae_restrict_iff hs]\n    exact eventually_of_forall fun x hxs => hxs\n  rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) \u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0 ** by_contra h ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) h : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0 \u22a2 False ** refine' (lt_self_iff_false (\u222b x in s, f x \u2202\u03bc)).mp (h_int_gt.trans_lt _) ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) h : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0 \u22a2 b * ENNReal.toReal (\u2191\u2191\u03bc s) < \u222b (x : \u03b1) in s, f x \u2202\u03bc ** refine' (mul_neg_iff.mpr (Or.inr \u27e8hb_neg, _\u27e9)).trans_le _ ** case refine'_1 \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) h : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0 \u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s)  case refine'_2 \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) h : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0 \u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc ** swap ** case refine'_1 \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) h : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0 \u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s) ** refine' ENNReal.toReal_nonneg.lt_of_ne fun h_eq => h _ ** case refine'_1 \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) h : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0 h_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s) \u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0 ** cases' (ENNReal.toReal_eq_zero_iff _).mp h_eq.symm with h\u03bcs_eq_zero h\u03bcs_eq_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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s \u22a2 \u2191\u2191\u03bc s < \u22a4 ** let c : \u211d\u22650 := \u27e8|b|, abs_nonneg _\u27e9 ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s c : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) } \u22a2 \u2191\u2191\u03bc s < \u22a4 ** have c_pos : (c : \u211d\u22650\u221e) \u2260 0 := by simpa [\u2190 NNReal.coe_eq_zero] using hb_neg.ne ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s c : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) } c_pos : \u2191c \u2260 0 \u22a2 \u2191\u2191\u03bc s < \u22a4 ** calc\n  \u03bc s \u2264 \u03bc {x | (c : \u211d\u22650\u221e) \u2264 \u2016f x\u2016\u208a} := by\n    apply measure_mono\n    intro x hx\n    simp only [Set.mem_setOf_eq] at hx\n    simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,\n      Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx\n  _ \u2264 (\u222b\u207b x, \u2016f x\u2016\u208a \u2202\u03bc) / c :=\n    (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)\n  _ < \u221e := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s c : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) } \u22a2 \u2191c \u2260 0 ** simpa [\u2190 NNReal.coe_eq_zero] using hb_neg.ne ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s c : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) } c_pos : \u2191c \u2260 0 \u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a} ** apply measure_mono ** case h \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s c : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) } c_pos : \u2191c \u2260 0 \u22a2 s \u2286 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a} ** intro x hx ** case h \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s c : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) } c_pos : \u2191c \u2260 0 x : \u03b1 hx : x \u2208 s \u22a2 x \u2208 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a} ** simp only [Set.mem_setOf_eq] at hx ** case h \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s c : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) } c_pos : \u2191c \u2260 0 x : \u03b1 hx : f x \u2264 b \u22a2 x \u2208 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a} ** simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,\n  Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) ** have h_const_le : (\u222b x in s, f x \u2202\u03bc) \u2264 \u222b _ in s, b \u2202\u03bc := by\n  refine'\n    set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _\n  rw [EventuallyLE, ae_restrict_iff hs]\n  exact eventually_of_forall fun x hxs => hxs ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_const_le : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, b \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) ** rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, b \u2202\u03bc ** refine'\n  set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _ ** \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 \u22a2 (fun x => f x) \u2264\u1d50[Measure.restrict \u03bc s] fun x => b ** rw [EventuallyLE, ae_restrict_iff 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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 b ** exact eventually_of_forall fun x hxs => hxs ** case refine'_2 \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) h : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0 \u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc ** exact hf_zero s hs mus ** case refine'_1.inl \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) h : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0 h_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s) h\u03bcs_eq_zero : \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0 ** exact h\u03bcs_eq_zero ** case refine'_1.inr \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 : \u03b1 \u2192 \u211d hfm : StronglyMeasurable f hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc b : \u211d hb_neg : b < 0 s : Set \u03b1 := {x | f x \u2264 b} hs : MeasurableSet s mus : \u2191\u2191\u03bc s < \u22a4 h_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s) h : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0 h_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s) h\u03bcs_eq_top : \u2191\u2191\u03bc s = \u22a4 \u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0 ** exact absurd h\u03bcs_eq_top mus.ne ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.snorm_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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** by_cases hp_ne_zero : p = 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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 \u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** by_cases hp_ne_top : p = \u221e ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 \u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8M, hM', hM\u27e9 := Mem\u2112p.integral_indicator_norm_ge_nonneg_le\n  (\u03bc := \u03bc) (hf.norm_rpow hp_ne_zero hp_ne_top) (Real.rpow_pos_of_pos h\u03b5 p.toReal) ** 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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) \u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e8M ^ (1 / p.toReal), _\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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) \u22a2 snorm (Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** rw [snorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top, \u2190 ENNReal.rpow_one (ENNReal.ofReal \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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) \u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^       (1 / ENNReal.toReal p) \u2264     ENNReal.ofReal \u03b5 ^ 1 ** conv_rhs => rw [\u2190 mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm] ** 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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) \u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^       (1 / ENNReal.toReal p) \u2264     ENNReal.ofReal \u03b5 ^ (ENNReal.toReal p * (1 / ENNReal.toReal p)) ** rw [ENNReal.rpow_mul,\n  ENNReal.rpow_le_rpow_iff (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top),\n  ENNReal.ofReal_rpow_of_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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) ** convert hM ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x\u271d : \u03b1 \u22a2 \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u271d\u2016\u208a ^ ENNReal.toReal p =     \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u271d\u2016\u208a ** rename_i x ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 \u22a2 \u2191\u2016Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a ^ ENNReal.toReal p =     \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a ** rw [ENNReal.coe_rpow_of_nonneg _ ENNReal.toReal_nonneg, nnnorm_indicator_eq_indicator_nnnorm,\n  nnnorm_indicator_eq_indicator_nnnorm] ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 \u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =     \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) x) ** have hiff : M ^ (1 / p.toReal) \u2264 \u2016f x\u2016\u208a \u2194 M \u2264 \u2016\u2016f x\u2016 ^ p.toReal\u2016\u208a := by\n  rw [coe_nnnorm, coe_nnnorm, Real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm,\n    \u2190 Real.rpow_le_rpow_iff hM' (Real.rpow_nonneg_of_nonneg (norm_nonneg _) _)\n      (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top), \u2190 Real.rpow_mul (norm_nonneg _),\n    mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm, Real.rpow_one] ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a \u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =     \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) x) ** by_cases hx : x \u2208 { x : \u03b1 | M ^ (1 / p.toReal) \u2264 \u2016f 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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : p = 0 \u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e81, hp_ne_zero.symm \u25b8 _\u27e9 ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : p = 0 \u22a2 snorm (Set.indicator {x | 1 \u2264 \u2191\u2016f x\u2016\u208a} f) 0 \u03bc \u2264 ENNReal.ofReal \u03b5 ** simp [snorm_exponent_zero] ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : p = \u22a4 \u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** subst hp_ne_top ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 f : \u03b1 \u2192 \u03b2 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hf : Mem\u2112p f \u22a4 hp_ne_zero : \u00ac\u22a4 = 0 \u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u22a4 \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8M, hM\u27e9 := hf.snormEssSup_indicator_norm_ge_eq_zero \u03bc hmeas ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 f : \u03b1 \u2192 \u03b2 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hf : Mem\u2112p f \u22a4 hp_ne_zero : \u00ac\u22a4 = 0 M : \u211d hM : snormEssSup (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc = 0 \u22a2 \u2203 M, snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u22a4 \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e8M, _\u27e9 ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 f : \u03b1 \u2192 \u03b2 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hf : Mem\u2112p f \u22a4 hp_ne_zero : \u00ac\u22a4 = 0 M : \u211d hM : snormEssSup (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc = 0 \u22a2 snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) \u22a4 \u03bc \u2264 ENNReal.ofReal \u03b5 ** simp only [snorm_exponent_top, hM, zero_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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 \u22a2 M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a ** rw [coe_nnnorm, coe_nnnorm, Real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm,\n  \u2190 Real.rpow_le_rpow_iff hM' (Real.rpow_nonneg_of_nonneg (norm_nonneg _) _)\n    (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top), \u2190 Real.rpow_mul (norm_nonneg _),\n  mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm, Real.rpow_one] ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =     \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) x) ** rw [Set.indicator_of_mem hx, Set.indicator_of_mem, Real.nnnorm_of_nonneg] ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 \u2191(\u2016f x\u2016\u208a ^ ENNReal.toReal p) = \u2191{ val := \u2016f x\u2016 ^ ENNReal.toReal p, property := ?pos\u271d }  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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 0 \u2264 \u2016f x\u2016 ^ ENNReal.toReal p  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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 x \u2208 {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} ** rfl ** 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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 x \u2208 {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} ** rw [Set.mem_setOf_eq] ** 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 : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : x \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a ** rwa [\u2190 hiff] ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 \u2191(Set.indicator {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016\u208a) x ^ ENNReal.toReal p) =     \u2191(Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun a => \u2016\u2016f a\u2016 ^ ENNReal.toReal p\u2016\u208a) x) ** rw [Set.indicator_of_not_mem hx, Set.indicator_of_not_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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 \u2191(0 ^ ENNReal.toReal p) = \u21910 ** simp [(ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm] ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 \u00acx \u2208 {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} ** rw [Set.mem_setOf_eq] ** 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 p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hp_ne_zero : \u00acp = 0 hp_ne_top : \u00acp = \u22a4 M : \u211d hM' : 0 \u2264 M hM :   \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a} (fun x => \u2016f x\u2016 ^ ENNReal.toReal p) x\u2016\u208a \u2202\u03bc \u2264     ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) x : \u03b1 hiff : M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a \u2194 M \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a hx : \u00acx \u2208 {x | M ^ (1 / ENNReal.toReal p) \u2264 \u2191\u2016f x\u2016\u208a} \u22a2 \u00acM \u2264 \u2191\u2016\u2016f x\u2016 ^ ENNReal.toReal p\u2016\u208a ** rwa [\u2190 hiff] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_of_tendsto_nnreal' ** \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 \u211d\u22650 g : \u03b1 \u2192 \u211d\u22650 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 ** simp_rw [\u2190 measurable_coe_nnreal_ennreal_iff] at hf \u22a2 ** \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 \u211d\u22650 g : \u03b1 \u2192 \u211d\u22650 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u lim : Tendsto f u (\ud835\udcdd g) hf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x) \u22a2 Measurable fun x => \u2191(g x) ** refine' measurable_of_tendsto_ennreal' u 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 \u211d\u22650 g : \u03b1 \u2192 \u211d\u22650 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u lim : Tendsto f u (\ud835\udcdd g) hf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x) \u22a2 Tendsto (fun i x => \u2191(f i x)) u (\ud835\udcdd fun x => \u2191(g x)) ** rw [tendsto_pi_nhds] at lim \u22a2 ** \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 \u211d\u22650 g : \u03b1 \u2192 \u211d\u22650 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u lim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x)) hf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x) \u22a2 \u2200 (x : \u03b1), Tendsto (fun i => \u2191(f i x)) u (\ud835\udcdd \u2191(g x)) ** exact fun x => (ENNReal.continuous_coe.tendsto (g x)).comp (lim x) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.setToL1S_indicatorConst ** \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 s : Set \u03b1 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 x : E \u22a2 setToL1S T (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x) = \u2191(T s) x ** have h_empty : T \u2205 = 0 := h_zero _ MeasurableSet.empty measure_empty ** \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 s : Set \u03b1 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 x : E h_empty : T \u2205 = 0 \u22a2 setToL1S T (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x) = \u2191(T s) x ** rw [setToL1S_eq_setToSimpleFunc] ** \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 s : Set \u03b1 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 x : E h_empty : T \u2205 = 0 \u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x)) = \u2191(T s) x ** refine' Eq.trans _ (SimpleFunc.setToSimpleFunc_indicator T h_empty hs 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\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 s : Set \u03b1 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 x : E h_empty : T \u2205 = 0 \u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x)) =     SimpleFunc.setToSimpleFunc T (SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 x) (SimpleFunc.const \u03b1 0)) ** 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 s : Set \u03b1 h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 x : E h_empty : T \u2205 = 0 \u22a2 \u2191(toSimpleFunc (indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) x)) =\u1d50[\u03bc]     \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 x) (SimpleFunc.const \u03b1 0)) ** exact toSimpleFunc_indicatorConst hs h\u03bcs.ne x ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_integral_tendsto ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc)) ** rw [tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds] ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u22a2 Tendsto (toFiniteMeasure \u2218 \u03bcs) F (\ud835\udcdd (toFiniteMeasure \u03bc)) \u2194     \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc)) ** rw [FiniteMeasure.tendsto_iff_forall_integral_tendsto] ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d),       Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191((toFiniteMeasure \u2218 \u03bcs) i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191(toFiniteMeasure \u03bc)))) \u2194     \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc)) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "String.foldr_eq ** \u03b1 : Type u_1 f : Char \u2192 \u03b1 \u2192 \u03b1 s : String a : \u03b1 \u22a2 foldr f a s = List.foldr f a s.data ** simpa using foldrAux_of_valid f [] s.1 [] a ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.inv_divInt ** n d : Int \u22a2 Rat.inv (n /. d) = d /. n ** if z : d = 0 then simp [z] else\ncases e : n /. d; rcases divInt_num_den z e with \u27e8g, zg, rfl, rfl\u27e9\nsimp [inv_def, divInt_mul_right zg] ** n d : Int z : d = 0 \u22a2 Rat.inv (n /. d) = d /. n ** simp [z] ** n d : Int z : \u00acd = 0 \u22a2 Rat.inv (n /. d) = d /. n ** cases e : n /. d ** case mk' n d : Int z : \u00acd = 0 num\u271d : Int den\u271d : Nat den_nz\u271d : den\u271d \u2260 0 reduced\u271d : Nat.Coprime (Int.natAbs num\u271d) den\u271d e : n /. d = mk' num\u271d den\u271d \u22a2 Rat.inv (mk' num\u271d den\u271d) = d /. n ** rcases divInt_num_den z e with \u27e8g, zg, rfl, rfl\u27e9 ** case mk'.intro.intro.intro num\u271d : Int den\u271d : Nat den_nz\u271d : den\u271d \u2260 0 reduced\u271d : Nat.Coprime (Int.natAbs num\u271d) den\u271d g : Int zg : g \u2260 0 z : \u00ac\u2191den\u271d * g = 0 e : num\u271d * g /. (\u2191den\u271d * g) = mk' num\u271d den\u271d \u22a2 Rat.inv (mk' num\u271d den\u271d) = \u2191den\u271d * g /. (num\u271d * g) ** simp [inv_def, divInt_mul_right zg] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.erase_append_left ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List \u03b1 h : a \u2208 l\u2081 \u22a2 List.erase (l\u2081 ++ l\u2082) a = List.erase l\u2081 a ++ l\u2082 ** simp [erase_eq_eraseP] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List \u03b1 h : a \u2208 l\u2081 \u22a2 eraseP (fun b => decide (a = b)) (l\u2081 ++ l\u2082) = eraseP (fun b => decide (a = b)) l\u2081 ++ l\u2082 ** exact eraseP_append_left (by exact decide_eq_true rfl) l\u2082 h ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List \u03b1 h : a \u2208 l\u2081 \u22a2 decide (a = a) = true ** exact decide_eq_true rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Function.Periodic.tendsto_atBot_intervalIntegral_of_pos ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d g : \u211d \u2192 \u211d hg : Periodic g T h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082 h\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x hT : 0 < T \u22a2 Tendsto (fun t => \u222b (x : \u211d) in 0 ..t, g x) atBot atBot ** apply tendsto_atBot_mono (hg.integral_le_sSup_add_zsmul_of_pos h_int hT) ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d g : \u211d \u2192 \u211d hg : Periodic g T h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082 h\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x hT : 0 < T \u22a2 Tendsto (fun n => sSup ((fun t => \u222b (x : \u211d) in 0 ..t, g x) '' Icc 0 T) + \u230an / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atBot     atBot ** apply atBot.tendsto_atBot_add_const_left (sSup <| (fun t => \u222b x in (0)..t, g x) '' Icc 0 T) ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d g : \u211d \u2192 \u211d hg : Periodic g T h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082 h\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x hT : 0 < T \u22a2 Tendsto (fun x => \u230ax / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atBot atBot ** apply Tendsto.atBot_zsmul_const h\u2080 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d g : \u211d \u2192 \u211d hg : Periodic g T h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082 h\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x hT : 0 < T \u22a2 Tendsto (fun x => \u230ax / T\u230b) atBot atBot ** exact tendsto_floor_atBot.comp (tendsto_id.atBot_mul_const (inv_pos.mpr hT)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.image_of_continuousOn_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 t\u03b3 : TopologicalSpace \u03b3 inst\u271d\u00b2 : PolishSpace \u03b3 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : BorelSpace \u03b3 hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s \u22a2 MeasurableSet (f '' s) ** obtain \u27e8t', t't, t'_polish, s_closed, _\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b3, t' \u2264 t\u03b3 \u2227 @PolishSpace \u03b3 t' \u2227 IsClosed[t'] s \u2227 IsOpen[t'] s :=\n  hs.isClopenable ** case intro.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 t\u03b3 : TopologicalSpace \u03b3 inst\u271d\u00b2 : PolishSpace \u03b3 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : BorelSpace \u03b3 hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s t' : TopologicalSpace \u03b3 t't : t' \u2264 t\u03b3 t'_polish : PolishSpace \u03b3 s_closed : IsClosed s right\u271d : IsOpen s \u22a2 MeasurableSet (f '' s) ** exact\n  @IsClosed.measurableSet_image_of_continuousOn_injOn \u03b3 t' t'_polish \u03b2 _ _ _ _ s s_closed f\n    (f_cont.mono_dom t't) f_inj ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.ediv_le_self ** a b : Int Ha : 0 \u2264 a \u22a2 a / b \u2264 a ** have := Int.le_trans le_natAbs (ofNat_le.2 <| natAbs_div_le_natAbs a b) ** a b : Int Ha : 0 \u2264 a this : a / b \u2264 \u2191(natAbs a) \u22a2 a / b \u2264 a ** rwa [natAbs_of_nonneg Ha] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "DFA.toNFA_correct ** \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : NFA \u03b1 \u03c3 M : DFA \u03b1 \u03c3 \u22a2 NFA.accepts (toNFA M) = accepts M ** ext x ** case h \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : NFA \u03b1 \u03c3 M : DFA \u03b1 \u03c3 x : List \u03b1 \u22a2 x \u2208 NFA.accepts (toNFA M) \u2194 x \u2208 accepts M ** rw [NFA.mem_accepts, toNFA_start, toNFA_evalFrom_match] ** case h \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : NFA \u03b1 \u03c3 M : DFA \u03b1 \u03c3 x : List \u03b1 \u22a2 (\u2203 S, S \u2208 (toNFA M).accept \u2227 S \u2208 {evalFrom M M.start x}) \u2194 x \u2208 accepts M ** constructor ** case h.mp \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : NFA \u03b1 \u03c3 M : DFA \u03b1 \u03c3 x : List \u03b1 \u22a2 (\u2203 S, S \u2208 (toNFA M).accept \u2227 S \u2208 {evalFrom M M.start x}) \u2192 x \u2208 accepts M ** rintro \u27e8S, hS\u2081, hS\u2082\u27e9 ** case h.mp.intro.intro \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : NFA \u03b1 \u03c3 M : DFA \u03b1 \u03c3 x : List \u03b1 S : \u03c3 hS\u2081 : S \u2208 (toNFA M).accept hS\u2082 : S \u2208 {evalFrom M M.start x} \u22a2 x \u2208 accepts M ** rwa [Set.mem_singleton_iff.mp hS\u2082] at hS\u2081 ** case h.mpr \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : NFA \u03b1 \u03c3 M : DFA \u03b1 \u03c3 x : List \u03b1 \u22a2 x \u2208 accepts M \u2192 \u2203 S, S \u2208 (toNFA M).accept \u2227 S \u2208 {evalFrom M M.start x} ** exact fun h => \u27e8M.eval x, h, rfl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_surjective ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 \u22a2 Surjective (image f) \u2194 Surjective f ** refine' \u27e8fun h y => _, Surjective.image_surjective\u27e9 ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Surjective (image f) y : \u03b2 \u22a2 \u2203 a, f a = y ** cases' h {y} with s hs ** case intro \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Surjective (image f) y : \u03b2 s : Set \u03b1 hs : f '' s = {y} \u22a2 \u2203 a, f a = y ** have := mem_singleton y ** case intro \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Surjective (image f) y : \u03b2 s : Set \u03b1 hs : f '' s = {y} this : y \u2208 {y} \u22a2 \u2203 a, f a = y ** rw [\u2190 hs] at this ** case intro \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Surjective (image f) y : \u03b2 s : Set \u03b1 hs : f '' s = {y} this : y \u2208 f '' s \u22a2 \u2203 a, f a = y ** rcases this with \u27e8x, _, hx\u27e9 ** case intro.intro.intro \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Surjective (image f) y : \u03b2 s : Set \u03b1 hs : f '' s = {y} x : \u03b1 left\u271d : x \u2208 s hx : f x = y \u22a2 \u2203 a, f a = y ** exact \u27e8x, hx\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.succ_ofInt' ** \u03b1 : Type u_1 n : \u2115 \u22a2 ZNum.ofInt' (\u2191n + 1) = ZNum.ofInt' \u2191n + 1 ** change ZNum.ofInt' (n + 1 : \u2115) = ZNum.ofInt' (n : \u2115) + 1 ** \u03b1 : Type u_1 n : \u2115 \u22a2 ZNum.ofInt' \u2191(n + 1) = ZNum.ofInt' \u2191n + 1 ** dsimp only [ZNum.ofInt', ZNum.ofInt'] ** \u03b1 : Type u_1 n : \u2115 \u22a2 toZNum (ofNat' (n + 1)) = toZNum (ofNat' n) + 1 ** rw [Num.ofNat'_succ, Num.add_one, toZNum_succ, ZNum.add_one] ** \u03b1 : Type u_1 \u22a2 ZNum.ofInt' (-[0+1] + 1) = ZNum.ofInt' -[0+1] + 1 ** change ZNum.ofInt' 0 = ZNum.ofInt' (-[0+1]) + 1 ** \u03b1 : Type u_1 \u22a2 ZNum.ofInt' 0 = ZNum.ofInt' -[0+1] + 1 ** dsimp only [ZNum.ofInt', ZNum.ofInt'] ** \u03b1 : Type u_1 \u22a2 toZNum (ofNat' 0) = toZNumNeg (ofNat' (0 + 1)) + 1 ** rw [ofNat'_succ, ofNat'_zero] ** \u03b1 : Type u_1 \u22a2 toZNum 0 = toZNumNeg (0 + 1) + 1 ** rfl ** \u03b1 : Type u_1 n : \u2115 \u22a2 ZNum.ofInt' (-[n + 1+1] + 1) = ZNum.ofInt' -[n + 1+1] + 1 ** change ZNum.ofInt' -[n+1] = ZNum.ofInt' -[(n + 1)+1] + 1 ** \u03b1 : Type u_1 n : \u2115 \u22a2 ZNum.ofInt' -[n+1] = ZNum.ofInt' -[n + 1+1] + 1 ** dsimp only [ZNum.ofInt', ZNum.ofInt'] ** \u03b1 : Type u_1 n : \u2115 \u22a2 toZNumNeg (ofNat' (n + 1)) = toZNumNeg (ofNat' (n + 1 + 1)) + 1 ** rw [@Num.ofNat'_succ (n + 1), Num.add_one, toZNumNeg_succ,\n  @ofNat'_succ n, Num.add_one, ZNum.add_one, pred_succ] ** Qed",
        "informal": ""
    },
    {
        "formal": "tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt_aux ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 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 0) ** refine' Metric.tendsto_nhds.2 fun \u03b5 \u03b5pos => _ ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5 ** obtain \u27e8\u03b4, h\u03b4, \u03b4pos\u27e9 : \u2203 \u03b4, (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 < \u03b5 \u2227 0 < \u03b4 := by\n  have A :\n    Tendsto (fun \u03b4 => (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd ((0 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + 0)) := by\n    apply Tendsto.mono_left _ nhdsWithin_le_nhds\n    exact (tendsto_id.mul tendsto_const_nhds).add tendsto_id\n  rw [zero_mul, zero_add] at A\n  exact (((tendsto_order.1 A).2 \u03b5 \u03b5pos).and self_mem_nhdsWithin).exists ** case intro.intro \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 \u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5 ** suffices \u2200\u1da0 i in l, \u2016\u222b x in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 by\n  filter_upwards [this] with i hi\n  simp only [dist_zero_right]\n  exact hi.trans_lt h\u03b4 ** case intro.intro \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 \u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 ** obtain \u27e8u, u_open, x\u2080u, hu\u27e9 : \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 x \u2208 u \u2229 s, g x \u2208 ball (g x\u2080) \u03b4 ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 \u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4  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\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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 \u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 ** exact mem_nhdsWithin.1 (hcg (ball_mem_nhds _ \u03b4pos)) ** 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\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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 \u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 ** filter_upwards [tendstoUniformlyOn_iff.1 (hl\u03c6 u u_open x\u2080u) \u03b4 \u03b4pos, hi\u03c6, hn\u03c6,\n  integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt hs hl\u03c6 hi\u03c6 hmg hcg] with i hi h'i\n  h\u03c6pos 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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 C : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc \u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 ** calc\n  \u2016\u222b x in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 =\n    \u2016(\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc) + \u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 := by\n    conv_lhs => rw [\u2190 diff_union_inter s u]\n    rw [integral_union (disjoint_sdiff_inter _ _) (hs.inter u_open.measurableSet)\n        (h''i.mono_set (diff_subset _ _)) (h''i.mono_set (inter_subset_left _ _))]\n  _ \u2264 \u2016\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 + \u2016\u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 := (norm_add_le _ _)\n  _ \u2264 (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 := add_le_add C B ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4 ** have A :\n  Tendsto (fun \u03b4 => (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd ((0 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + 0)) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  exact (tendsto_id.mul tendsto_const_nhds).add tendsto_id ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 A : Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0)) \u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4 ** rw [zero_mul, 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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 A : Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4 ** exact (((tendsto_order.1 A).2 \u03b5 \u03b5pos).and self_mem_nhdsWithin).exists ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0)) ** apply Tendsto.mono_left _ nhdsWithin_le_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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0)) ** exact (tendsto_id.mul tendsto_const_nhds).add tendsto_id ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 this : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 \u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5 ** filter_upwards [this] with i hi ** 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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 this : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 i : \u03b9 hi : \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 \u22a2 dist (\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) 0 < \u03b5 ** simp only [dist_zero_right] ** 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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 this : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 i : \u03b9 hi : \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 \u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 < \u03b5 ** exact hi.trans_lt h\u03b4 ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 \u222b (x : \u03b1) in s \u2229 u, \u2016\u03c6 i x \u2022 g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s \u2229 u, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc ** refine' set_integral_mono_on _ _ (hs.inter u_open.measurableSet) fun x hx => _ ** case refine'_3 \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s x : \u03b1 hx : x \u2208 s \u2229 u \u22a2 \u2016\u03c6 i x \u2022 g x\u2016 \u2264 \u2016\u03c6 i x\u2016 * \u03b4 ** rw [norm_smul] ** case refine'_3 \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s x : \u03b1 hx : x \u2208 s \u2229 u \u22a2 \u2016\u03c6 i x\u2016 * \u2016g x\u2016 \u2264 \u2016\u03c6 i x\u2016 * \u03b4 ** apply mul_le_mul_of_nonneg_left _ (norm_nonneg _) ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s x : \u03b1 hx : x \u2208 s \u2229 u \u22a2 \u2016g x\u2016 \u2264 \u03b4 ** rw [inter_comm, h'g] at hu ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 s \u2229 u \u2192 g x \u2208 ball 0 \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s x : \u03b1 hx : x \u2208 s \u2229 u \u22a2 \u2016g x\u2016 \u2264 \u03b4 ** exact (mem_ball_zero_iff.1 (hu x hx)).le ** case refine'_1 \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 IntegrableOn (fun x => \u2016\u03c6 i x \u2022 g x\u2016) (s \u2229 u) ** exact IntegrableOn.mono_set h''i.norm (inter_subset_left _ _) ** case refine'_2 \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 IntegrableOn (fun x => \u2016\u03c6 i x\u2016 * \u03b4) (s \u2229 u) ** exact\n  IntegrableOn.mono_set ((integrable_of_integral_eq_one h'i).norm.mul_const _)\n    (inter_subset_left _ _) ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 \u222b (x : \u03b1) in s \u2229 u, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc ** apply set_integral_mono_set ** case hfi \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 IntegrableOn (fun x => \u2016\u03c6 i x\u2016 * \u03b4) s ** exact (integrable_of_integral_eq_one h'i).norm.mul_const _ ** case hf \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2016\u03c6 i x\u2016 * \u03b4 ** exact eventually_of_forall fun x => mul_nonneg (norm_nonneg _) \u03b4pos.le ** case hst \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 s \u2229 u \u2264\u1da0[ae \u03bc] s ** apply eventually_of_forall ** case hst.hp \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 \u2200 (x : \u03b1), (s \u2229 u) x \u2264 s x ** exact inter_subset_left s u ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 \u222b (x : \u03b1) in s, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc = \u222b (x : \u03b1) in s, \u03c6 i x * \u03b4 \u2202\u03bc ** apply set_integral_congr hs fun x hx => ?_ ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s x : \u03b1 hx : x \u2208 s \u22a2 \u2016\u03c6 i x\u2016 * \u03b4 = \u03c6 i x * \u03b4 ** rw [Real.norm_of_nonneg (h\u03c6pos _ hx)] ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u22a2 \u222b (x : \u03b1) in s, \u03c6 i x * \u03b4 \u2202\u03bc = \u03b4 ** rw [integral_mul_right, h'i, one_mul] ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 \u22a2 \u222b (x : \u03b1) in s \\ u, \u2016\u03c6 i x \u2022 g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s \\ u, \u03b4 * \u2016g x\u2016 \u2202\u03bc ** refine' set_integral_mono_on _ _ (hs.diff u_open.measurableSet) fun x hx => _ ** case refine'_3 \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 x : \u03b1 hx : x \u2208 s \\ u \u22a2 \u2016\u03c6 i x \u2022 g x\u2016 \u2264 \u03b4 * \u2016g x\u2016 ** rw [norm_smul] ** case refine'_3 \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 x : \u03b1 hx : x \u2208 s \\ u \u22a2 \u2016\u03c6 i x\u2016 * \u2016g x\u2016 \u2264 \u03b4 * \u2016g x\u2016 ** apply mul_le_mul_of_nonneg_right _ (norm_nonneg _) ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 x : \u03b1 hx : x \u2208 s \\ u \u22a2 \u2016\u03c6 i x\u2016 \u2264 \u03b4 ** simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le ** case refine'_1 \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 \u22a2 IntegrableOn (fun x => \u2016\u03c6 i x \u2022 g x\u2016) (s \\ u) ** exact IntegrableOn.mono_set h''i.norm (diff_subset _ _) ** case refine'_2 \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 \u22a2 IntegrableOn (fun x => \u03b4 * \u2016g x\u2016) (s \\ u) ** exact IntegrableOn.mono_set (hmg.norm.const_mul _) (diff_subset _ _) ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 \u22a2 \u222b (x : \u03b1) in s \\ u, \u03b4 * \u2016g x\u2016 \u2202\u03bc \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc ** rw [integral_mul_left] ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 \u22a2 \u03b4 * \u222b (a : \u03b1) in s \\ u, \u2016g a\u2016 \u2202\u03bc \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc ** apply mul_le_mul_of_nonneg_left (set_integral_mono_set hmg.norm _ _) \u03b4pos.le ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 \u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun a => \u2016g a\u2016 ** exact eventually_of_forall fun x => norm_nonneg _ ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 \u22a2 s \\ u \u2264\u1da0[ae \u03bc] s ** apply eventually_of_forall ** case hp \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 \u22a2 \u2200 (x : \u03b1), (s \\ u) x \u2264 s x ** exact diff_subset s u ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 C : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc \u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 = \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc + \u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 ** conv_lhs => rw [\u2190 diff_union_inter s u] ** \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 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 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s h'g : g x\u2080 = 0 hcg : ContinuousWithinAt g s x\u2080 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u03b4 : \u211d h\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u03b4pos : 0 < \u03b4 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 h\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x h''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s B : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 C : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc \u22a2 \u2016\u222b (x : \u03b1) in s \\ u \u222a s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 =     \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc + \u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 ** rw [integral_union (disjoint_sdiff_inter _ _) (hs.inter u_open.measurableSet)\n    (h''i.mono_set (diff_subset _ _)) (h''i.mono_set (inter_subset_left _ _))] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.card_fintype_Icc_of_le ** a b : \u2124 h : a \u2264 b + 1 \u22a2 \u2191(Fintype.card \u2191(Set.Icc a b)) = b + 1 - a ** rw [card_fintype_Icc, toNat_sub_of_le h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_bot' ** \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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** by_cases h\u03bc_finite : IsFiniteMeasure \u03bc ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : \u00acIsFiniteMeasure \u03bc \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** 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 : \u03b1 \u2192 F' s : Set \u03b1 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** by_cases hf : Integrable f \u03bc ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : Integrable f \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : \u00acIntegrable f \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** 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 : \u03b1 \u2192 F' s : Set \u03b1 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : Integrable f \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** have h_meas : StronglyMeasurable[\u22a5] (\u03bc[f|\u22a5]) := stronglyMeasurable_condexp ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : Integrable f h_meas : StronglyMeasurable (\u03bc[f|\u22a5]) \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** obtain \u27e8c, h_eq\u27e9 := stronglyMeasurable_bot_iff.mp h_meas ** case pos.intro \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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : Integrable f h_meas : StronglyMeasurable (\u03bc[f|\u22a5]) c : F' h_eq : \u03bc[f|\u22a5] = fun x => c \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** rw [h_eq] ** case pos.intro \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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : Integrable f h_meas : StronglyMeasurable (\u03bc[f|\u22a5]) c : F' h_eq : \u03bc[f|\u22a5] = fun x => c \u22a2 (fun x => c) = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** have h_integral : \u222b x, (\u03bc[f|\u22a5]) x \u2202\u03bc = \u222b x, f x \u2202\u03bc := integral_condexp bot_le hf ** case pos.intro \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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : Integrable f h_meas : StronglyMeasurable (\u03bc[f|\u22a5]) c : F' h_eq : \u03bc[f|\u22a5] = fun x => c h_integral : \u222b (x : \u03b1), (\u03bc[f|\u22a5]) x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc \u22a2 (fun x => c) = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** simp_rw [h_eq, integral_const] at h_integral ** case pos.intro \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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : Integrable f h_meas : StronglyMeasurable (\u03bc[f|\u22a5]) c : F' h_eq : \u03bc[f|\u22a5] = fun x => c h_integral : ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 c = \u222b (x : \u03b1), f x \u2202\u03bc \u22a2 (fun x => c) = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** rw [\u2190 h_integral, \u2190 smul_assoc, smul_eq_mul, inv_mul_cancel, one_smul] ** case pos.intro \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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : Integrable f h_meas : StronglyMeasurable (\u03bc[f|\u22a5]) c : F' h_eq : \u03bc[f|\u22a5] = fun x => c h_integral : ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 c = \u222b (x : \u03b1), f x \u2202\u03bc \u22a2 ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2260 0 ** rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or] ** case pos.intro \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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : Integrable f h_meas : StronglyMeasurable (\u03bc[f|\u22a5]) c : F' h_eq : \u03bc[f|\u22a5] = fun x => c h_integral : ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 c = \u222b (x : \u03b1), f x \u2202\u03bc \u22a2 \u00ac\u2191\u2191\u03bc Set.univ = 0 \u2227 \u00ac\u2191\u2191\u03bc Set.univ = \u22a4 ** exact \u27e8NeZero.ne _, measure_ne_top \u03bc Set.univ\u27e9 ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : \u00acIsFiniteMeasure \u03bc \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** have h : \u00acSigmaFinite (\u03bc.trim bot_le) := by rwa [sigmaFinite_trim_bot_iff] ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : \u00acIsFiniteMeasure \u03bc h : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** rw [not_isFiniteMeasure_iff] at h\u03bc_finite ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : \u2191\u2191\u03bc Set.univ = \u22a4 h : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** rw [condexp_of_not_sigmaFinite bot_le h] ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : \u2191\u2191\u03bc Set.univ = \u22a4 h : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) \u22a2 0 = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** simp only [h\u03bc_finite, ENNReal.top_toReal, inv_zero, zero_smul] ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : \u2191\u2191\u03bc Set.univ = \u22a4 h : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) \u22a2 0 = fun x => 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 : \u03b1 \u2192 F' s : Set \u03b1 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : \u00acIsFiniteMeasure \u03bc \u22a2 \u00acSigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) ** rwa [sigmaFinite_trim_bot_iff] ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : \u00acIntegrable f \u22a2 \u03bc[f|\u22a5] = fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** rw [integral_undef hf, smul_zero, condexp_undef hf] ** 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 h\u03bc : NeZero \u03bc f : \u03b1 \u2192 F' h\u03bc_finite : IsFiniteMeasure \u03bc hf : \u00acIntegrable f \u22a2 0 = fun x => 0 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Partrec.sum_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 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 g : \u03b1 \u2192 \u03b2 \u2192. \u03c3 h : \u03b1 \u2192 \u03b3 \u2192. \u03c3 hf : Computable f hg : Partrec\u2082 g hh : Partrec\u2082 h a : \u03b1 \u22a2 (bif Sum.casesOn (f a) (fun b => true) fun b => false then       Sum.casesOn (f a) (fun b => Part.map Option.some (g (a, b).1 (a, b).2)) fun c => Part.some Option.none     else Sum.casesOn (f a) (fun b => Part.some Option.none) fun b => Part.map Option.some (h (a, b).1 (a, b).2)) =     Part.map Option.some (Sum.casesOn (f a) (g a) (h a)) ** cases f a <;> simp only [Bool.cond_true, Bool.cond_false] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.countP_cons_of_pos ** \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 pa : p a = true \u22a2 countP p (a :: l) = countP p l + 1 ** have : countP.go p (a :: l) 0 = countP.go p l 1 := show cond .. = _ by rw [pa]; rfl ** \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 pa : p a = true this : countP.go p (a :: l) 0 = countP.go p l 1 \u22a2 countP p (a :: l) = countP p l + 1 ** unfold countP ** \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 pa : p a = true this : countP.go p (a :: l) 0 = countP.go p l 1 \u22a2 countP.go p (a :: l) 0 = countP.go p l 0 + 1 ** rw [this, Nat.add_comm, List.countP_go_eq_add] ** \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 pa : p a = true \u22a2 (bif p a then countP.go p l (0 + 1) else countP.go p l 0) = countP.go p l 1 ** rw [pa] ** \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 pa : p a = true \u22a2 (bif true then countP.go p l (0 + 1) else countP.go p l 0) = countP.go p l 1 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mem\u2112p_finset_sum ** \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 \u03b9 : Type u_5 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 E hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p \u22a2 Mem\u2112p (fun a => \u2211 i in s, f i a) p ** haveI : DecidableEq \u03b9 := Classical.decEq _ ** \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 \u03b9 : Type u_5 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 E hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p this : DecidableEq \u03b9 \u22a2 Mem\u2112p (fun a => \u2211 i in s, f i a) p ** revert hf ** \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 \u03b9 : Type u_5 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 E this : DecidableEq \u03b9 \u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p ** refine' Finset.induction_on s _ _ ** case refine'_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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b9 : Type u_5 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 E this : DecidableEq \u03b9 \u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in \u2205, f i a) p ** simp only [zero_mem_\u2112p', Finset.sum_empty, imp_true_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 : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b9 : Type u_5 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 E this : DecidableEq \u03b9 \u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},     \u00aca \u2208 s \u2192       ((\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p) \u2192         (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a_3 => \u2211 i in insert a s, f i a_3) p ** intro i s his ih hf ** case refine'_2 \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 \u03b9 : Type u_5 s\u271d : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 E this : DecidableEq \u03b9 i : \u03b9 s : Finset \u03b9 his : \u00aci \u2208 s ih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p hf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2112p (f i_1) p \u22a2 Mem\u2112p (fun a => \u2211 i in insert i s, f i a) p ** simp only [his, Finset.sum_insert, 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 : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b9 : Type u_5 s\u271d : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 E this : DecidableEq \u03b9 i : \u03b9 s : Finset \u03b9 his : \u00aci \u2208 s ih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p hf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2112p (f i_1) p \u22a2 Mem\u2112p (fun a => f i a + \u2211 i in s, f i a) p ** exact (hf i (s.mem_insert_self i)).add (ih fun j hj => hf j (Finset.mem_insert_of_mem hj)) ** Qed",
        "informal": ""
    },
    {
        "formal": "generateFrom_piiUnionInter_measurableSet ** \u03b1 : Type u_1 \u03b9 : Type u_2 m : \u03b9 \u2192 MeasurableSpace \u03b1 S : Set \u03b9 \u22a2 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) = \u2a06 i \u2208 S, m i ** refine' le_antisymm _ _ ** case refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 m : \u03b9 \u2192 MeasurableSpace \u03b1 S : Set \u03b9 \u22a2 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) \u2264 \u2a06 i \u2208 S, m i ** rw [\u2190 @generateFrom_measurableSet \u03b1 (\u2a06 i \u2208 S, m i)] ** case refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 m : \u03b9 \u2192 MeasurableSpace \u03b1 S : Set \u03b9 \u22a2 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) \u2264 generateFrom {s | MeasurableSet s} ** exact generateFrom_mono (measurableSet_iSup_of_mem_piiUnionInter m S) ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 m : \u03b9 \u2192 MeasurableSpace \u03b1 S : Set \u03b9 \u22a2 \u2a06 i \u2208 S, m i \u2264 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) ** refine' iSup\u2082_le fun i hi => _ ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 m : \u03b9 \u2192 MeasurableSpace \u03b1 S : Set \u03b9 i : \u03b9 hi : i \u2208 S \u22a2 m i \u2264 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) ** rw [\u2190 @generateFrom_measurableSet \u03b1 (m i)] ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 m : \u03b9 \u2192 MeasurableSpace \u03b1 S : Set \u03b9 i : \u03b9 hi : i \u2208 S \u22a2 generateFrom {s | MeasurableSet s} \u2264 generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) ** exact generateFrom_mono (mem_piiUnionInter_of_measurableSet m hi) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.finStronglyMeasurable_zero ** \u03b1\u271d : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 \u03b1 : Type u_5 \u03b2 : Type u_6 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Zero \u03b2 inst\u271d : TopologicalSpace \u03b2 \u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(OfNat.ofNat 0 n)) < \u22a4 ** simp only [Pi.zero_apply, SimpleFunc.coe_zero, support_zero', measure_empty,\n  WithTop.zero_lt_top, forall_const] ** Qed",
        "informal": ""
    },
    {
        "formal": "IntervalIntegrable.iff_comp_neg ** \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 \u22a2 IntervalIntegrable f volume a b \u2194 IntervalIntegrable (fun x => f (-x)) volume (-a) (-b) ** rw [\u2190 comp_mul_left_iff (neg_ne_zero.2 one_ne_zero)] ** \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 \u22a2 IntervalIntegrable (fun x => f (-1 * x)) volume (a / -1) (b / -1) \u2194     IntervalIntegrable (fun x => f (-x)) volume (-a) (-b) ** simp [div_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsStoppingTime.measurableSet_min_const_iff ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 h\u03c4 : IsStoppingTime f \u03c4 s : Set \u03a9 i : \u03b9 \u22a2 MeasurableSet s \u2194 MeasurableSet s \u2227 MeasurableSet s ** apply MeasurableSpace.measurableSet_inf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.le_sup_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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e \u22a2 lintegral (map Prod.fst (pair f g)) \u03bc \u2294 lintegral (map Prod.snd (pair f g)) \u03bc \u2264     \u2211 x in SimpleFunc.range (pair f g), (x.1 \u2294 x.2) * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x}) ** rw [map_lintegral, map_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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e \u22a2 (\u2211 x in SimpleFunc.range (pair f g), x.1 * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x})) \u2294       \u2211 x in SimpleFunc.range (pair f g), x.2 * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x}) \u2264     \u2211 x in SimpleFunc.range (pair f g), (x.1 \u2294 x.2) * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x}) ** refine' sup_le _ _ <;> refine' Finset.sum_le_sum fun a _ => mul_le_mul_right' _ _ ** 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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e x\u271d : a \u2208 SimpleFunc.range (pair f g) \u22a2 a.1 \u2264 a.1 \u2294 a.2 ** exact le_sup_left ** 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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e x\u271d : a \u2208 SimpleFunc.range (pair f g) \u22a2 a.2 \u2264 a.1 \u2294 a.2 ** exact le_sup_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 g : \u03b1 \u2192\u209b \u211d\u22650\u221e \u22a2 \u2211 x in SimpleFunc.range (pair f g), (x.1 \u2294 x.2) * \u2191\u2191\u03bc (\u2191(pair f g) \u207b\u00b9' {x}) = lintegral (f \u2294 g) \u03bc ** rw [sup_eq_map\u2082, map_lintegral] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.set_lintegral_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\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** by_cases h\u03c1_zero : \u03c1.fst.restrict s = 0 ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 \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) ** have h :\n  \u222b\u207b a in s, ENNReal.ofReal (condCdf \u03c1 a x) \u2202\u03c1.fst =\n    \u222b\u207b a in s, ENNReal.ofReal (\u2a05 r : { r' : \u211a // x < r' }, condCdf \u03c1 a r) \u2202\u03c1.fst := by\n  congr with a : 1\n  rw [\u2190 (condCdf \u03c1 a).iInf_rat_gt_eq x] ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 \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) ** have h_nonempty : Nonempty { r' : \u211a // x < \u2191r' } := by\n  obtain \u27e8r, hrx\u27e9 := exists_rat_gt x\n  exact \u27e8\u27e8r, hrx\u27e9\u27e9 ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } \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) ** rw [h] ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } \u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** simp_rw [ENNReal.ofReal_cinfi] ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } \u22a2 \u222b\u207b (a : \u03b1) in s, \u2a05 i, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** have h_coe : \u2200 b : { r' : \u211a // x < \u2191r' }, (b : \u211d) = ((b : \u211a) : \u211d) := fun _ => by congr ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u222b\u207b (a : \u03b1) in s, \u2a05 i, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** rw [lintegral_iInf_directed_of_measurable h\u03c1_zero fun q : { r' : \u211a // x < \u2191r' } =>\n    (measurable_condCdf \u03c1 q).ennreal_ofReal] ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u2a05 b, \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)  case neg.hf_int \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u2200 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4  case neg.h_directed \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 Directed (fun x x_1 => x \u2265 x_1) fun q x_1 => ENNReal.ofReal (\u2191(condCdf \u03c1 x_1) \u2191\u2191q) ** rotate_left ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u2a05 b, \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** simp_rw [set_lintegral_condCdf_rat \u03c1 _ hs] ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u2a05 b, \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191b) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** rw [\u2190 measure_iInter_eq_iInf] ** case pos \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\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0 \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) ** rw [h\u03c1_zero, lintegral_zero_measure] ** case pos \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\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0 \u22a2 0 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** refine' le_antisymm (zero_le _) _ ** case pos \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\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0 \u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) \u2264 0 ** calc\n  \u03c1 (s \u00d7\u02e2 Iic x) \u2264 \u03c1 (Prod.fst \u207b\u00b9' s) := measure_mono (prod_subset_preimage_fst s (Iic x))\n  _ = \u03c1.fst s := by rw [Measure.fst_apply hs]\n  _ = \u03c1.fst.restrict s univ := by rw [Measure.restrict_apply_univ]\n  _ = 0 := by simp only [h\u03c1_zero, Measure.coe_zero, Pi.zero_apply] ** \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\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0 \u22a2 \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s) = \u2191\u2191(Measure.fst \u03c1) s ** rw [Measure.fst_apply 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\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0 \u22a2 \u2191\u2191(Measure.fst \u03c1) s = \u2191\u2191(Measure.restrict (Measure.fst \u03c1) s) univ ** rw [Measure.restrict_apply_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 s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0 \u22a2 \u2191\u2191(Measure.restrict (Measure.fst \u03c1) s) univ = 0 ** simp only [h\u03c1_zero, Measure.coe_zero, Pi.zero_apply] ** \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 \u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 ** congr with a : 1 ** case e_f.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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 a : \u03b1 \u22a2 ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) = ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) ** rw [\u2190 (condCdf \u03c1 a).iInf_rat_gt_eq x] ** \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 \u22a2 Nonempty { r' // x < \u2191r' } ** 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) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 r : \u211a hrx : x < \u2191r \u22a2 Nonempty { r' // x < \u2191r' } ** exact \u27e8\u27e8r, hrx\u27e9\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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } x\u271d : { r' // x < \u2191r' } \u22a2 \u2191\u2191x\u271d = \u2191\u2191x\u271d ** congr ** case neg.hf_int \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u2200 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4 ** intro b ** case neg.hf_int \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b b : { r' // x < \u2191r' } \u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4 ** rw [set_lintegral_condCdf_rat \u03c1 _ hs] ** case neg.hf_int \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b b : { r' // x < \u2191r' } \u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191b) \u2260 \u22a4 ** exact measure_ne_top \u03c1 _ ** case neg.h_directed \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 Directed (fun x x_1 => x \u2265 x_1) fun q x_1 => ENNReal.ofReal (\u2191(condCdf \u03c1 x_1) \u2191\u2191q) ** refine' Monotone.directed_ge fun i j hij a => ENNReal.ofReal_le_ofReal ((condCdf \u03c1 a).mono _) ** case neg.h_directed \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b i j : { r' // x < \u2191r' } hij : i \u2264 j a : \u03b1 \u22a2 \u2191\u2191i \u2264 \u2191\u2191j ** exact_mod_cast hij ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** rw [\u2190 prod_iInter] ** case 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 x : \u211d s : Set \u03b1 hs : MeasurableSet s h\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c2 i, Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** congr with y ** case neg.e_a.e_a.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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b y : \u211d \u22a2 y \u2208 \u22c2 i, Iic \u2191\u2191i \u2194 y \u2208 Iic x ** simp only [mem_iInter, mem_Iic, Subtype.forall, Subtype.coe_mk] ** case neg.e_a.e_a.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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b y : \u211d \u22a2 (\u2200 (a : \u211a), x < \u2191a \u2192 y \u2264 \u2191a) \u2194 y \u2264 x ** exact \u27e8le_of_forall_lt_rat_imp_le, fun hyx q hq => hyx.trans hq.le\u27e9 ** case neg.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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u2200 (i : { r' // x < \u2191r' }), MeasurableSet (s \u00d7\u02e2 Iic \u2191\u2191i) ** exact fun i => hs.prod measurableSet_Iic ** case neg.hd \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 Directed (fun x x_1 => x \u2287 x_1) fun b => s \u00d7\u02e2 Iic \u2191\u2191b ** refine' Monotone.directed_ge fun i j hij => _ ** case neg.hd \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b i j : { r' // x < \u2191r' } hij : i \u2264 j \u22a2 s \u00d7\u02e2 Iic \u2191\u2191i \u2264 s \u00d7\u02e2 Iic \u2191\u2191j ** refine' prod_subset_prod_iff.mpr (Or.inl \u27e8subset_rfl, Iic_subset_Iic.mpr _\u27e9) ** case neg.hd \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b i j : { r' // x < \u2191r' } hij : i \u2264 j \u22a2 \u2191\u2191i \u2264 \u2191\u2191j ** exact_mod_cast hij ** case neg.hfin \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\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0 h :   \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 h_nonempty : Nonempty { r' // x < \u2191r' } h_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b \u22a2 \u2203 i, \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191i) \u2260 \u22a4 ** exact \u27e8h_nonempty.some, measure_ne_top _ _\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.join_map_join ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 \u03bc : Measure (Measure (Measure \u03b1)) \u22a2 join (map join \u03bc) = join (join \u03bc) ** show bind \u03bc join = join (join \u03bc) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 \u03bc : Measure (Measure (Measure \u03b1)) \u22a2 bind \u03bc join = join (join \u03bc) ** rw [join_eq_bind, join_eq_bind, bind_bind measurable_id measurable_id] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 \u03bc : Measure (Measure (Measure \u03b1)) \u22a2 bind \u03bc join = bind \u03bc fun a => bind (id a) id ** apply congr_arg (bind \u03bc) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 \u03bc : Measure (Measure (Measure \u03b1)) \u22a2 join = fun a => bind (id a) id ** funext \u03bd ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 \u03bc : Measure (Measure (Measure \u03b1)) \u03bd : Measure (Measure \u03b1) \u22a2 join \u03bd = bind (id \u03bd) id ** exact join_eq_bind \u03bd ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.singularPart_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 singularPart 0 \u03bc = 0 ** refine' (eq_singularPart 0 0 VectorMeasure.MutuallySingular.zero_left _).symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 0 = 0 + withDensity\u1d65 \u03bc 0 ** rw [zero_add, withDensity\u1d65_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrable_of_summable_norm_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : Countable \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc f : C(\u03b1, E) s : \u03b2 \u2192 Compacts \u03b1 hf : Summable fun i => \u2016ContinuousMap.restrict (\u2191(s i)) f\u2016 * ENNReal.toReal (\u2191\u2191\u03bc \u2191(s i)) hs : \u22c3 i, \u2191(s i) = univ \u22a2 Integrable \u2191f ** simpa only [hs, integrableOn_univ] using integrableOn_iUnion_of_summable_norm_restrict hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.disjSups_inter_subset_left ** 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\u2081 \u2229 s\u2082) \u25cb t \u2286 s\u2081 \u25cb t \u2229 s\u2082 \u25cb t ** simpa only [disjSups, inter_product, filter_inter_distrib] using image_inter_subset _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_lintegral_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 => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) ** rw [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 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))) \u2194     \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc)) ** simp_rw [toWeakDualBCNN_apply _ _, \u2190 testAgainstNN_coe_eq, ENNReal.tendsto_coe,\n  ENNReal.toNNReal_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_to_bool ** \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 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 Bool h : MeasurableSet (f \u207b\u00b9' {true}) \u22a2 Measurable f ** apply measurable_to_countable' ** 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\u03b9 s t u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 Bool h : MeasurableSet (f \u207b\u00b9' {true}) \u22a2 \u2200 (x : Bool), MeasurableSet (f \u207b\u00b9' {x}) ** rintro (- | -) ** case h.true \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 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 Bool h : MeasurableSet (f \u207b\u00b9' {true}) \u22a2 MeasurableSet (f \u207b\u00b9' {true}) ** exact h ** case h.false \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 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 Bool h : MeasurableSet (f \u207b\u00b9' {true}) \u22a2 MeasurableSet (f \u207b\u00b9' {false}) ** convert h.compl ** case h.e'_3 \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 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 Bool h : MeasurableSet (f \u207b\u00b9' {true}) \u22a2 f \u207b\u00b9' {false} = (f \u207b\u00b9' {true})\u1d9c ** rw [\u2190 preimage_compl, Bool.compl_singleton, Bool.not_true] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.snorm_compMeasurePreserving ** \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\u271d : 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\u271d : \u03b2 \u2192 E \u03bd : Measure \u03b2 g : \u03b2 \u2192\u2098[\u03bd] E hf : MeasurePreserving f \u22a2 snorm (\u2191(compMeasurePreserving g f hf)) p \u03bc = snorm (\u2191g) p \u03bd ** rw [snorm_congr_ae (g.coeFn_compMeasurePreserving _)] ** \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\u271d : 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\u271d : \u03b2 \u2192 E \u03bd : Measure \u03b2 g : \u03b2 \u2192\u2098[\u03bd] E hf : MeasurePreserving f \u22a2 snorm (\u2191g \u2218 f) p \u03bc = snorm (\u2191g) p \u03bd ** exact snorm_comp_measurePreserving g.aestronglyMeasurable hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.dvd_of_dvd_mul_right_of_gcd_one ** a b c : \u2124 habc : a \u2223 b * c hab : gcd a b = 1 \u22a2 a \u2223 c ** rw [mul_comm] at habc ** a b c : \u2124 habc : a \u2223 c * b hab : gcd a b = 1 \u22a2 a \u2223 c ** exact dvd_of_dvd_mul_left_of_gcd_one habc hab ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.union_comm_of_disjoint ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u2081\u271d s\u2082\u271d : Finmap \u03b2 s\u2081 s\u2082 : AList \u03b2 \u22a2 Disjoint \u27e6s\u2081\u27e7 \u27e6s\u2082\u27e7 \u2192 \u27e6s\u2081\u27e7 \u222a \u27e6s\u2082\u27e7 = \u27e6s\u2082\u27e7 \u222a \u27e6s\u2081\u27e7 ** intro h ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s\u2081\u271d s\u2082\u271d : Finmap \u03b2 s\u2081 s\u2082 : AList \u03b2 h : Disjoint \u27e6s\u2081\u27e7 \u27e6s\u2082\u27e7 \u22a2 \u27e6s\u2081\u27e7 \u222a \u27e6s\u2082\u27e7 = \u27e6s\u2082\u27e7 \u222a \u27e6s\u2081\u27e7 ** simp only [AList.toFinmap_eq, union_toFinmap, AList.union_comm_of_disjoint h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.re ** \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.re (f x)) p ** have : \u2200 x, \u2016IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 := by\n  intro x\n  rw [one_mul]\n  exact IsROrC.norm_re_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.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 \u22a2 Mem\u2112p (fun x => \u2191IsROrC.re (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.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 \u22a2 AEStronglyMeasurable (fun x => \u2191IsROrC.re (f x)) \u03bc ** exact IsROrC.continuous_re.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.re (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.re (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.re (f x)\u2016 \u2264 \u2016f x\u2016 ** exact IsROrC.norm_re_le_norm (f x) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.measurable_condCdf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) x : \u211d \u22a2 Measurable fun a => \u2191(condCdf \u03c1 a) x ** have : (fun a => condCdf \u03c1 a x) = fun a => \u2a05 r : { r' : \u211a // x < r' }, condCdfRat \u03c1 a \u2191r := by\n  ext1 a\n  rw [\u2190 StieltjesFunction.iInf_rat_gt_eq]\n  congr with q\n  rw [condCdf_eq_condCdfRat] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) x : \u211d this : (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r \u22a2 Measurable fun a => \u2191(condCdf \u03c1 a) x ** rw [this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) x : \u211d this : (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r \u22a2 Measurable fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r ** exact measurable_iInf (fun q => measurable_condCdfRat \u03c1 q) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) x : \u211d \u22a2 (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 r, condCdfRat \u03c1 a \u2191r ** ext1 a ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) x : \u211d a : \u03b1 \u22a2 \u2191(condCdf \u03c1 a) x = \u2a05 r, condCdfRat \u03c1 a \u2191r ** rw [\u2190 StieltjesFunction.iInf_rat_gt_eq] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) x : \u211d a : \u03b1 \u22a2 \u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r ** congr with q ** case h.e_s.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) x : \u211d a : \u03b1 q : { r' // x < \u2191r' } \u22a2 \u2191(condCdf \u03c1 a) \u2191\u2191q = condCdfRat \u03c1 a \u2191q ** rw [condCdf_eq_condCdfRat] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.fst_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 \u03b2 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(fst \u03ba) a) s = \u2191\u2191(\u2191\u03ba a) {p | p.1 \u2208 s} ** rw [fst_apply, Measure.map_apply measurable_fst 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 \u03b2 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191\u03ba a) (Prod.fst \u207b\u00b9' s) = \u2191\u2191(\u2191\u03ba a) {p | p.1 \u2208 s} ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** let npf := (\u222b\u207b c : \u03b1, f c ^ p \u2202\u03bc) ^ (1 / p) ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** let nqg := (\u222b\u207b c : \u03b1, g c ^ q \u2202\u03bc) ^ (1 / q) ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** calc\n  (\u222b\u207b a : \u03b1, (f * g) a \u2202\u03bc) =\n      \u222b\u207b a : \u03b1, (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) \u2202\u03bc := by\n    refine' lintegral_congr fun a => _\n    rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_snorm f hf_nonzero hf_nontop,\n      fun_eq_funMulInvSnorm_mul_snorm g hg_nonzero hg_nontop, Pi.mul_apply]\n    ring\n  _ \u2264 npf * nqg := by\n    rw [lintegral_mul_const' (npf * nqg) _\n        (by simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top])]\n    refine' mul_le_of_le_one_left' _\n    have hf1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.pos hf_nonzero hf_nontop\n    have hg1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.symm.pos hg_nonzero hg_nontop\n    exact lintegral_mul_le_one_of_lintegral_rpow_eq_one hpq (hf.mul_const _) hf1 hg1 ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) \u2202\u03bc ** refine' lintegral_congr fun a => _ ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) a : \u03b1 \u22a2 (f * g) a = (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) ** rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_snorm f hf_nonzero hf_nontop,\n  fun_eq_funMulInvSnorm_mul_snorm g hg_nonzero hg_nontop, Pi.mul_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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) a : \u03b1 \u22a2 funMulInvSnorm f p \u03bc a * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) *       (funMulInvSnorm g q \u03bc a * (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)) =     funMulInvSnorm f p \u03bc a * funMulInvSnorm g q \u03bc a * (npf * nqg) ** ring ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) \u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) \u2202\u03bc \u2264 npf * nqg ** rw [lintegral_mul_const' (npf * nqg) _\n    (by simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top])] ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) \u22a2 (\u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc) * (npf * nqg) \u2264 npf * nqg ** refine' mul_le_of_le_one_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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) \u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1 ** have hf1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.pos hf_nonzero hf_nontop ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) hf1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => f a) p \u03bc c ^ p \u2202\u03bc = 1 \u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1 ** have hg1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.symm.pos hg_nonzero hg_nontop ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) hf1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => f a) p \u03bc c ^ p \u2202\u03bc = 1 hg1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => g a) q \u03bc c ^ q \u2202\u03bc = 1 \u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1 ** exact lintegral_mul_le_one_of_lintegral_rpow_eq_one hpq (hf.mul_const _) hf1 hg1 ** \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_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4 hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0 npf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) nqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q) \u22a2 npf * nqg \u2260 \u22a4 ** simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top] ** Qed",
        "informal": ""
    },
    {
        "formal": "PEquiv.self_trans_symm ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 \u22a2 PEquiv.trans f (PEquiv.symm f) = ofSet {a | isSome (\u2191f a) = true} ** ext ** case h.a \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 x\u271d a\u271d : \u03b1 \u22a2 a\u271d \u2208 \u2191(PEquiv.trans f (PEquiv.symm f)) x\u271d \u2194 a\u271d \u2208 \u2191(ofSet {a | isSome (\u2191f a) = true}) x\u271d ** dsimp [PEquiv.trans] ** case h.a \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 x\u271d a\u271d : \u03b1 \u22a2 a\u271d \u2208 Option.bind (\u2191f x\u271d) \u2191(PEquiv.symm f) \u2194 a\u271d \u2208 \u2191(ofSet {a | isSome (\u2191f a) = true}) x\u271d ** simp only [eq_some_iff f, Option.isSome_iff_exists, Option.mem_def, bind_eq_some',\n  ofSet_eq_some_iff] ** case h.a \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 x\u271d a\u271d : \u03b1 \u22a2 (\u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a) \u2194 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1} ** constructor ** case h.a.mp \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 x\u271d a\u271d : \u03b1 \u22a2 (\u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a) \u2192 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1} ** rintro \u27e8b, hb\u2081, hb\u2082\u27e9 ** case h.a.mp.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 x\u271d a\u271d : \u03b1 b : \u03b2 hb\u2081 : \u2191f x\u271d = some b hb\u2082 : \u2191f a\u271d = some b \u22a2 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1} ** exact \u27e8PEquiv.inj _ hb\u2082 hb\u2081, b, hb\u2082\u27e9 ** case h.a.mpr \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 x\u271d a\u271d : \u03b1 \u22a2 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1} \u2192 \u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a ** simp (config := { contextual := true }) ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.degrees_map_of_injective ** 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 : R \u2192+* S hf : Injective \u2191f \u22a2 degrees (\u2191(map f) p) = degrees p ** simp only [degrees, MvPolynomial.support_map_of_injective _ hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.DList.toList_cons ** \u03b1 : Type u x : \u03b1 l : DList \u03b1 \u22a2 toList (cons x l) = x :: toList l ** cases l ** case mk \u03b1 : Type u x : \u03b1 apply\u271d : List \u03b1 \u2192 List \u03b1 invariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l \u22a2 toList (cons x { apply := apply\u271d, invariant := invariant\u271d }) =     x :: toList { apply := apply\u271d, invariant := invariant\u271d } ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div ** \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 : LocallyIntegrable 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) ** rcases hf.exists_nat_integrableOn with \u27e8u, u_open, u_univ, hu\u27e9 ** case intro.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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) \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) ** have : \u2200 n, \u2200\u1d50 x \u2202\u03bc,\n    Tendsto (fun a => (\u222b\u207b y in a, \u2016(u n).indicator f y - (u n).indicator f x\u2016\u208a \u2202\u03bc) / \u03bc a)\n    (v.filterAt x) (\ud835\udcdd 0) := by\n  intro n\n  apply ae_tendsto_lintegral_nnnorm_sub_div_of_integrable\n  exact (integrable_indicator_iff (u_open n).measurableSet).2 (hu n) ** case intro.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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) this :   \u2200 (n : \u2115),     \u2200\u1d50 (x : \u03b1) \u2202\u03bc,       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)         (\ud835\udcdd 0) \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 [ae_all_iff.2 this] with x hx ** 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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) this :   \u2200 (n : \u2115),     \u2200\u1d50 (x : \u03b1) \u2202\u03bc,       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)         (\ud835\udcdd 0) x : \u03b1 hx :   \u2200 (i : \u2115),     Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) \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) ** obtain \u27e8n, hn\u27e9 : \u2203 n, x \u2208 u n := by simpa only [\u2190 u_univ, mem_iUnion] using mem_univ x ** case h.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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) this :   \u2200 (n : \u2115),     \u2200\u1d50 (x : \u03b1) \u2202\u03bc,       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)         (\ud835\udcdd 0) x : \u03b1 hx :   \u2200 (i : \u2115),     Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) n : \u2115 hn : x \u2208 u n \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 Tendsto.congr' _ (hx n) ** \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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) this :   \u2200 (n : \u2115),     \u2200\u1d50 (x : \u03b1) \u2202\u03bc,       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)         (\ud835\udcdd 0) x : \u03b1 hx :   \u2200 (i : \u2115),     Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) n : \u2115 hn : x \u2208 u n \u22a2 (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) =\u1da0[filterAt v x] fun a =>     (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a ** filter_upwards [v.eventually_filterAt_subset_of_nhds ((u_open n).mem_nhds hn),\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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) this :   \u2200 (n : \u2115),     \u2200\u1d50 (x : \u03b1) \u2202\u03bc,       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)         (\ud835\udcdd 0) x : \u03b1 hx :   \u2200 (i : \u2115),     Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) n : \u2115 hn : x \u2208 u n a : Set \u03b1 ha : a \u2286 u n h'a : MeasurableSet a \u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a =     (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) this :   \u2200 (n : \u2115),     \u2200\u1d50 (x : \u03b1) \u2202\u03bc,       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)         (\ud835\udcdd 0) x : \u03b1 hx :   \u2200 (i : \u2115),     Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) n : \u2115 hn : x \u2208 u n a : Set \u03b1 ha : a \u2286 u n h'a : MeasurableSet a \u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc = \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc ** refine' set_lintegral_congr_fun h'a (eventually_of_forall (fun y hy \u21a6 _)) ** 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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) this :   \u2200 (n : \u2115),     \u2200\u1d50 (x : \u03b1) \u2202\u03bc,       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)         (\ud835\udcdd 0) x : \u03b1 hx :   \u2200 (i : \u2115),     Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) n : \u2115 hn : x \u2208 u n a : Set \u03b1 ha : a \u2286 u n h'a : MeasurableSet a y : \u03b1 hy : y \u2208 a \u22a2 \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a = \u2191\u2016f y - f x\u2016\u208a ** rw [indicator_of_mem (ha hy) f, indicator_of_mem hn 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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) \u22a2 \u2200 (n : \u2115),     \u2200\u1d50 (x : \u03b1) \u2202\u03bc,       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)         (\ud835\udcdd 0) ** intro n ** \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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) n : \u2115 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) ** apply ae_tendsto_lintegral_nnnorm_sub_div_of_integrable ** 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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) n : \u2115 \u22a2 Integrable fun y => indicator (u n) f y ** exact (integrable_indicator_iff (u_open n).measurableSet).2 (hu n) ** \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 : LocallyIntegrable f u : \u2115 \u2192 Set \u03b1 u_open : \u2200 (n : \u2115), IsOpen (u n) u_univ : \u22c3 n, u n = univ hu : \u2200 (n : \u2115), IntegrableOn f (u n) this :   \u2200 (n : \u2115),     \u2200\u1d50 (x : \u03b1) \u2202\u03bc,       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u n) f y - indicator (u n) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x)         (\ud835\udcdd 0) x : \u03b1 hx :   \u2200 (i : \u2115),     Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016indicator (u i) f y - indicator (u i) f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) \u22a2 \u2203 n, x \u2208 u n ** simpa only [\u2190 u_univ, mem_iUnion] using mem_univ x ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.dvd_iff_mod_eq_zero ** m n : ZNum \u22a2 m \u2223 n \u2194 n % m = 0 ** rw [\u2190 dvd_to_int, Int.dvd_iff_emod_eq_zero, \u2190 to_int_inj, mod_to_int] ** m n : ZNum \u22a2 \u2191n % \u2191m = 0 \u2194 \u2191n % \u2191m = \u21910 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.normalize_eq ** num : Int den : Nat den_nz : den \u2260 0 \u22a2 normalize num den = mk' (num / \u2191(Nat.gcd (Int.natAbs num) den)) (den / Nat.gcd (Int.natAbs num) den) ** simp only [normalize, maybeNormalize_eq,\n  Int.div_eq_ediv_of_dvd (Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.meas_ge_le_variance_div_sq ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 c : \u211d hc : 0 < c \u22a2 \u2191\u2191\u2119 {\u03c9 | c \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|} \u2264 ENNReal.ofReal (variance X \u2119 / c ^ 2) ** rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.ofReal_variance_eq] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 c : \u211d hc : 0 < c \u22a2 \u2191\u2191\u2119 {\u03c9 | c \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|} \u2264 evariance X \u2119 / ENNReal.ofReal (c ^ 2) ** convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 c.toNNReal (by simp [hc]) using 1 ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 c : \u211d hc : 0 < c \u22a2 Real.toNNReal c \u2260 0 ** simp [hc] ** case h.e'_3 \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 c : \u211d hc : 0 < c \u22a2 \u2191\u2191\u2119 {\u03c9 | c \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|} = \u2191\u2191\u2119 {\u03c9 | \u2191(Real.toNNReal c) \u2264 |X \u03c9 - \u222b (a : \u03a9), X a|} ** simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true_iff] ** case h.e'_4 \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 c : \u211d hc : 0 < c \u22a2 evariance X \u2119 / ENNReal.ofReal (c ^ 2) = evariance X \u2119 / \u2191(Real.toNNReal c ^ 2) ** rw [ENNReal.ofReal_pow hc.le, ENNReal.coe_pow] ** case h.e'_4 \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 c : \u211d hc : 0 < c \u22a2 evariance X \u2119 / ENNReal.ofReal c ^ 2 = evariance X \u2119 / \u2191(Real.toNNReal c) ^ 2 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.indepFun_iff_integral_comp_mul ** \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b2 : Type u_2 \u03b2' : Type u_3 m\u03b2 : MeasurableSpace \u03b2 m\u03b2' : MeasurableSpace \u03b2' f : \u03a9 \u2192 \u03b2 g : \u03a9 \u2192 \u03b2' hfm : Measurable f hgm : Measurable g \u22a2 IndepFun f g \u2194     \u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},       Measurable \u03c6 \u2192         Measurable \u03c8 \u2192           Integrable (\u03c6 \u2218 f) \u2192 Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g) ** refine' \u27e8fun hfg _ _ h\u03c6 h\u03c8 => IndepFun.integral_mul_of_integrable (hfg.comp h\u03c6 h\u03c8), _\u27e9 ** \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b2 : Type u_2 \u03b2' : Type u_3 m\u03b2 : MeasurableSpace \u03b2 m\u03b2' : MeasurableSpace \u03b2' f : \u03a9 \u2192 \u03b2 g : \u03a9 \u2192 \u03b2' hfm : Measurable f hgm : Measurable g \u22a2 (\u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},       Measurable \u03c6 \u2192         Measurable \u03c8 \u2192           Integrable (\u03c6 \u2218 f) \u2192             Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g)) \u2192     IndepFun f g ** rw [IndepFun_iff] ** \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b2 : Type u_2 \u03b2' : Type u_3 m\u03b2 : MeasurableSpace \u03b2 m\u03b2' : MeasurableSpace \u03b2' f : \u03a9 \u2192 \u03b2 g : \u03a9 \u2192 \u03b2' hfm : Measurable f hgm : Measurable g \u22a2 (\u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},       Measurable \u03c6 \u2192         Measurable \u03c8 \u2192           Integrable (\u03c6 \u2218 f) \u2192             Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g)) \u2192     \u2200 (t1 t2 : Set \u03a9), MeasurableSet t1 \u2192 MeasurableSet t2 \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2 ** rintro h _ _ \u27e8A, hA, rfl\u27e9 \u27e8B, hB, rfl\u27e9 ** case intro.intro.intro.intro \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b2 : Type u_2 \u03b2' : Type u_3 m\u03b2 : MeasurableSpace \u03b2 m\u03b2' : MeasurableSpace \u03b2' f : \u03a9 \u2192 \u03b2 g : \u03a9 \u2192 \u03b2' hfm : Measurable f hgm : Measurable g h :   \u2200 {\u03c6 : \u03b2 \u2192 \u211d} {\u03c8 : \u03b2' \u2192 \u211d},     Measurable \u03c6 \u2192       Measurable \u03c8 \u2192         Integrable (\u03c6 \u2218 f) \u2192 Integrable (\u03c8 \u2218 g) \u2192 integral \u03bc (\u03c6 \u2218 f * \u03c8 \u2218 g) = integral \u03bc (\u03c6 \u2218 f) * integral \u03bc (\u03c8 \u2218 g) A : Set \u03b2 hA : MeasurableSet A B : Set \u03b2' hB : MeasurableSet B \u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' A \u2229 g \u207b\u00b9' B) = \u2191\u2191\u03bc (f \u207b\u00b9' A) * \u2191\u2191\u03bc (g \u207b\u00b9' B) ** specialize\n  h (measurable_one.indicator hA) (measurable_one.indicator hB)\n    ((integrable_const 1).indicator (hfm.comp measurable_id hA))\n    ((integrable_const 1).indicator (hgm.comp measurable_id hB)) ** case intro.intro.intro.intro \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b2 : Type u_2 \u03b2' : Type u_3 m\u03b2 : MeasurableSpace \u03b2 m\u03b2' : MeasurableSpace \u03b2' f : \u03a9 \u2192 \u03b2 g : \u03a9 \u2192 \u03b2' hfm : Measurable f hgm : Measurable g A : Set \u03b2 hA : MeasurableSet A B : Set \u03b2' hB : MeasurableSet B h : integral \u03bc (indicator A 1 \u2218 f * indicator B 1 \u2218 g) = integral \u03bc (indicator A 1 \u2218 f) * integral \u03bc (indicator B 1 \u2218 g) \u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' A \u2229 g \u207b\u00b9' B) = \u2191\u2191\u03bc (f \u207b\u00b9' A) * \u2191\u2191\u03bc (g \u207b\u00b9' B) ** rwa [\u2190 ENNReal.toReal_eq_toReal (measure_ne_top \u03bc _), ENNReal.toReal_mul, \u2190\n  integral_indicator_one ((hfm hA).inter (hgm hB)), \u2190 integral_indicator_one (hfm hA), \u2190\n  integral_indicator_one (hgm hB), Set.inter_indicator_one] ** case intro.intro.intro.intro \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d g\u271d : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b2 : Type u_2 \u03b2' : Type u_3 m\u03b2 : MeasurableSpace \u03b2 m\u03b2' : MeasurableSpace \u03b2' f : \u03a9 \u2192 \u03b2 g : \u03a9 \u2192 \u03b2' hfm : Measurable f hgm : Measurable g A : Set \u03b2 hA : MeasurableSet A B : Set \u03b2' hB : MeasurableSet B h : integral \u03bc (indicator A 1 \u2218 f * indicator B 1 \u2218 g) = integral \u03bc (indicator A 1 \u2218 f) * integral \u03bc (indicator B 1 \u2218 g) \u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' A) * \u2191\u2191\u03bc (g \u207b\u00b9' B) \u2260 \u22a4 ** exact ENNReal.mul_ne_top (measure_ne_top \u03bc _) (measure_ne_top \u03bc _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Setoid.classes_inj ** \u03b1 : Type u_1 r\u2081 r\u2082 : Setoid \u03b1 h : classes r\u2081 = classes r\u2082 a b : \u03b1 \u22a2 Rel r\u2081 a b \u2194 Rel r\u2082 a b ** simp only [rel_iff_exists_classes, exists_prop, h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.ringHom_ext ** 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 A : Type u_2 inst\u271d : Semiring A f g : MvPolynomial \u03c3 R \u2192+* A hC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r) hX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i) \u22a2 f = g ** refine AddMonoidAlgebra.ringHom_ext' ?_ ?_ ** case refine_1 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 A : Type u_2 inst\u271d : Semiring A f g : MvPolynomial \u03c3 R \u2192+* A hC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r) hX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i) \u22a2 RingHom.comp f singleZeroRingHom = RingHom.comp g singleZeroRingHom ** ext x ** case refine_1.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 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R A : Type u_2 inst\u271d : Semiring A f g : MvPolynomial \u03c3 R \u2192+* A hC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r) hX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i) x : R \u22a2 \u2191(RingHom.comp f singleZeroRingHom) x = \u2191(RingHom.comp g singleZeroRingHom) x ** exact hC _ ** case refine_2 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 A : Type u_2 inst\u271d : Semiring A f g : MvPolynomial \u03c3 R \u2192+* A hC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r) hX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i) \u22a2 MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115)) = MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115)) ** apply Finsupp.mulHom_ext' ** case refine_2.H 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 A : Type u_2 inst\u271d : Semiring A f g : MvPolynomial \u03c3 R \u2192+* A hC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r) hX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i) \u22a2 \u2200 (x : \u03c3),     MonoidHom.comp (MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)) =       MonoidHom.comp (MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)) ** intros x ** case refine_2.H 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 A : Type u_2 inst\u271d : Semiring A f g : MvPolynomial \u03c3 R \u2192+* A hC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r) hX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i) x : \u03c3 \u22a2 MonoidHom.comp (MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)) =     MonoidHom.comp (MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)) ** apply MonoidHom.ext_mnat ** case refine_2.H.h 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 A : Type u_2 inst\u271d : Semiring A f g : MvPolynomial \u03c3 R \u2192+* A hC : \u2200 (r : R), \u2191f (\u2191C r) = \u2191g (\u2191C r) hX : \u2200 (i : \u03c3), \u2191f (X i) = \u2191g (X i) x : \u03c3 \u22a2 \u2191(MonoidHom.comp (MonoidHom.comp (\u2191f) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)))       (\u2191Multiplicative.ofAdd 1) =     \u2191(MonoidHom.comp (MonoidHom.comp (\u2191g) (of R (\u03c3 \u2192\u2080 \u2115))) (\u2191AddMonoidHom.toMultiplicative (singleAddHom x)))       (\u2191Multiplicative.ofAdd 1) ** exact hX _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.finset_sum_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 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 m : MeasurableSpace \u03b1 I : Finset \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 s : Set \u03b1 \u22a2 \u2191\u2191(\u2211 i in I, \u03bc i) s = \u2211 i in I, \u2191\u2191(\u03bc i) s ** rw [coe_finset_sum, Finset.sum_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "\u03b5NFA.evalFrom_empty ** \u03b1 : Type u \u03c3 \u03c3' : Type v M : \u03b5NFA \u03b1 \u03c3 S : Set \u03c3 x\u271d : List \u03b1 s : \u03c3 a : \u03b1 x : List \u03b1 \u22a2 evalFrom M \u2205 x = \u2205 ** induction' x using List.reverseRecOn with x a ih ** case H0 \u03b1 : Type u \u03c3 \u03c3' : Type v M : \u03b5NFA \u03b1 \u03c3 S : Set \u03c3 x : List \u03b1 s : \u03c3 a : \u03b1 \u22a2 evalFrom M \u2205 [] = \u2205 ** rw [evalFrom_nil, \u03b5Closure_empty] ** case H1 \u03b1 : Type u \u03c3 \u03c3' : Type v M : \u03b5NFA \u03b1 \u03c3 S : Set \u03c3 x\u271d : List \u03b1 s : \u03c3 a\u271d : \u03b1 x : List \u03b1 a : \u03b1 ih : evalFrom M \u2205 x = \u2205 \u22a2 evalFrom M \u2205 (x ++ [a]) = \u2205 ** rw [evalFrom_append_singleton, ih, stepSet_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 f\u271d g : \u03b1 \u2192 \u03b2\u271d inst\u271d\u00b2 : TopologicalSpace \u03b2\u271d \u03b2 : Type u_5 f : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedSpace \u211d \u03b2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hf : StronglyMeasurable f c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(approxBounded hf c n) x) atTop (\ud835\udcdd (f x)) ** filter_upwards [hf_bound] with x hfx using tendsto_approxBounded_of_norm_le hf hfx ** Qed",
        "informal": ""
    },
    {
        "formal": "meas_lt_essInf ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2 x : \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : FirstCountableTopology \u03b2 inst\u271d : OrderTopology \u03b2 hf : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f) _auto\u271d \u22a2 \u2191\u2191\u03bc {y | f y < essInf f \u03bc} = 0 ** simp_rw [\u2190 not_le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2 x : \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : FirstCountableTopology \u03b2 inst\u271d : OrderTopology \u03b2 hf : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f) _auto\u271d \u22a2 \u2191\u2191\u03bc {y | \u00acessInf f \u03bc \u2264 f y} = 0 ** exact ae_essInf_le hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.integral_comp_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 f : E \u2192 F R : \u211d \u22a2 \u222b (x : E), f (R \u2022 x) \u2202\u03bc = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** rcases eq_or_ne R 0 with (rfl | hR) ** 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 : Set E f : E \u2192 F \u22a2 \u222b (x : E), f (0 \u2022 x) \u2202\u03bc = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** simp only [zero_smul, integral_const] ** 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 : Set E f : E \u2192 F \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** rcases Nat.eq_zero_or_pos (finrank \u211d E) with (hE | hE) ** case inl.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 f : E \u2192 F hE : finrank \u211d E = 0 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** have : Subsingleton E := finrank_zero_iff.1 hE ** case inl.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 f : E \u2192 F hE : finrank \u211d E = 0 this : Subsingleton E \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** have : f = fun _ => f 0 := by ext x; rw [Subsingleton.elim x 0] ** case inl.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 f : E \u2192 F hE : finrank \u211d E = 0 this\u271d : Subsingleton E this : f = fun x => f 0 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** conv_rhs => rw [this] ** case inl.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 f : E \u2192 F hE : finrank \u211d E = 0 this\u271d : Subsingleton E this : f = fun x => f 0 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), (fun x => f 0) x \u2202\u03bc ** simp only [hE, pow_zero, inv_one, abs_one, one_smul, integral_const] ** 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 f : E \u2192 F hE : finrank \u211d E = 0 this : Subsingleton E \u22a2 f = fun x => f 0 ** 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 s : Set E f : E \u2192 F hE : finrank \u211d E = 0 this : Subsingleton E x : E \u22a2 f x = f 0 ** rw [Subsingleton.elim x 0] ** case inl.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 f : E \u2192 F hE : finrank \u211d E > 0 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** have : Nontrivial E := finrank_pos_iff.1 hE ** case inl.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 f : E \u2192 F hE : finrank \u211d E > 0 this : Nontrivial E \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 f 0 = |(0 ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** simp only [zero_pow hE, measure_univ_of_isAddLeftInvariant, ENNReal.top_toReal, zero_smul,\n  inv_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 s : Set E f : E \u2192 F R : \u211d hR : R \u2260 0 \u22a2 \u222b (x : E), f (R \u2022 x) \u2202\u03bc = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** calc\n  (\u222b x, f (R \u2022 x) \u2202\u03bc) = \u222b y, f y \u2202Measure.map (fun x => R \u2022 x) \u03bc :=\n    (integral_map_equiv (Homeomorph.smul (isUnit_iff_ne_zero.2 hR).unit).toMeasurableEquiv\n        f).symm\n  _ = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b x, f x \u2202\u03bc := by\n    simp only [map_addHaar_smul \u03bc hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg] ** 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 f : E \u2192 F R : \u211d hR : R \u2260 0 \u22a2 \u222b (y : E), f y \u2202map (fun x => R \u2022 x) \u03bc = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b (x : E), f x \u2202\u03bc ** simp only [map_addHaar_smul \u03bc hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg] ** Qed",
        "informal": ""
    },
    {
        "formal": "AEMeasurable.sum_measure ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u22a2 AEMeasurable f ** nontriviality \u03b2 ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 \u22a2 AEMeasurable f ** inhabit \u03b2 ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 \u22a2 AEMeasurable f ** set s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) { x | f x \u2260 (h i).mk f x } ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} \u22a2 AEMeasurable f ** have hs\u03bc : \u2200 i, \u03bc i (s i) = 0 := by\n  intro i\n  rw [measure_toMeasurable]\n  exact (h i).ae_eq_mk ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 \u22a2 AEMeasurable f ** have hsm : MeasurableSet (\u22c2 i, s i) :=\n  MeasurableSet.iInter fun i => measurableSet_toMeasurable _ _ ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) \u22a2 AEMeasurable f ** have hs : \u2200 i x, x \u2209 s i \u2192 f x = (h i).mk f x := by\n  intro i x hx\n  contrapose! hx\n  exact subset_toMeasurable _ _ hx ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x \u22a2 AEMeasurable f ** set g : \u03b1 \u2192 \u03b2 := (\u22c2 i, s i).piecewise (const \u03b1 default) f ** \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f \u22a2 AEMeasurable f ** refine' \u27e8g, measurable_of_restrict_of_restrict_compl hsm _ _, ae_sum_iff.mpr fun i => _\u27e9 ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} \u22a2 \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 ** intro i ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} i : \u03b9 \u22a2 \u2191\u2191(\u03bc i) (s i) = 0 ** rw [measure_toMeasurable] ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} i : \u03b9 \u22a2 \u2191\u2191(\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} = 0 ** exact (h i).ae_eq_mk ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) \u22a2 \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x ** intro i x hx ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) i : \u03b9 x : \u03b1 hx : \u00acx \u2208 s i \u22a2 f x = mk f (_ : AEMeasurable f) x ** contrapose! hx ** \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 g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) i : \u03b9 x : \u03b1 hx : f x \u2260 mk f (_ : AEMeasurable f) x \u22a2 x \u2208 s i ** exact subset_toMeasurable _ _ hx ** case refine'_1 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f \u22a2 Measurable (Set.restrict (\u22c2 i, s i) g) ** rw [restrict_piecewise] ** case refine'_1 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f \u22a2 Measurable (Set.restrict (\u22c2 i, s i) (const \u03b1 default)) ** simp only [Set.restrict, const] ** case refine'_1 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f \u22a2 Measurable fun x => default ** exact measurable_const ** case refine'_2 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f \u22a2 Measurable (Set.restrict (\u22c2 i, s i)\u1d9c g) ** rw [restrict_piecewise_compl, compl_iInter] ** case refine'_2 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f \u22a2 Measurable (Set.restrict (\u22c3 i, (s i)\u1d9c) f) ** intro t ht ** case refine'_2 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f t : Set \u03b2 ht : MeasurableSet t \u22a2 MeasurableSet (Set.restrict (\u22c3 i, (s i)\u1d9c) f \u207b\u00b9' t) ** refine \u27e8\u22c3 i, (h i).mk f \u207b\u00b9' t \u2229 (s i)\u1d9c, MeasurableSet.iUnion fun i \u21a6\n  (measurable_mk _ ht).inter (measurableSet_toMeasurable _ _).compl, ?_\u27e9 ** case refine'_2 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f t : Set \u03b2 ht : MeasurableSet t \u22a2 Subtype.val \u207b\u00b9' \u22c3 i, mk f (_ : AEMeasurable f) \u207b\u00b9' t \u2229 (s i)\u1d9c = Set.restrict (\u22c3 i, (s i)\u1d9c) f \u207b\u00b9' t ** ext \u27e8x, hx\u27e9 ** case refine'_2.h.mk \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f t : Set \u03b2 ht : MeasurableSet t x : \u03b1 hx : x \u2208 \u22c3 i, (s i)\u1d9c \u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' \u22c3 i, mk f (_ : AEMeasurable f) \u207b\u00b9' t \u2229 (s i)\u1d9c \u2194     { val := x, property := hx } \u2208 Set.restrict (\u22c3 i, (s i)\u1d9c) f \u207b\u00b9' t ** simp only [mem_preimage, mem_iUnion, Subtype.coe_mk, Set.restrict, mem_inter_iff,\n  mem_compl_iff] at hx \u22a2 ** case refine'_2.h.mk \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f t : Set \u03b2 ht : MeasurableSet t x : \u03b1 hx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} \u22a2 (\u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}) \u2194 f x \u2208 t ** constructor ** case refine'_2.h.mk.mp \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f t : Set \u03b2 ht : MeasurableSet t x : \u03b1 hx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} \u22a2 (\u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x}) \u2192 f x \u2208 t ** rintro \u27e8i, hxt, hxs\u27e9 ** case refine'_2.h.mk.mp.intro.intro \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f t : Set \u03b2 ht : MeasurableSet t x : \u03b1 hx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} i : \u03b9 hxt : mk f (_ : AEMeasurable f) x \u2208 t hxs : \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} \u22a2 f x \u2208 t ** rwa [hs _ _ hxs] ** case refine'_2.h.mk.mpr \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f t : Set \u03b2 ht : MeasurableSet t x : \u03b1 hx : \u2203 i, \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} \u22a2 f x \u2208 t \u2192 \u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} ** rcases hx with \u27e8i, hi\u27e9 ** case refine'_2.h.mk.mpr.intro \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f t : Set \u03b2 ht : MeasurableSet t x : \u03b1 i : \u03b9 hi : \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} \u22a2 f x \u2208 t \u2192 \u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} ** rw [hs _ _ hi] ** case refine'_2.h.mk.mpr.intro \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f t : Set \u03b2 ht : MeasurableSet t x : \u03b1 i : \u03b9 hi : \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} \u22a2 mk f (_ : AEMeasurable f) x \u2208 t \u2192     \u2203 i, mk f (_ : AEMeasurable f) x \u2208 t \u2227 \u00acx \u2208 toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} ** exact fun h => \u27e8i, h, hi\u27e9 ** case refine'_3 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f i : \u03b9 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc i, f x = g x ** refine' measure_mono_null (fun x (hx : f x \u2260 g x) => _) (hs\u03bc i) ** case refine'_3 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f i : \u03b9 x : \u03b1 hx : f x \u2260 g x \u22a2 x \u2208 s i ** contrapose! hx ** case refine'_3 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f i : \u03b9 x : \u03b1 hx : \u00acx \u2208 s i \u22a2 f x = g x ** refine' (piecewise_eq_of_not_mem _ _ _ _).symm ** case refine'_3 \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 g\u271d : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 inst\u271d : Countable \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable f \u271d : Nontrivial \u03b2 inhabited_h : Inhabited \u03b2 s : \u03b9 \u2192 Set \u03b1 := fun i => toMeasurable (\u03bc i) {x | f x \u2260 mk f (_ : AEMeasurable f) x} hs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0 hsm : MeasurableSet (\u22c2 i, s i) hs : \u2200 (i : \u03b9) (x : \u03b1), \u00acx \u2208 s i \u2192 f x = mk f (_ : AEMeasurable f) x g : \u03b1 \u2192 \u03b2 := piecewise (\u22c2 i, s i) (const \u03b1 default) f i : \u03b9 x : \u03b1 hx : \u00acx \u2208 s i \u22a2 \u00acx \u2208 \u22c2 i, s i ** exact fun h => hx (mem_iInter.1 h i) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_Lp_of_tendsto_ae ** \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), AEStronglyMeasurable (f n) \u03bc 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) ** have : \u2200 n, snorm (f n - g) p \u03bc = snorm ((hf n).mk (f n) - hg.1.mk g) p \u03bc :=\n  fun n => snorm_congr_ae ((hf n).ae_eq_mk.sub hg.1.ae_eq_mk) ** \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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc \u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) ** simp_rw [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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc \u22a2 Tendsto     (fun n =>       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc)     atTop (\ud835\udcdd 0) ** refine' tendsto_Lp_of_tendsto_ae_of_meas \u03bc hp hp' (fun n => (hf n).stronglyMeasurable_mk)\n  hg.1.stronglyMeasurable_mk (hg.ae_eq hg.1.ae_eq_mk) (hui.ae_eq fun n => (hf n).ae_eq_mk) _ ** \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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop       (\ud835\udcdd (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x)) ** have h_ae_forall_eq : \u2200\u1d50 x \u2202\u03bc, \u2200 n, f n x = (hf n).mk (f n) x := by\n  rw [ae_all_iff]\n  exact fun n => (hf n).ae_eq_mk ** \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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc h_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop       (\ud835\udcdd (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x)) ** filter_upwards [hfg, h_ae_forall_eq, hg.1.ae_eq_mk] with x hx_tendsto hxf_eq hxg_eq ** case h \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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc h_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x x : \u03b1 hx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) hxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x hxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x \u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop     (\ud835\udcdd (AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x)) ** rw [\u2190 hxg_eq] ** case h \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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc h_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x x : \u03b1 hx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) hxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x hxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x \u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop (\ud835\udcdd (g x)) ** convert hx_tendsto 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\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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc h_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x x : \u03b1 hx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) hxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x hxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x \u22a2 (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) = fun n => f n x ** ext1 n ** case h.e'_3.h \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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc h_ae_forall_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x x : \u03b1 hx_tendsto : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) hxf_eq : \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x hxg_eq : g x = AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc) x n : \u2115 \u22a2 AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x ** exact (hxf_eq n).symm ** \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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f n x = AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x ** rw [ae_all_iff] ** \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), AEStronglyMeasurable (f n) \u03bc 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)) this :   \u2200 (n : \u2115),     snorm (f n - g) p \u03bc =       snorm         (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) -           AEStronglyMeasurable.mk g (_ : AEStronglyMeasurable g \u03bc))         p \u03bc \u22a2 \u2200 (i : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f i a = AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) \u03bc) a ** exact fun n => (hf n).ae_eq_mk ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image_add_right_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 + c) (Ioo a b) = Ioo (a + c) (b + c) ** rw [\u2190 map_add_right_Ioo, map_eq_image, addRightEmbedding, Embedding.coeFn_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.coeff_C_mul ** 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 m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R m : \u03c3 \u2192\u2080 \u2115 a : R p : MvPolynomial \u03c3 R \u22a2 coeff m (\u2191C a * p) = a * coeff m p ** rw [mul_def, sum_C] ** 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 m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R m : \u03c3 \u2192\u2080 \u2115 a : R p : MvPolynomial \u03c3 R \u22a2 (sum p fun n b => \u2191(monomial (0 + n)) (0 * b)) = 0 ** simp ** 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 m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R m : \u03c3 \u2192\u2080 \u2115 a : R p : MvPolynomial \u03c3 R \u22a2 coeff m (sum p fun n b => \u2191(monomial (0 + n)) (a * b)) = a * coeff m p ** simp (config := { contextual := true }) [sum_def, coeff_sum] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.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 comp\u2082Measurable g hg f\u2081 f\u2082 =     mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) ** rw [comp\u2082Measurable_eq_pair, pair_eq_mk, compMeasurable_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 mk (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) =     mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.mod_to_nat ** \u22a2 \u2191(0 % 0) = \u21910 % \u21910 ** simp ** 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \ud835\udd5c x : F \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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \ud835\udd5c x : F \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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \ud835\udd5c x : F \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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \ud835\udd5c x : F \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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \ud835\udd5c x : F \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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \ud835\udd5c x : F \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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \ud835\udd5c x : F 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": "MeasureTheory.condexpIndL1_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) x y : G \u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y ** 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 y : G hs : MeasurableSet s \u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y  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 y : G hs : \u00acMeasurableSet s \u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y ** 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 y : G hs : MeasurableSet s \u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y ** 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 y : G hs : \u00acMeasurableSet s \u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y ** 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 y : G hs : \u00acMeasurableSet s \u22a2 0 = 0 + 0 ** rw [zero_add] ** 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 y : G hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = \u22a4 \u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y ** 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) x y : G hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = \u22a4 \u22a2 0 = 0 + 0 ** rw [zero_add] ** 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 y : G hs : MeasurableSet s h\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4 \u22a2 condexpIndL1 hm \u03bc s (x + y) = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc s y ** 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) x y : G hs : MeasurableSet s h\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4 \u22a2 condexpIndL1Fin hm hs h\u03bcs (x + y) = condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y ** exact condexpIndL1Fin_add hs h\u03bcs x y ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_lintegral_strict_mono ** \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 s : Set \u03b1 hsm : MeasurableSet s hs : \u2191\u2191\u03bc s \u2260 0 hg : Measurable g hfi : \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2260 \u22a4 h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x < g x \u22a2 Measure.restrict \u03bc s \u2260 0 ** simp [hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R \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) ** set t' := R\u207b\u00b9 \u2022 t with ht' ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t \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) ** set u' := R\u207b\u00b9 \u2022 u with hu' ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi (R * 0)] (R * 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) ** rw [mul_zero] at B ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 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 (A.comp B).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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0) \u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) =\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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0) \u22a2 \u2200 (a : \u211d),     a \u2208 Ioi 0 \u2192       ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) a =         \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 u) ** rintro 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0) r : \u211d \u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) r =     \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** have T : (R * r) \u2022 t' = r \u2022 t := by\n  rw [mul_comm, ht', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one] ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0) r : \u211d T : (R * r) \u2022 t' = r \u2022 t \u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) r =     \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** have U : (R * r) \u2022 u' = r \u2022 u := by\n  rw [mul_comm, hu', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one] ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0) r : \u211d T : (R * r) \u2022 t' = r \u2022 t U : (R * r) \u2022 u' = r \u2022 u \u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) r =     \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** dsimp ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0) r : \u211d T : (R * r) \u2022 t' = r \u2022 t U : (R * r) \u2022 u' = r \u2022 u \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + (R * r) \u2022 R\u207b\u00b9 \u2022 t)) / \u2191\u2191\u03bc ({x} + (R * r) \u2022 R\u207b\u00b9 \u2022 u) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** rw [T, U] ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 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) ** apply tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 \u03bc s x h t' u' ** case h'u 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u \u22a2 \u2191\u2191\u03bc u' \u2260 0 ** simp only [h'u, (pow_pos Rpos _).ne', abs_nonpos_iff, addHaar_smul, not_false_iff,\n  ENNReal.ofReal_eq_zero, inv_eq_zero, inv_pow, Ne.def, or_self_iff, mul_eq_zero] ** case t_bound 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u \u22a2 t' \u2286 closedBall 0 1 ** refine (smul_set_mono t_bound).trans_eq ?_ ** case t_bound 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u \u22a2 R\u207b\u00b9 \u2022 closedBall 0 R = closedBall 0 1 ** rw [smul_closedBall _ _ Rpos.le, smul_zero, Real.norm_of_nonneg (inv_nonneg.2 Rpos.le),\n  inv_mul_cancel Rpos.ne'] ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) \u22a2 Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi (R * 0)] (R * 0)) ** apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within ** case h1 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) \u22a2 Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (R * 0)) ** exact (tendsto_const_nhds.mul tendsto_id).mono_left nhdsWithin_le_nhds ** case h2 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) \u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi 0] 0, R * x \u2208 Ioi (R * 0) ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) \u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 R * a \u2208 Ioi (R * 0) ** intro r rpos ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) r : \u211d rpos : r \u2208 Ioi 0 \u22a2 R * r \u2208 Ioi (R * 0) ** rw [mul_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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) r : \u211d rpos : r \u2208 Ioi 0 \u22a2 R * r \u2208 Ioi 0 ** exact mul_pos Rpos rpos ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0) r : \u211d \u22a2 (R * r) \u2022 t' = r \u2022 t ** rw [mul_comm, ht', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one] ** 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 R : \u211d Rpos : 0 < R t_bound : t \u2286 closedBall 0 R t' : Set E := R\u207b\u00b9 \u2022 t ht' : t' = R\u207b\u00b9 \u2022 t u' : Set E := R\u207b\u00b9 \u2022 u hu' : u' = R\u207b\u00b9 \u2022 u A : 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) B : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0) r : \u211d T : (R * r) \u2022 t' = r \u2022 t \u22a2 (R * r) \u2022 u' = r \u2022 u ** rw [mul_comm, hu', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.DominatedFinMeasAdditive.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\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' inst\u271d\u00b3 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d\u00b2 : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c \u03b2 hT : DominatedFinMeasAdditive \u03bc T C c : \ud835\udd5c \u22a2 DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C) ** refine' \u27e8hT.1.smul c, 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\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' inst\u271d\u00b3 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d\u00b2 : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c \u03b2 hT : DominatedFinMeasAdditive \u03bc T C c : \ud835\udd5c s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2016(fun s => c \u2022 T s) s\u2016 \u2264 \u2016c\u2016 * C * ENNReal.toReal (\u2191\u2191\u03bc s) ** dsimp only ** \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\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' inst\u271d\u00b3 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d\u00b2 : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c \u03b2 hT : DominatedFinMeasAdditive \u03bc T C c : \ud835\udd5c s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2016c \u2022 T s\u2016 \u2264 \u2016c\u2016 * C * ENNReal.toReal (\u2191\u2191\u03bc s) ** rw [norm_smul, mul_assoc] ** \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\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' inst\u271d\u00b3 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d\u00b2 : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c \u03b2 hT : DominatedFinMeasAdditive \u03bc T C c : \ud835\udd5c s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2016c\u2016 * \u2016T s\u2016 \u2264 \u2016c\u2016 * (C * ENNReal.toReal (\u2191\u2191\u03bc s)) ** exact mul_le_mul le_rfl (hT.2 s hs h\u03bcs) (norm_nonneg _) (norm_nonneg _) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_mul_lintegral_eq ** 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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 \u2191\u2191\u03bc s * \u222b\u207b (y : G), f y \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc ** rw [\u2190 set_lintegral_one, \u2190 lintegral_indicator _ sm,\n  \u2190 lintegral_lintegral_mul (measurable_const.indicator sm).aemeasurable hf.aemeasurable,\n  \u2190 lintegral_lintegral_mul_inv \u03bc \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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =     \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc  case hf 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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 AEMeasurable (uncurry fun x y => indicator s (fun x => 1) x * f y) ** swap ** 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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =     \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc ** have ms :\n  \u2200 x : G, Measurable fun y => ((fun z => z * x) \u207b\u00b9' s).indicator (fun _ => (1 : \u211d\u22650\u221e)) y :=\n  fun x => measurable_const.indicator (measurable_mul_const _ 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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f ms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y \u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =     \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc ** have : \u2200 x y, s.indicator (fun _ : G => (1 : \u211d\u22650\u221e)) (y * x) =\n    ((fun z => z * x) \u207b\u00b9' s).indicator (fun b : G => 1) y := by\n  intro x y; symm; convert indicator_comp_right (M := \u211d\u22650\u221e) fun y => y * x using 2; ext1; rfl ** 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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f ms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y this : \u2200 (x y : G), indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y \u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =     \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc ** simp_rw [this, lintegral_mul_const _ (ms _), lintegral_indicator _ (measurable_mul_const _ sm),\n  set_lintegral_one] ** case hf 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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 AEMeasurable (uncurry fun x y => indicator s (fun x => 1) x * f y) ** exact (((measurable_const.indicator sm).comp measurable_fst).mul\n  (hf.comp measurable_snd)).aemeasurable ** 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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f ms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y \u22a2 \u2200 (x y : G), indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y ** intro x y ** 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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f ms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y x y : G \u22a2 indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y ** symm ** 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 f : G \u2192 \u211d\u22650\u221e hf : Measurable f ms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y x y : G \u22a2 indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y = indicator s (fun x => 1) (y * x) ** convert indicator_comp_right (M := \u211d\u22650\u221e) fun y => y * x using 2 ** case h.e'_2.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 sm : MeasurableSet s f : G \u2192 \u211d\u22650\u221e hf : Measurable f ms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y x y : G \u22a2 (fun b => 1) = (fun x => 1) \u2218 fun y => y * x ** ext1 ** case h.e'_2.h.e'_5.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 sm : MeasurableSet s f : G \u2192 \u211d\u22650\u221e hf : Measurable f ms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y x y x\u271d : G \u22a2 1 = ((fun x => 1) \u2218 fun y => y * x) x\u271d ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_top_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 hT : DominatedFinMeasAdditive (\u22a4 \u2022 \u03bc) T C f : \u03b1 \u2192 E \u22a2 setToFun (\u22a4 \u2022 \u03bc) T hT f = 0 ** refine' setToFun_measure_zero' hT fun s _ h\u03bcs => _ ** \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 (\u22a4 \u2022 \u03bc) T C f : \u03b1 \u2192 E s : Set \u03b1 x\u271d : MeasurableSet s h\u03bcs : \u2191\u2191(\u22a4 \u2022 \u03bc) s < \u22a4 \u22a2 \u2191\u2191(\u22a4 \u2022 \u03bc) s = 0 ** rw [lt_top_iff_ne_top] at h\u03bcs ** \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 (\u22a4 \u2022 \u03bc) T C f : \u03b1 \u2192 E s : Set \u03b1 x\u271d : MeasurableSet s h\u03bcs : \u2191\u2191(\u22a4 \u2022 \u03bc) s \u2260 \u22a4 \u22a2 \u2191\u2191(\u22a4 \u2022 \u03bc) s = 0 ** simp only [true_and_iff, Measure.smul_apply, ENNReal.mul_eq_top, eq_self_iff_true,\n  top_ne_zero, Ne.def, not_false_iff, not_or, Classical.not_not, smul_eq_mul] at h\u03bcs ** \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 (\u22a4 \u2022 \u03bc) T C f : \u03b1 \u2192 E s : Set \u03b1 x\u271d : MeasurableSet s h\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4 \u2227 \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191(\u22a4 \u2022 \u03bc) s = 0 ** simp only [h\u03bcs.right, Measure.smul_apply, mul_zero, smul_eq_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_add_adjacent_intervals_cancel ** \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 c..a, f x \u2202\u03bc = 0 ** have hac := hab.trans 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 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 hac : IntervalIntegrable f \u03bc a 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 c..a, f x \u2202\u03bc = 0 ** simp only [intervalIntegral, sub_add_sub_comm, sub_eq_zero] ** \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 hac : IntervalIntegrable f \u03bc a c \u22a2 \u222b (x : \u211d) in Ioc a b, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc b c, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc c a, f x \u2202\u03bc =     \u222b (x : \u211d) in Ioc b a, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc c b, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc a c, f x \u2202\u03bc ** iterate 4 rw [\u2190 integral_union] ** case hst \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 hac : IntervalIntegrable f \u03bc a c \u22a2 Disjoint (Ioc b a \u222a Ioc c b) (Ioc a c)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 MeasurableSet (Ioc a c)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc b a \u222a Ioc c b)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc a c)  case hst \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 hac : IntervalIntegrable f \u03bc a c \u22a2 Disjoint (Ioc b a) (Ioc c b)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 MeasurableSet (Ioc c 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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc b 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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc c b)  case hst \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 hac : IntervalIntegrable f \u03bc a c \u22a2 Disjoint (Ioc a b \u222a Ioc b c) (Ioc c a)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 MeasurableSet (Ioc c a)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc a b \u222a Ioc b c)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc c a)  case hst \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 hac : IntervalIntegrable f \u03bc a c \u22a2 Disjoint (Ioc a b) (Ioc b c)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 MeasurableSet (Ioc b c)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc a b)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc b c) ** all_goals\n  simp [*, MeasurableSet.union, measurableSet_Ioc, Ioc_disjoint_Ioc_same,\n    Ioc_disjoint_Ioc_same.symm, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] ** \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 hac : IntervalIntegrable f \u03bc a c \u22a2 \u222b (x : \u211d) in Ioc a b \u222a Ioc b c \u222a Ioc c a, f x \u2202\u03bc =     \u222b (x : \u211d) in Ioc b a \u222a Ioc c b, f x \u2202\u03bc + \u222b (x : \u211d) in Ioc a c, f x \u2202\u03bc  case hst \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 hac : IntervalIntegrable f \u03bc a c \u22a2 Disjoint (Ioc b a) (Ioc c b)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 MeasurableSet (Ioc c 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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc b 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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc c b)  case hst \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 hac : IntervalIntegrable f \u03bc a c \u22a2 Disjoint (Ioc a b \u222a Ioc b c) (Ioc c a)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 MeasurableSet (Ioc c a)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc a b \u222a Ioc b c)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc c a)  case hst \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 hac : IntervalIntegrable f \u03bc a c \u22a2 Disjoint (Ioc a b) (Ioc b c)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 MeasurableSet (Ioc b c)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc a b)  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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc b c) ** rw [\u2190 integral_union] ** \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 hac : IntervalIntegrable f \u03bc a c \u22a2 \u222b (x : \u211d) in Ioc a b \u222a Ioc b c \u222a Ioc c a, f x \u2202\u03bc = \u222b (x : \u211d) in Ioc b a \u222a Ioc c b \u222a Ioc a c, f x \u2202\u03bc ** suffices Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc b a \u222a Ioc c b \u222a Ioc a c by rw [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 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 hac : IntervalIntegrable f \u03bc a c \u22a2 Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc b a \u222a Ioc c b \u222a Ioc a c ** rw [Ioc_union_Ioc_union_Ioc_cycle, union_right_comm, Ioc_union_Ioc_union_Ioc_cycle,\n  min_left_comm, max_left_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 hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c this : Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc b a \u222a Ioc c b \u222a Ioc a c \u22a2 \u222b (x : \u211d) in Ioc a b \u222a Ioc b c \u222a Ioc c a, f x \u2202\u03bc = \u222b (x : \u211d) in Ioc b a \u222a Ioc c b \u222a Ioc a c, f x \u2202\u03bc ** rw [this] ** 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 b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c hac : IntervalIntegrable f \u03bc a c \u22a2 IntegrableOn (fun x => f x) (Ioc b c) ** simp [*, MeasurableSet.union, measurableSet_Ioc, Ioc_disjoint_Ioc_same,\n  Ioc_disjoint_Ioc_same.symm, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.get_cons_succ ** n : \u2115 \u03b1 : Type u_1 a : \u03b1 v : Vector \u03b1 n i : Fin n \u22a2 get (a ::\u1d65 v) (Fin.succ i) = get v i ** rw [\u2190 get_tail_succ, tail_cons] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.bindOnSupport_bindOnSupport ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 \u22a2 bindOnSupport (bindOnSupport p f) g =     bindOnSupport p fun a ha => bindOnSupport (f a ha) fun b hb => g b (_ : b \u2208 support (bindOnSupport p f)) ** refine' PMF.ext fun a => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 \u22a2 \u2191(bindOnSupport (bindOnSupport p f) g) a =     \u2191(bindOnSupport p fun a ha => bindOnSupport (f a ha) fun b hb => g b (_ : b \u2208 support (bindOnSupport p f))) a ** dsimp only [bindOnSupport_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 \u22a2 (\u2211' (a_1 : \u03b2),       (\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) *         if h : (\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) = 0 then 0 else \u2191(g a_1 h) a) =     \u2211' (a_1 : \u03b1),       \u2191p a_1 *         if h : \u2191p a_1 = 0 then 0         else           \u2211' (a_2 : \u03b2),             \u2191(f a_1 h) a_2 * if h : \u2191(f a_1 h) a_2 = 0 then 0 else \u2191(g a_2 (_ : a_2 \u2208 support (bindOnSupport p f))) a ** simp only [\u2190 tsum_dite_right, ENNReal.tsum_mul_left.symm, ENNReal.tsum_mul_right.symm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 \u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),       (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) a_1) *         if h : (\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) = 0 then 0 else \u2191(g a_1 h) a) =     \u2211' (a_1 : \u03b1) (i : \u03b2),       \u2191p a_1 *         if h : \u2191p a_1 = 0 then 0         else \u2191(f a_1 h) i * if h : \u2191(f a_1 h) i = 0 then 0 else \u2191(g i (_ : i \u2208 support (bindOnSupport p f))) a ** simp only [ENNReal.tsum_eq_zero, dite_eq_left_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 \u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),       (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) a_1) *         if h : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) a_1) = 0 then 0         else \u2191(g a_1 (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) a_1) = 0)) a) =     \u2211' (a_1 : \u03b1) (i : \u03b2),       \u2191p a_1 *         if h : \u2191p a_1 = 0 then 0         else \u2191(f a_1 h) i * if h : \u2191(f a_1 h) i = 0 then 0 else \u2191(g i (_ : i \u2208 support (bindOnSupport p f))) a ** refine' ENNReal.tsum_comm.trans (tsum_congr fun a' => tsum_congr fun b => _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 \u22a2 ((\u2191p a' * if h : \u2191p a' = 0 then 0 else \u2191(f a' h) b) *       if h : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 then 0       else \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a) =     \u2191p a' *       if h : \u2191p a' = 0 then 0       else \u2191(f a' h) b * if h : \u2191(f a' h) b = 0 then 0 else \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a ** split_ifs with h _ h_1 _ h_2 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u2191p a' = 0 _ : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 \u22a2 \u2191p a' * 0 * 0 = \u2191p a' * 0  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u2191p a' = 0 _ : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 \u22a2 \u2191p a' * 0 * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a = \u2191p a' * 0  case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u00ac\u2191p a' = 0 h_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 _ : \u2191(f a' h) b = 0 \u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * 0)  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u00ac\u2191p a' = 0 h_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 _ : \u00ac\u2191(f a' h) b = 0 \u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a)  case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u00ac\u2191p a' = 0 h_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 h_2 : \u2191(f a' h) b = 0 \u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =     \u2191p a' * (\u2191(f a' h) b * 0)  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u00ac\u2191p a' = 0 h_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 h_2 : \u00ac\u2191(f a' h) b = 0 \u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =     \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a) ** any_goals ring1 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u00ac\u2191p a' = 0 h_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 h_2 : \u00ac\u2191(f a' h) b = 0 \u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =     \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a) ** ring1 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u00ac\u2191p a' = 0 h_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 _ : \u00ac\u2191(f a' h) b = 0 \u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a) ** have := h_1 a' ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u00ac\u2191p a' = 0 h_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 _ : \u00ac\u2191(f a' h) b = 0 this : (\u2191p a' * if h : \u2191p a' = 0 then 0 else \u2191(f a' h) b) = 0 \u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a) ** simp [h] at this ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u00ac\u2191p a' = 0 h_1 : \u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 _ : \u00ac\u2191(f a' h) b = 0 this : \u2191(f a' (_ : \u00ac\u2191p a' = 0)) b = 0 \u22a2 \u2191p a' * \u2191(f a' h) b * 0 = \u2191p a' * (\u2191(f a' h) b * \u2191(g b (_ : b \u2208 support (bindOnSupport p f))) a) ** contradiction ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 g : (b : \u03b2) \u2192 b \u2208 support (bindOnSupport p f) \u2192 PMF \u03b3 a : \u03b3 a' : \u03b1 b : \u03b2 h : \u00ac\u2191p a' = 0 h_1 : \u00ac\u2200 (i : \u03b1), (\u2191p i * if h : \u2191p i = 0 then 0 else \u2191(f i h) b) = 0 h_2 : \u2191(f a' h) b = 0 \u22a2 \u2191p a' * \u2191(f a' h) b * \u2191(g b (_ : \u00ac(\u2211' (a : \u03b1), \u2191p a * if h : \u2191p a = 0 then 0 else \u2191(f a h) b) = 0)) a =     \u2191p a' * (\u2191(f a' h) b * 0) ** simp [h_2] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.fold_max_add ** \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 inst\u271d\u00b2 : LinearOrder \u03b2 c : \u03b2 inst\u271d\u00b9 : Add \u03b2 inst\u271d : CovariantClass \u03b2 \u03b2 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 n : WithBot \u03b2 s : Finset \u03b1 \u22a2 fold max \u22a5 (fun x => \u2191(f x) + n) s = fold max \u22a5 (WithBot.some \u2218 f) s + n ** classical\n  induction' s using Finset.induction_on with a s _ ih <;> simp [*, max_add_add_right] ** \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 inst\u271d\u00b2 : LinearOrder \u03b2 c : \u03b2 inst\u271d\u00b9 : Add \u03b2 inst\u271d : CovariantClass \u03b2 \u03b2 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 n : WithBot \u03b2 s : Finset \u03b1 \u22a2 fold max \u22a5 (fun x => \u2191(f x) + n) s = fold max \u22a5 (WithBot.some \u2218 f) s + n ** induction' s using Finset.induction_on with a s _ ih <;> simp [*, max_add_add_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_pi_iff ** \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 g : \u03b1 \u2192 (a : \u03b4) \u2192 \u03c0 a \u22a2 Measurable g \u2194 \u2200 (a : \u03b4), Measurable fun x => g x a ** simp_rw [measurable_iff_comap_le, MeasurableSpace.pi, MeasurableSpace.comap_iSup,\n  MeasurableSpace.comap_comp, Function.comp, iSup_le_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.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 \u22a2 \u222b (a : \u03b1), \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic x)) ** rw [\u2190 set_integral_condCdf \u03c1 _ MeasurableSet.univ, Measure.restrict_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.sign_eq_zero_iff_zero ** a : Int h : a = 0 \u22a2 sign a = 0 ** rw [h, sign_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.data_take ** s : String n : Nat \u22a2 (take s n).data = List.take n s.data ** rw [take_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.bitwise_or ** \u22a2 bitwise or = lor ** funext m n ** case h.h m n : \u2124 \u22a2 bitwise or m n = lor 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, cond_true, lor, Nat.ldiff,\n    negSucc.injEq, Bool.true_or, Nat.land] ** case h.h.ofNat.negSucc m n : \u2115 \u22a2 Nat.bitwise (fun x y => !(x || !y)) m n = Nat.bitwise (fun a b => a && !b) n m ** rw [Nat.bitwise_swap, Function.swap] ** case h.h.ofNat.negSucc m n : \u2115 \u22a2 Nat.bitwise (fun y x => !(x || !y)) n m = Nat.bitwise (fun a b => a && !b) n m ** congr ** case h.h.ofNat.negSucc.e_f m n : \u2115 \u22a2 (fun y x => !(x || !y)) = fun a b => a && !b ** funext x y ** case h.h.ofNat.negSucc.e_f.h.h m n : \u2115 x y : Bool \u22a2 (!(y || !x)) = (x && !y) ** cases x <;> cases y <;> rfl ** case h.h.negSucc.ofNat m n : \u2115 \u22a2 Nat.bitwise (fun x y => !(!x || y)) m n = Nat.bitwise (fun a b => a && !b) m n ** congr ** case h.h.negSucc.ofNat.e_f m n : \u2115 \u22a2 (fun x y => !(!x || y)) = fun a b => a && !b ** funext x y ** case h.h.negSucc.ofNat.e_f.h.h m n : \u2115 x y : Bool \u22a2 (!(!x || y)) = (x && !y) ** cases x <;> cases y <;> rfl ** case h.h.negSucc.negSucc m n : \u2115 \u22a2 Nat.bitwise (fun x y => !(!x || !y)) m n = m &&& n ** congr ** case h.h.negSucc.negSucc.e_f m n : \u2115 \u22a2 (fun x y => !(!x || !y)) = and ** funext x y ** case h.h.negSucc.negSucc.e_f.h.h m n : \u2115 x y : Bool \u22a2 (!(!x || !y)) = (x && y) ** cases x <;> cases y <;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.cond_eq_inv_mul_cond_mul ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 s t : Set \u03a9 inst\u271d : IsFiniteMeasure \u03bc hms : MeasurableSet s hmt : MeasurableSet t \u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t ** by_cases ht : \u03bc t = 0 ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 s t : Set \u03a9 inst\u271d : IsFiniteMeasure \u03bc hms : MeasurableSet s hmt : MeasurableSet t ht : \u2191\u2191\u03bc t = 0 \u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t ** simp [cond, ht, Measure.restrict_apply hmt, Or.inr (measure_inter_null_of_null_left s ht)] ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 s t : Set \u03a9 inst\u271d : IsFiniteMeasure \u03bc hms : MeasurableSet s hmt : MeasurableSet t ht : \u00ac\u2191\u2191\u03bc t = 0 \u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t ** rw [mul_assoc, cond_mul_eq_inter \u03bc hmt ht s, Set.inter_comm, cond_apply _ hms] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.sub_add_eq_max ** a b : Nat \u22a2 a - b + b = max a b ** match a.le_total b with\n| .inl hl => rw [Nat.max_eq_right hl, Nat.sub_eq_zero_iff_le.mpr hl, Nat.zero_add]\n| .inr hr => rw [Nat.max_eq_left hr, Nat.sub_add_cancel hr] ** a b : Nat hl : a \u2264 b \u22a2 a - b + b = max a b ** rw [Nat.max_eq_right hl, Nat.sub_eq_zero_iff_le.mpr hl, Nat.zero_add] ** a b : Nat hr : b \u2264 a \u22a2 a - b + b = max a b ** rw [Nat.max_eq_left hr, Nat.sub_add_cancel hr] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.natAbs_valMinAbs_add_le ** n\u271d a\u271d n : \u2115 a b : ZMod n \u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b) ** cases' n with n ** case succ n\u271d a\u271d n : \u2115 a b : ZMod (Nat.succ n) \u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b) ** apply natAbs_min_of_le_div_two n.succ ** case zero n a\u271d : \u2115 a b : ZMod Nat.zero \u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b) ** rfl ** case succ.he n\u271d a\u271d n : \u2115 a b : ZMod (Nat.succ n) \u22a2 \u2191(valMinAbs (a + b)) = \u2191(valMinAbs a + valMinAbs b) ** simp_rw [Int.cast_add, coe_valMinAbs] ** case succ.hl n\u271d a\u271d n : \u2115 a b : ZMod (Nat.succ n) \u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Nat.succ n / 2 ** apply natAbs_valMinAbs_le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.StronglyMeasurable.measurableSet_mulSupport ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 f g : \u03b1 \u2192 \u03b2 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : One \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : MetrizableSpace \u03b2 hf : StronglyMeasurable f \u22a2 MeasurableSet (mulSupport f) ** borelize \u03b2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 f g : \u03b1 \u2192 \u03b2 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : One \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : MetrizableSpace \u03b2 hf : StronglyMeasurable f this\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d : BorelSpace \u03b2 \u22a2 MeasurableSet (mulSupport f) ** exact measurableSet_mulSupport hf.measurable ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.aestronglyMeasurable'_condexpL1 ** \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 : \u03b1 \u2192 F' \u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc ** by_cases hf : Integrable f \u03bc ** 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\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 : \u03b1 \u2192 F' hf : Integrable f \u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc ** rw [condexpL1_eq hf] ** 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\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 : \u03b1 \u2192 F' hf : Integrable f \u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hf))) \u03bc ** exact aestronglyMeasurable'_condexpL1Clm _ ** 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\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 : \u03b1 \u2192 F' hf : \u00acIntegrable f \u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc ** rw [condexpL1_undef hf] ** 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\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 : \u03b1 \u2192 F' hf : \u00acIntegrable f \u22a2 AEStronglyMeasurable' m (\u2191\u21910) \u03bc ** refine AEStronglyMeasurable'.congr ?_ (coeFn_zero _ _ _).symm ** 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\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 : \u03b1 \u2192 F' hf : \u00acIntegrable f \u22a2 AEStronglyMeasurable' m 0 \u03bc ** exact StronglyMeasurable.aeStronglyMeasurable' (@stronglyMeasurable_zero _ _ m _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_mul_le_Lp_mul_Lq_of_nonneg ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) \u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** have h_left : \u222b a, f a * g a \u2202\u03bc = \u222b a, \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc := by\n  refine' integral_congr_ae _\n  filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg\n  rw [Real.norm_of_nonneg hxf, Real.norm_of_nonneg hxg] ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc \u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** have h_right_f : \u222b a, f a ^ p \u2202\u03bc = \u222b a, \u2016f a\u2016 ^ p \u2202\u03bc := by\n  refine' integral_congr_ae _\n  filter_upwards [hf_nonneg] with x hxf\n  rw [Real.norm_of_nonneg hxf] ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc h_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc \u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** have h_right_g : \u222b a, g a ^ q \u2202\u03bc = \u222b a, \u2016g a\u2016 ^ q \u2202\u03bc := by\n  refine' integral_congr_ae _\n  filter_upwards [hg_nonneg] with x hxg\n  rw [Real.norm_of_nonneg hxg] ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc h_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc h_right_g : \u222b (a : \u03b1), g a ^ q \u2202\u03bc = \u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc \u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc \u2264 (\u222b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) ** rw [h_left, h_right_f, h_right_g] ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc h_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc h_right_g : \u222b (a : \u03b1), g a ^ q \u2202\u03bc = \u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc \u22a2 \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc \u2264 (\u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc) ^ (1 / p) * (\u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc) ^ (1 / q) ** exact integral_mul_norm_le_Lp_mul_Lq hpq hf hg ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) \u22a2 \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc ** refine' integral_congr_ae _ ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) \u22a2 (fun a => f a * g a) =\u1d50[\u03bc] fun a => \u2016f a\u2016 * \u2016g a\u2016 ** filter_upwards [hf_nonneg, hg_nonneg] with x hxf hxg ** 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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) x : \u03b1 hxf : OfNat.ofNat 0 x \u2264 f x hxg : OfNat.ofNat 0 x \u2264 g x \u22a2 f x * g x = \u2016f x\u2016 * \u2016g x\u2016 ** rw [Real.norm_of_nonneg hxf, Real.norm_of_nonneg hxg] ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc \u22a2 \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc ** refine' integral_congr_ae _ ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc \u22a2 (fun a => f a ^ p) =\u1d50[\u03bc] fun a => \u2016f a\u2016 ^ p ** filter_upwards [hf_nonneg] with x hxf ** 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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc x : \u03b1 hxf : OfNat.ofNat 0 x \u2264 f x \u22a2 f x ^ p = \u2016f x\u2016 ^ p ** rw [Real.norm_of_nonneg hxf] ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc h_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc \u22a2 \u222b (a : \u03b1), g a ^ q \u2202\u03bc = \u222b (a : \u03b1), \u2016g a\u2016 ^ q \u2202\u03bc ** refine' integral_congr_ae _ ** \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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc h_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc \u22a2 (fun a => g a ^ q) =\u1d50[\u03bc] fun a => \u2016g a\u2016 ^ q ** filter_upwards [hg_nonneg] with x hxg ** 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\u271d : \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 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d hf_nonneg : 0 \u2264\u1d50[\u03bc] f hg_nonneg : 0 \u2264\u1d50[\u03bc] g hf : Mem\u2112p f (ENNReal.ofReal p) hg : Mem\u2112p g (ENNReal.ofReal q) h_left : \u222b (a : \u03b1), f a * g a \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 * \u2016g a\u2016 \u2202\u03bc h_right_f : \u222b (a : \u03b1), f a ^ p \u2202\u03bc = \u222b (a : \u03b1), \u2016f a\u2016 ^ p \u2202\u03bc x : \u03b1 hxg : OfNat.ofNat 0 x \u2264 g x \u22a2 g x ^ q = \u2016g x\u2016 ^ q ** rw [Real.norm_of_nonneg hxg] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.mem_sup ** F : Type u_1 \u03b1\u271d : Type u_2 \u03b2\u271d : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 \u03b1 : Type u_7 \u03b2 : Type u_8 inst\u271d : DecidableEq \u03b2 s : Finset \u03b1 f : \u03b1 \u2192 Finset \u03b2 x : \u03b2 \u22a2 x \u2208 sup s f \u2194 \u2203 v, v \u2208 s \u2227 x \u2208 f v ** change _ \u2194 \u2203 v \u2208 s, x \u2208 (f v).val ** F : Type u_1 \u03b1\u271d : Type u_2 \u03b2\u271d : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 \u03b1 : Type u_7 \u03b2 : Type u_8 inst\u271d : DecidableEq \u03b2 s : Finset \u03b1 f : \u03b1 \u2192 Finset \u03b2 x : \u03b2 \u22a2 x \u2208 sup s f \u2194 \u2203 v, v \u2208 s \u2227 x \u2208 (f v).val ** rw [\u2190 Multiset.mem_sup, \u2190 Multiset.mem_toFinset, sup_toFinset] ** F : Type u_1 \u03b1\u271d : Type u_2 \u03b2\u271d : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 \u03b1 : Type u_7 \u03b2 : Type u_8 inst\u271d : DecidableEq \u03b2 s : Finset \u03b1 f : \u03b1 \u2192 Finset \u03b2 x : \u03b2 \u22a2 x \u2208 sup s f \u2194 x \u2208 sup s fun x => Multiset.toFinset (f x).val ** simp_rw [val_toFinset] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haarMeasure_unique ** 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 : SigmaFinite \u03bc inst\u271d : IsMulLeftInvariant \u03bc K\u2080 : PositiveCompacts G \u22a2 (\u2191\u2191\u03bc \u2191K\u2080 / \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080) \u2022 haarMeasure K\u2080 = \u2191\u2191\u03bc \u2191K\u2080 \u2022 haarMeasure K\u2080 ** rw [haarMeasure_self, div_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.range_orderEmbOfFin ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : LinearOrder \u03b1 s : Finset \u03b1 k : \u2115 h : card s = k \u22a2 Set.range \u2191(orderEmbOfFin s h) = \u2191s ** simp only [orderEmbOfFin, Set.range_comp ((\u2191) : _ \u2192 \u03b1) (s.orderIsoOfFin h),\nRelEmbedding.coe_trans, Set.image_univ, Finset.orderEmbOfFin, RelIso.range_eq,\n  OrderEmbedding.subtype_apply, OrderIso.coe_toOrderEmbedding, eq_self_iff_true,\n  Subtype.range_coe_subtype, Finset.setOf_mem, Finset.coe_inj] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.AssocList.modify_toList ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 inst\u271d : BEq \u03b1 a : \u03b1 l : AssocList \u03b1 \u03b2 \u22a2 toList (modify a f l) =     List.replaceF       (fun x =>         match x with         | (k, v) => bif k == a then some (a, f k v) else none)       (toList l) ** simp [cond] ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 inst\u271d : BEq \u03b1 a : \u03b1 l : AssocList \u03b1 \u03b2 \u22a2 toList (modify a f l) =     List.replaceF       (fun x =>         match x.fst == a with         | true => some (a, f x.fst x.snd)         | false => none)       (toList l) ** induction l with simp [List.replaceF]\n| cons k v es ih => cases k == a <;> simp [ih] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 inst\u271d : BEq \u03b1 a k : \u03b1 v : \u03b2 es : AssocList \u03b1 \u03b2 ih :   toList (modify a f es) =     List.replaceF       (fun x =>         match x.fst == a with         | true => some (a, f x.fst x.snd)         | false => none)       (toList es) \u22a2 toList       (match k == a with       | true => cons a (f k v) es       | false => cons k v (modify a f es)) =     match       match k == a with       | true => some (a, f k v)       | false => none with     | none =>       (k, v) ::         List.replaceF           (fun x =>             match x.fst == a with             | true => some (a, f x.fst x.snd)             | false => none)           (toList es)     | some a => a :: toList es ** cases k == a <;> simp [ih] ** Qed",
        "informal": ""
    },
    {
        "formal": "QPF.Fix.dest_mk ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F x : F (Fix F) \u22a2 dest (mk x) = x ** unfold Fix.dest ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F x : F (Fix F) \u22a2 rec (Functor.map mk) (mk x) = x ** rw [Fix.rec_eq, \u2190 Fix.dest, \u2190 comp_map] ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F x : F (Fix F) \u22a2 (mk \u2218 dest) <$> x = x ** conv =>\n  rhs\n  rw [\u2190 id_map x] ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F x : F (Fix F) \u22a2 (mk \u2218 dest) <$> x = id <$> x ** congr with x ** case e_a.h F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F x\u271d : F (Fix F) x : Fix F \u22a2 (mk \u2218 dest) x = id x ** apply Fix.mk_dest ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snormEssSup_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 snormEssSup 0 \u03bc = 0 ** simp_rw [snormEssSup, Pi.zero_apply, nnnorm_zero, ENNReal.coe_zero, \u2190 ENNReal.bot_eq_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 essSup (fun x => \u22a5) \u03bc = \u22a5 ** exact essSup_const_bot ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.le_mkMetric ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e \u03bc : Measure X \u03b5 : \u211d\u22650\u221e h\u2080 : 0 < \u03b5 h : \u2200 (s : Set X), diam s \u2264 \u03b5 \u2192 \u2191\u2191\u03bc s \u2264 m (diam s) \u22a2 \u03bc \u2264 mkMetric m ** rw [\u2190 toOuterMeasure_le, mkMetric_toOuterMeasure] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e \u03bc : Measure X \u03b5 : \u211d\u22650\u221e h\u2080 : 0 < \u03b5 h : \u2200 (s : Set X), diam s \u2264 \u03b5 \u2192 \u2191\u2191\u03bc s \u2264 m (diam s) \u22a2 \u2191\u03bc \u2264 OuterMeasure.mkMetric m ** exact OuterMeasure.le_mkMetric m \u03bc.toOuterMeasure \u03b5 h\u2080 h ** Qed",
        "informal": ""
    },
    {
        "formal": "NFA.pumping_lemma ** \u03b1 : Type u \u03c3 \u03c3' : Type v M : NFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 hx : x \u2208 accepts M hlen : Fintype.card (Set \u03c3) \u2264 List.length x \u22a2 \u2203 a b c,     x = a ++ b ++ c \u2227 List.length a + List.length b \u2264 Fintype.card (Set \u03c3) \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 accepts M ** rw [\u2190 toDFA_correct] at hx \u22a2 ** \u03b1 : Type u \u03c3 \u03c3' : Type v M : NFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 hx : x \u2208 DFA.accepts (toDFA M) hlen : Fintype.card (Set \u03c3) \u2264 List.length x \u22a2 \u2203 a b c,     x = a ++ b ++ c \u2227       List.length a + List.length b \u2264 Fintype.card (Set \u03c3) \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 DFA.accepts (toDFA M) ** exact M.toDFA.pumping_lemma hx hlen ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.disjoint_ordT5Nhd ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x y z : \u03b1 \u22a2 Disjoint (ordT5Nhd s t) (ordT5Nhd t s) ** rw [disjoint_iff_inf_le] ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x y z : \u03b1 \u22a2 ordT5Nhd s t \u2293 ordT5Nhd t s \u2264 \u22a5 ** rintro x \u27e8hx\u2081, hx\u2082\u27e9 ** case intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x : \u03b1 hx\u2081 : x \u2208 ordT5Nhd s t hx\u2082 : x \u2208 ordT5Nhd t s \u22a2 x \u2208 \u22a5 ** rcases mem_iUnion\u2082.1 hx\u2081 with \u27e8a, has, ha\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x : \u03b1 hx\u2081 : x \u2208 ordT5Nhd s t hx\u2082 : x \u2208 ordT5Nhd t s a : \u03b1 has : a \u2208 s ha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a \u22a2 x \u2208 \u22a5 ** clear hx\u2081 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x : \u03b1 hx\u2082 : x \u2208 ordT5Nhd t s a : \u03b1 has : a \u2208 s ha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a \u22a2 x \u2208 \u22a5 ** rcases mem_iUnion\u2082.1 hx\u2082 with \u27e8b, hbt, hb\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x : \u03b1 hx\u2082 : x \u2208 ordT5Nhd t s a : \u03b1 has : a \u2208 s ha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a b : \u03b1 hbt : b \u2208 t hb : x \u2208 ordConnectedComponent (s\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet t s))\u1d9c) b \u22a2 x \u2208 \u22a5 ** clear hx\u2082 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x a : \u03b1 has : a \u2208 s ha : x \u2208 ordConnectedComponent (t\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet s t))\u1d9c) a b : \u03b1 hbt : b \u2208 t hb : x \u2208 ordConnectedComponent (s\u1d9c \u2229 (ordConnectedSection (ordSeparatingSet t s))\u1d9c) b \u22a2 x \u2208 \u22a5 ** rw [mem_ordConnectedComponent, subset_inter_iff] at ha hb ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x a : \u03b1 has : a \u2208 s ha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c b : \u03b1 hbt : b \u2208 t hb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c \u22a2 x \u2208 \u22a5 ** cases' le_total a b with hab hab ** case intro.intro.intro.intro.intro.inl \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x a : \u03b1 has : a \u2208 s ha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c b : \u03b1 hbt : b \u2208 t hb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c hab : a \u2264 b \u22a2 x \u2208 \u22a5  case intro.intro.intro.intro.intro.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x a : \u03b1 has : a \u2208 s ha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c b : \u03b1 hbt : b \u2208 t hb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c hab : b \u2264 a \u22a2 x \u2208 \u22a5 ** on_goal 2 => swap_var a \u2194 b, s \u2194 t, ha \u2194 hb, has \u2194 hbt ** case intro.intro.intro.intro.intro.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x a : \u03b1 has : a \u2208 s ha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c b : \u03b1 hbt : b \u2208 t hb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c hab : b \u2264 a \u22a2 x \u2208 \u22a5 ** swap_var a \u2194 b, s \u2194 t, ha \u2194 hb, has \u2194 hbt ** case intro.intro.intro.intro.intro.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t hb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c a : \u03b1 has : a \u2208 s ha : [[a, x]] \u2286 t\u1d9c \u2227 [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hab : a \u2264 b \u22a2 x \u2208 \u22a5 ** cases' ha with ha ha' ** case intro.intro.intro.intro.intro.inr.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t hb : [[b, x]] \u2286 s\u1d9c \u2227 [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c ha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c \u22a2 x \u2208 \u22a5 ** cases' hb with hb hb' ** case intro.intro.intro.intro.intro.inr.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c ha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hb : [[b, x]] \u2286 s\u1d9c hb' : [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c \u22a2 x \u2208 \u22a5 ** have hsub : [[a, b]] \u2286 (ordSeparatingSet s t).ordConnectedSection\u1d9c := by\n  rw [ordSeparatingSet_comm, uIcc_comm] at hb'\n  calc\n    [[a, b]] \u2286 [[a, x]] \u222a [[x, b]] := uIcc_subset_uIcc_union_uIcc\n    _ \u2286 (ordSeparatingSet s t).ordConnectedSection\u1d9c := union_subset ha' hb' ** case intro.intro.intro.intro.intro.inr.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c ha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hb : [[b, x]] \u2286 s\u1d9c hb' : [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c \u22a2 x \u2208 \u22a5 ** clear ha' hb' ** case intro.intro.intro.intro.intro.inr.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c \u22a2 x \u2208 \u22a5 ** cases' le_total x a with hxa hax ** case intro.intro.intro.intro.intro.inr.intro.intro.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hax : a \u2264 x \u22a2 x \u2208 \u22a5 ** cases' le_total b x with hbx hxb ** case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hax : a \u2264 x hxb : x \u2264 b \u22a2 x \u2208 \u22a5 ** have h' : x \u2208 ordSeparatingSet s t := \u27e8mem_iUnion\u2082.2 \u27e8a, has, ha\u27e9, mem_iUnion\u2082.2 \u27e8b, hbt, hb\u27e9\u27e9 ** case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hax : a \u2264 x hxb : x \u2264 b h' : x \u2208 ordSeparatingSet s t \u22a2 x \u2208 \u22a5 ** suffices ordConnectedComponent (ordSeparatingSet s t) x \u2286 [[a, b]] from\n  hsub (this <| ordConnectedProj_mem_ordConnectedComponent _ \u27e8x, h'\u27e9) (mem_range_self _) ** case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hax : a \u2264 x hxb : x \u2264 b h' : x \u2208 ordSeparatingSet s t \u22a2 ordConnectedComponent (ordSeparatingSet s t) x \u2286 [[a, b]] ** rintro y (hy : [[x, y]] \u2286 ordSeparatingSet s t) ** case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y\u271d z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hax : a \u2264 x hxb : x \u2264 b h' : x \u2208 ordSeparatingSet s t y : \u03b1 hy : [[x, y]] \u2286 ordSeparatingSet s t \u22a2 y \u2208 [[a, b]] ** rw [uIcc_of_le hab, mem_Icc, \u2190 not_lt, \u2190 not_lt] ** case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y\u271d z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hax : a \u2264 x hxb : x \u2264 b h' : x \u2208 ordSeparatingSet s t y : \u03b1 hy : [[x, y]] \u2286 ordSeparatingSet s t \u22a2 \u00acy < a \u2227 \u00acb < y ** have sol1 := fun (hya : y < a) =>\n    (disjoint_left (t := ordSeparatingSet s t)).1 disjoint_left_ordSeparatingSet has\n      (hy <| Icc_subset_uIcc' \u27e8hya.le, hax\u27e9) ** case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y\u271d z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hax : a \u2264 x hxb : x \u2264 b h' : x \u2208 ordSeparatingSet s t y : \u03b1 hy : [[x, y]] \u2286 ordSeparatingSet s t sol1 : y < a \u2192 False \u22a2 \u00acy < a \u2227 \u00acb < y ** have sol2 := fun (hby : b < y) =>\n    (disjoint_left (t := ordSeparatingSet s t)).1 disjoint_right_ordSeparatingSet hbt\n      (hy <| Icc_subset_uIcc \u27e8hxb, hby.le\u27e9) ** case intro.intro.intro.intro.intro.inr.intro.intro.inr.inr \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y\u271d z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hax : a \u2264 x hxb : x \u2264 b h' : x \u2208 ordSeparatingSet s t y : \u03b1 hy : [[x, y]] \u2286 ordSeparatingSet s t sol1 : y < a \u2192 False sol2 : b < y \u2192 False \u22a2 \u00acy < a \u2227 \u00acb < y ** exact \u27e8sol1, sol2\u27e9 ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c ha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hb : [[b, x]] \u2286 s\u1d9c hb' : [[b, x]] \u2286 (ordConnectedSection (ordSeparatingSet t s))\u1d9c \u22a2 [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c ** rw [ordSeparatingSet_comm, uIcc_comm] at hb' ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c ha' : [[a, x]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hb : [[b, x]] \u2286 s\u1d9c hb' : [[x, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c \u22a2 [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c ** calc\n  [[a, b]] \u2286 [[a, x]] \u222a [[x, b]] := uIcc_subset_uIcc_union_uIcc\n  _ \u2286 (ordSeparatingSet s t).ordConnectedSection\u1d9c := union_subset ha' hb' ** case intro.intro.intro.intro.intro.inr.intro.intro.inl \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hxa : x \u2264 a \u22a2 x \u2208 \u22a5 ** exact hb (Icc_subset_uIcc' \u27e8hxa, hab\u27e9) has ** case intro.intro.intro.intro.intro.inr.intro.intro.inr.inl \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 t s : Set \u03b1 x\u271d y z x b : \u03b1 hbt : b \u2208 t a : \u03b1 has : a \u2208 s hab : a \u2264 b ha : [[a, x]] \u2286 t\u1d9c hb : [[b, x]] \u2286 s\u1d9c hsub : [[a, b]] \u2286 (ordConnectedSection (ordSeparatingSet s t))\u1d9c hax : a \u2264 x hbx : b \u2264 x \u22a2 x \u2208 \u22a5 ** exact ha (Icc_subset_uIcc \u27e8hab, hbx\u27e9) hbt ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.compProd_eq_tsum_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 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 = \u2211' (n : \u2115) (m : \u2115), \u2191\u2191(\u2191(seq \u03ba n \u2297\u2096 seq \u03b7 m) a) s ** simp_rw [compProd_apply_eq_compProdFun _ _ _ 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 = \u2211' (n : \u2115) (m : \u2115), compProdFun (seq \u03ba n) (seq \u03b7 m) a s ** exact compProdFun_eq_tsum \u03ba \u03b7 a hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.continuous_mass ** \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 \u22a2 Continuous fun \u03bc => mass \u03bc ** simp_rw [\u2190 testAgainstNN_one] ** \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 \u22a2 Continuous fun \u03bc => testAgainstNN \u03bc 1 ** exact continuous_testAgainstNN_eval 1 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_absolutelyContinuous_isFiniteMeasure ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u22a2 \u2203 \u03bd, IsFiniteMeasure \u03bd \u2227 \u03bc \u226a \u03bd ** obtain \u27e8g, gpos, gmeas, hg\u27e9 :\n  \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x : \u03b1, 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b x : \u03b1, \u2191(g x) \u2202\u03bc < 1 :=\n  exists_pos_lintegral_lt_of_sigmaFinite \u03bc one_ne_zero ** case intro.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc g : \u03b1 \u2192 \u211d\u22650 gpos : \u2200 (x : \u03b1), 0 < g x gmeas : Measurable g hg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1 \u22a2 \u2203 \u03bd, IsFiniteMeasure \u03bd \u2227 \u03bc \u226a \u03bd ** refine' \u27e8\u03bc.withDensity fun x => g x, isFiniteMeasure_withDensity hg.ne_top, _\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc g : \u03b1 \u2192 \u211d\u22650 gpos : \u2200 (x : \u03b1), 0 < g x gmeas : Measurable g hg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1 \u22a2 \u03bc \u226a withDensity \u03bc fun x => \u2191(g x) ** have : \u03bc = (\u03bc.withDensity fun x => g x).withDensity fun x => (g x)\u207b\u00b9 := by\n  have A : ((fun x : \u03b1 => (g x : \u211d\u22650\u221e)) * fun x : \u03b1 => (g x : \u211d\u22650\u221e)\u207b\u00b9) = 1 := by\n    ext1 x\n    exact ENNReal.mul_inv_cancel (ENNReal.coe_ne_zero.2 (gpos x).ne') ENNReal.coe_ne_top\n  rw [\u2190 withDensity_mul _ gmeas.coe_nnreal_ennreal gmeas.coe_nnreal_ennreal.inv, A,\n    withDensity_one] ** case intro.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc g : \u03b1 \u2192 \u211d\u22650 gpos : \u2200 (x : \u03b1), 0 < g x gmeas : Measurable g hg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1 this : \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9 \u22a2 \u03bc \u226a withDensity \u03bc fun x => \u2191(g x) ** nth_rw 1 [this] ** case intro.intro.intro \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc g : \u03b1 \u2192 \u211d\u22650 gpos : \u2200 (x : \u03b1), 0 < g x gmeas : Measurable g hg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1 this : \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9 \u22a2 (withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9) \u226a withDensity \u03bc fun x => \u2191(g x) ** exact withDensity_absolutelyContinuous _ _ ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc g : \u03b1 \u2192 \u211d\u22650 gpos : \u2200 (x : \u03b1), 0 < g x gmeas : Measurable g hg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1 \u22a2 \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9 ** have A : ((fun x : \u03b1 => (g x : \u211d\u22650\u221e)) * fun x : \u03b1 => (g x : \u211d\u22650\u221e)\u207b\u00b9) = 1 := by\n  ext1 x\n  exact ENNReal.mul_inv_cancel (ENNReal.coe_ne_zero.2 (gpos x).ne') ENNReal.coe_ne_top ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc g : \u03b1 \u2192 \u211d\u22650 gpos : \u2200 (x : \u03b1), 0 < g x gmeas : Measurable g hg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1 A : ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) = 1 \u22a2 \u03bc = withDensity (withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9 ** rw [\u2190 withDensity_mul _ gmeas.coe_nnreal_ennreal gmeas.coe_nnreal_ennreal.inv, A,\n  withDensity_one] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc g : \u03b1 \u2192 \u211d\u22650 gpos : \u2200 (x : \u03b1), 0 < g x gmeas : Measurable g hg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1 \u22a2 ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) = 1 ** ext1 x ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc g : \u03b1 \u2192 \u211d\u22650 gpos : \u2200 (x : \u03b1), 0 < g x gmeas : Measurable g hg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1 x : \u03b1 \u22a2 ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) x = OfNat.ofNat 1 x ** exact ENNReal.mul_inv_cancel (ENNReal.coe_ne_zero.2 (gpos x).ne') ENNReal.coe_ne_top ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.all_iff ** \u03b1 : Type u_1 p : \u03b1 \u2192 Bool t : RBNode \u03b1 \u22a2 all p t = true \u2194 All (fun x => p x = true) t ** induction t <;> simp [*, all, All, and_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.withDensity_mul\u2080 ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g \u22a2 withDensity \u03bc (f * g) = withDensity (withDensity \u03bc f) g ** ext1 s hs ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(withDensity \u03bc (f * g)) s = \u2191\u2191(withDensity (withDensity \u03bc f) g) s ** rw [withDensity_apply _ hs, withDensity_apply _ hs, restrict_withDensity hs,\n  lintegral_withDensity_eq_lintegral_mul\u2080 hf.restrict hg.restrict] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1to1.tr_supports ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (tr enc dec M q) ** suffices \u2200 q, SupportsStmt S q \u2192 (\u2200 q' \u2208 writes q, q' \u2208 trSupp M S) \u2192\n    SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n    \u2200 q' \u2208 writes q, SupportsStmt (trSupp M S) (tr enc dec M q') by\n  rcases Finset.mem_biUnion.1 h with \u27e8l, hl, h\u27e9\n  have :=\n    this _ (ss.2 _ hl) fun q' hq \u21a6 Finset.mem_biUnion.2 \u27e8_, hl, Finset.mem_insert_of_mem hq\u27e9\n  rcases Finset.mem_insert.1 h with (rfl | h)\n  exacts [this.1, this.2 _ h] ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S \u22a2 \u2200 (q : Stmt\u2081),     SupportsStmt S q \u2192       (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192         SupportsStmt (trSupp M S) (trNormal dec q) \u2227           \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** intro q hs hw ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S q : Stmt\u2081 hs : SupportsStmt S q hw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** induction q ** case move \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d\u00b9 : Dir a\u271d : Stmt\u2081 a_ih\u271d :   SupportsStmt S a\u271d \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.move a\u271d\u00b9 a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move a\u271d\u00b9 a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case write \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393 a\u271d : Stmt\u2081 a_ih\u271d :   SupportsStmt S a\u271d \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.write a\u271d\u00b9 a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case load \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3 a\u271d : Stmt\u2081 a_ih\u271d :   SupportsStmt S a\u271d \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.load a\u271d\u00b9 a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case branch \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   SupportsStmt S a\u271d\u00b9 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') a_ih\u271d :   SupportsStmt S a\u271d \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case goto \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b hs : SupportsStmt S (Stmt.goto a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S hs : SupportsStmt S Stmt.halt hw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** case move d q IH =>\n  unfold writes at hw \u22a2\n  replace IH := IH hs hw; refine' \u27e8_, IH.2\u27e9\n  cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH] ** case load \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3 a\u271d : Stmt\u2081 a_ih\u271d :   SupportsStmt S a\u271d \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.load a\u271d\u00b9 a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case branch \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   SupportsStmt S a\u271d\u00b9 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') a_ih\u271d :   SupportsStmt S a\u271d \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case goto \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b hs : SupportsStmt S (Stmt.goto a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S hs : SupportsStmt S Stmt.halt hw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** case load a q IH =>\n  unfold writes at hw \u22a2\n  replace IH := IH hs hw\n  refine' \u27e8supportsStmt_read _ fun _ \u21a6 IH.1, IH.2\u27e9 ** case branch \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   SupportsStmt S a\u271d\u00b9 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') a_ih\u271d :   SupportsStmt S a\u271d \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case goto \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b hs : SupportsStmt S (Stmt.goto a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S hs : SupportsStmt S Stmt.halt hw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** case branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n  unfold writes at hw \u22a2\n  simp only [Finset.mem_union] at hw \u22a2\n  replace IH\u2081 := IH\u2081 hs.1 fun q hq \u21a6 hw q (Or.inl hq)\n  replace IH\u2082 := IH\u2082 hs.2 fun q hq \u21a6 hw q (Or.inr hq)\n  exact \u27e8supportsStmt_read _ fun _ \u21a6 \u27e8IH\u2081.1, IH\u2082.1\u27e9, fun q \u21a6 Or.rec (IH\u2081.2 _) (IH\u2082.2 _)\u27e9 ** case goto \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b hs : SupportsStmt S (Stmt.goto a\u271d) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')  case halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S hs : SupportsStmt S Stmt.halt hw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** case goto l =>\n  simp only [writes, Finset.not_mem_empty]; refine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n  refine' supportsStmt_read _ fun a _ s \u21a6 _\n  exact Finset.mem_biUnion.2 \u27e8_, hs _ _, Finset.mem_insert_self _ _\u27e9 ** case halt \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S hs : SupportsStmt S Stmt.halt hw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** case halt =>\n  simp only [writes, Finset.not_mem_empty]; refine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n  simp only [SupportsStmt, supportsStmt_move, trNormal] ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S this :   \u2200 (q : Stmt\u2081),     SupportsStmt S q \u2192       (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192         SupportsStmt (trSupp M S) (trNormal dec q) \u2227           \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') \u22a2 SupportsStmt (trSupp M S) (tr enc dec M q) ** rcases Finset.mem_biUnion.1 h with \u27e8l, hl, h\u27e9 ** case intro.intro \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h\u271d : q \u2208 trSupp M S this :   \u2200 (q : Stmt\u2081),     SupportsStmt S q \u2192       (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192         SupportsStmt (trSupp M S) (trNormal dec q) \u2227           \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') l : \u039b hl : l \u2208 S h : q \u2208 insert (\u039b'.normal l) (writes (M l)) \u22a2 SupportsStmt (trSupp M S) (tr enc dec M q) ** have :=\n  this _ (ss.2 _ hl) fun q' hq \u21a6 Finset.mem_biUnion.2 \u27e8_, hl, Finset.mem_insert_of_mem hq\u27e9 ** case intro.intro \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h\u271d : q \u2208 trSupp M S this\u271d :   \u2200 (q : Stmt\u2081),     SupportsStmt S q \u2192       (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192         SupportsStmt (trSupp M S) (trNormal dec q) \u2227           \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') l : \u039b hl : l \u2208 S h : q \u2208 insert (\u039b'.normal l) (writes (M l)) this :   SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') \u22a2 SupportsStmt (trSupp M S) (tr enc dec M q) ** rcases Finset.mem_insert.1 h with (rfl | h) ** case intro.intro.inl \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S this\u271d :   \u2200 (q : Stmt\u2081),     SupportsStmt S q \u2192       (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192         SupportsStmt (trSupp M S) (trNormal dec q) \u2227           \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') l : \u039b hl : l \u2208 S this :   SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') h\u271d : \u039b'.normal l \u2208 trSupp M S h : \u039b'.normal l \u2208 insert (\u039b'.normal l) (writes (M l)) \u22a2 SupportsStmt (trSupp M S) (tr enc dec M (\u039b'.normal l))  case intro.intro.inr \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h\u271d\u00b9 : q \u2208 trSupp M S this\u271d :   \u2200 (q : Stmt\u2081),     SupportsStmt S q \u2192       (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192         SupportsStmt (trSupp M S) (trNormal dec q) \u2227           \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') l : \u039b hl : l \u2208 S h\u271d : q \u2208 insert (\u039b'.normal l) (writes (M l)) this :   SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') h : q \u2208 writes (M l) \u22a2 SupportsStmt (trSupp M S) (tr enc dec M q) ** exacts [this.1, this.2 _ h] ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S d : Dir q : Stmt\u2081 IH :   SupportsStmt S q \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.move d q) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move d q) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move d q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** unfold writes at hw \u22a2 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S d : Dir q : Stmt\u2081 IH :   SupportsStmt S q \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.move d q) hw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** replace IH := IH hs hw ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S d : Dir q : Stmt\u2081 hs : SupportsStmt S (Stmt.move d q) hw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S IH :   SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** refine' \u27e8_, IH.2\u27e9 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S d : Dir q : Stmt\u2081 hs : SupportsStmt S (Stmt.move d q) hw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S IH :   SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) ** cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH] ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   SupportsStmt S q \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.write f q) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write f q) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write f q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** unfold writes at hw \u22a2 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   SupportsStmt S q \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.write f q) hw : \u2200 (q' : \u039b'), q' \u2208 Finset.image (fun a => \u039b'.write a q) Finset.univ \u222a writes q \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227     \u2200 (q' : \u039b'),       q' \u2208 Finset.image (fun a => \u039b'.write a q) Finset.univ \u222a writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** simp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and_iff]\n  at hw \u22a2 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   SupportsStmt S q \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.write f q) hw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227     \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** replace IH := IH hs fun q hq \u21a6 hw q (Or.inr hq) ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 hs : SupportsStmt S (Stmt.write f q) hw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S IH :   SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227     \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** refine' \u27e8supportsStmt_read _ fun a _ s \u21a6 hw _ (Or.inl \u27e8_, rfl\u27e9), fun q' hq \u21a6 _\u27e9 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 hs : SupportsStmt S (Stmt.write f q) hw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S IH :   SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') q' : \u039b' hq : (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u22a2 SupportsStmt (trSupp M S) (tr enc dec M q') ** rcases hq with (\u27e8a, q\u2082, rfl\u27e9 | hq) ** case inl.intro.refl \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 hs : SupportsStmt S (Stmt.write f q) hw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S IH :   SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') a : \u0393 \u22a2 SupportsStmt (trSupp M S) (tr enc dec M (\u039b'.write a q)) ** simp only [tr, supportsStmt_write, supportsStmt_move, IH.1] ** case inr \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 hs : SupportsStmt S (Stmt.write f q) hw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S IH :   SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') q' : \u039b' hq : q' \u2208 writes q \u22a2 SupportsStmt (trSupp M S) (tr enc dec M q') ** exact IH.2 _ hq ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S a : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   SupportsStmt S q \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.load a q) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a q) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** unfold writes at hw \u22a2 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S a : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   SupportsStmt S q \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.load a q) hw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** replace IH := IH hs hw ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q\u271d : \u039b' h : q\u271d \u2208 trSupp M S a : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 hs : SupportsStmt S (Stmt.load a q) hw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S IH :   SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** refine' \u27e8supportsStmt_read _ fun _ \u21a6 IH.1, IH.2\u27e9 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   SupportsStmt S q\u2081 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') IH\u2082 :   SupportsStmt S q\u2082 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch p q\u2081 q\u2082) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch p q\u2081 q\u2082) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** unfold writes at hw \u22a2 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   SupportsStmt S q\u2081 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') IH\u2082 :   SupportsStmt S q\u2082 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082) hw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u222a writes q\u2082 \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u222a writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** simp only [Finset.mem_union] at hw \u22a2 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   SupportsStmt S q\u2081 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') IH\u2082 :   SupportsStmt S q\u2082 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082) hw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** replace IH\u2081 := IH\u2081 hs.1 fun q hq \u21a6 hw q (Or.inl hq) ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2082 :   SupportsStmt S q\u2082 \u2192     (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192       SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227         \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') hs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082) hw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S IH\u2081 :   SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** replace IH\u2082 := IH\u2082 hs.2 fun q hq \u21a6 hw q (Or.inr hq) ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 hs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082) hw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S IH\u2081 :   SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') IH\u2082 :   SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** exact \u27e8supportsStmt_read _ fun _ \u21a6 \u27e8IH\u2081.1, IH\u2082.1\u27e9, fun q \u21a6 Or.rec (IH\u2081.2 _) (IH\u2082.2 _)\u27e9 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S l : \u0393 \u2192 \u03c3 \u2192 \u039b hs : SupportsStmt S (Stmt.goto l) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l)) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** simp only [writes, Finset.not_mem_empty] ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S l : \u0393 \u2192 \u03c3 \u2192 \u039b hs : SupportsStmt S (Stmt.goto l) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l)) \u2227     \u2200 (q' : \u039b'), False \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** refine' \u27e8_, fun _ \u21a6 False.elim\u27e9 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S l : \u0393 \u2192 \u03c3 \u2192 \u039b hs : SupportsStmt S (Stmt.goto l) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l)) ** refine' supportsStmt_read _ fun a _ s \u21a6 _ ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S l : \u0393 \u2192 \u03c3 \u2192 \u039b hs : SupportsStmt S (Stmt.goto l) hw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S a : \u0393 x\u271d : Bool s : \u03c3 \u22a2 (fun x s => \u039b'.normal (l a s)) x\u271d s \u2208 trSupp M S ** exact Finset.mem_biUnion.2 \u27e8_, hs _ _, Finset.mem_insert_self _ _\u27e9 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S hs : SupportsStmt S Stmt.halt hw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227     \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** simp only [writes, Finset.not_mem_empty] ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S hs : SupportsStmt S Stmt.halt hw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227 \u2200 (q' : \u039b'), False \u2192 SupportsStmt (trSupp M S) (tr enc dec M q') ** refine' \u27e8_, fun _ \u21a6 False.elim\u27e9 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b2 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d\u00b9 : 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 inst\u271d : Fintype \u0393 S : Finset \u039b ss : Supports M S q : \u039b' h : q \u2208 trSupp M S hs : SupportsStmt S Stmt.halt hw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S \u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) ** simp only [SupportsStmt, supportsStmt_move, trNormal] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.PairingHeapImp.Heap.noSibling_tail ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 \u22a2 NoSibling (tail le s) ** simp only [Heap.tail] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 \u22a2 NoSibling (Option.getD (tail? le s) nil) ** match eq : s.tail? le with\n| none => cases s with cases eq | nil => constructor\n| some tl => exact Heap.noSibling_tail? eq ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 eq : tail? le s = none \u22a2 NoSibling (Option.getD none nil) ** cases s with cases eq | nil => constructor ** case nil.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool \u22a2 NoSibling (Option.getD none nil) ** constructor ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s tl : Heap \u03b1 eq : tail? le s = some tl \u22a2 NoSibling (Option.getD (some tl) nil) ** exact Heap.noSibling_tail? eq ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.natAbs_mul ** a b : Int \u22a2 natAbs (a * b) = natAbs a * natAbs b ** cases a <;> cases b <;>\n  simp only [\u2190 Int.mul_def, Int.mul, natAbs_negOfNat] <;> simp only [natAbs] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpIndSMul_empty ** \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\u2079 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2078 : NormedAddCommGroup E inst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2076 : CompleteSpace E inst\u271d\u00b9\u2075 : NormedAddCommGroup E' inst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b3 : CompleteSpace E' inst\u271d\u00b9\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedAddCommGroup G' inst\u271d\u2077 : NormedSpace \u211d G' inst\u271d\u2076 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2075 : IsROrC \ud835\udd5c' inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b2 : CompleteSpace E'' inst\u271d\u00b9 : NormedSpace \u211d E'' inst\u271d : NormedSpace \u211d G hm : m \u2264 m0 x : G \u22a2 condexpIndSMul hm (_ : MeasurableSet \u2205) (_ : \u2191\u2191\u03bc \u2205 \u2260 \u22a4) x = 0 ** rw [condexpIndSMul, indicatorConstLp_empty] ** \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\u2079 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2078 : NormedAddCommGroup E inst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2076 : CompleteSpace E inst\u271d\u00b9\u2075 : NormedAddCommGroup E' inst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b3 : CompleteSpace E' inst\u271d\u00b9\u00b2 : NormedSpace \u211d E' inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedAddCommGroup G' inst\u271d\u2077 : NormedSpace \u211d G' inst\u271d\u2076 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2075 : IsROrC \ud835\udd5c' inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b2 : CompleteSpace E'' inst\u271d\u00b9 : NormedSpace \u211d E'' inst\u271d : NormedSpace \u211d G hm : m \u2264 m0 x : G \u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) 0) = 0 ** simp only [Submodule.coe_zero, ContinuousLinearMap.map_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.measure_ge_le_exp_mul_mgf ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-t * \u03b5) * mgf X \u03bc t ** cases' ht.eq_or_lt with ht_zero_eq ht_pos ** case inr \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_pos : 0 < t \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-t * \u03b5) * mgf X \u03bc t ** calc\n  (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal = (\u03bc {\u03c9 | exp (t * \u03b5) \u2264 exp (t * X \u03c9)}).toReal := by\n    congr with \u03c9\n    simp only [Set.mem_setOf_eq, exp_le_exp, gt_iff_lt]\n    exact \u27e8fun h => mul_le_mul_of_nonneg_left h ht_pos.le,\n      fun h => le_of_mul_le_mul_left h ht_pos\u27e9\n  _ \u2264 (exp (t * \u03b5))\u207b\u00b9 * \u03bc[fun \u03c9 => exp (t * X \u03c9)] := by\n    have : exp (t * \u03b5) * (\u03bc {\u03c9 | exp (t * \u03b5) \u2264 exp (t * X \u03c9)}).toReal \u2264\n        \u03bc[fun \u03c9 => exp (t * X \u03c9)] :=\n      mul_meas_ge_le_integral_of_nonneg (ae_of_all _ fun x => (exp_pos _).le) h_int _\n    rwa [mul_comm (exp (t * \u03b5))\u207b\u00b9, \u2190 div_eq_mul_inv, le_div_iff' (exp_pos _)]\n  _ = exp (-t * \u03b5) * mgf X \u03bc t := by rw [neg_mul, exp_neg]; rfl ** case inl \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_zero_eq : 0 = t \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-t * \u03b5) * mgf X \u03bc t ** rw [ht_zero_eq.symm] ** case inl \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_zero_eq : 0 = t \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 rexp (-0 * \u03b5) * mgf X \u03bc 0 ** simp only [neg_zero, zero_mul, exp_zero, mgf_zero', one_mul] ** case inl \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_zero_eq : 0 = t \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) \u2264 ENNReal.toReal (\u2191\u2191\u03bc Set.univ) ** rw [ENNReal.toReal_le_toReal (measure_ne_top \u03bc _) (measure_ne_top \u03bc _)] ** case inl \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_zero_eq : 0 = t \u22a2 \u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9} \u2264 \u2191\u2191\u03bc Set.univ ** exact measure_mono (Set.subset_univ _) ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_pos : 0 < t \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}) = ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}) ** congr with \u03c9 ** case e_a.e_a.h \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_pos : 0 < t \u03c9 : \u03a9 \u22a2 \u03c9 \u2208 {\u03c9 | \u03b5 \u2264 X \u03c9} \u2194 \u03c9 \u2208 {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)} ** simp only [Set.mem_setOf_eq, exp_le_exp, gt_iff_lt] ** case e_a.e_a.h \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_pos : 0 < t \u03c9 : \u03a9 \u22a2 \u03b5 \u2264 X \u03c9 \u2194 t * \u03b5 \u2264 t * X \u03c9 ** exact \u27e8fun h => mul_le_mul_of_nonneg_left h ht_pos.le,\n  fun h => le_of_mul_le_mul_left h ht_pos\u27e9 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_pos : 0 < t \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}) \u2264     (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc ** have : exp (t * \u03b5) * (\u03bc {\u03c9 | exp (t * \u03b5) \u2264 exp (t * X \u03c9)}).toReal \u2264\n    \u03bc[fun \u03c9 => exp (t * X \u03c9)] :=\n  mul_meas_ge_le_integral_of_nonneg (ae_of_all _ fun x => (exp_pos _).le) h_int _ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_pos : 0 < t this :   rexp (t * \u03b5) * ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}) \u2264 \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | rexp (t * \u03b5) \u2264 rexp (t * X \u03c9)}) \u2264     (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc ** rwa [mul_comm (exp (t * \u03b5))\u207b\u00b9, \u2190 div_eq_mul_inv, le_div_iff' (exp_pos _)] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_pos : 0 < t \u22a2 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc = rexp (-t * \u03b5) * mgf X \u03bc t ** rw [neg_mul, exp_neg] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsFiniteMeasure \u03bc \u03b5 : \u211d ht : 0 \u2264 t h_int : Integrable fun \u03c9 => rexp (t * X \u03c9) ht_pos : 0 < t \u22a2 (rexp (t * \u03b5))\u207b\u00b9 * \u222b (x : \u03a9), (fun \u03c9 => rexp (t * X \u03c9)) x \u2202\u03bc = (rexp (t * \u03b5))\u207b\u00b9 * mgf X \u03bc t ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.singularPart_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 r : \u211d \u22a2 singularPart (r \u2022 s) \u03bc = r \u2022 singularPart s \u03bc ** by_cases hr : 0 \u2264 r ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 r : \u211d hr : 0 \u2264 r \u22a2 singularPart (r \u2022 s) \u03bc = r \u2022 singularPart s \u03bc ** lift r to \u211d\u22650 using hr ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 \u22a2 singularPart (\u2191r \u2022 s) \u03bc = \u2191r \u2022 singularPart s \u03bc ** exact singularPart_smul_nnreal s \u03bc r ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 r : \u211d hr : \u00ac0 \u2264 r \u22a2 singularPart (r \u2022 s) \u03bc = r \u2022 singularPart s \u03bc ** rw [singularPart, singularPart] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 r : \u211d hr : \u00ac0 \u2264 r \u22a2 toSignedMeasure (Real.toNNReal (-r) \u2022 Measure.singularPart (toJordanDecomposition s).negPart \u03bc) -       toSignedMeasure (Real.toNNReal (-r) \u2022 Measure.singularPart (toJordanDecomposition s).posPart \u03bc) =     r \u2022       (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -         toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)) ** rw [toSignedMeasure_smul, toSignedMeasure_smul, \u2190 neg_sub, \u2190 smul_sub] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 r : \u211d hr : \u00ac0 \u2264 r \u22a2 -(Real.toNNReal (-r) \u2022         (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -           toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))) =     r \u2022       (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -         toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)) ** change -(((-r).toNNReal : \u211d) \u2022 (_ : SignedMeasure \u03b1)) = _ ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s\u271d t s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 r : \u211d hr : \u00ac0 \u2264 r \u22a2 -(\u2191(Real.toNNReal (-r)) \u2022         (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -           toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc))) =     r \u2022       (toSignedMeasure (Measure.singularPart (toJordanDecomposition s).posPart \u03bc) -         toSignedMeasure (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)) ** rw [\u2190 neg_smul, Real.coe_toNNReal _ (le_of_lt (neg_pos.mpr (not_le.1 hr))), neg_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_smul_measure_of_ne_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 : \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 F c : \u211d\u22650\u221e hc : c \u2260 0 \u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 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\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 F c : \u211d\u22650\u221e hc : c \u2260 0 hp0 : \u00acp = 0 \u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 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\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 F c : \u211d\u22650\u221e hc : c \u2260 0 hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc ** exact snorm_smul_measure_of_ne_zero_of_ne_top hp0 hp_top c ** 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 F c : \u211d\u22650\u221e hc : c \u2260 0 hp0 : p = 0 \u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 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\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 F c : \u211d\u22650\u221e hc : c \u2260 0 hp0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc ** simp [hp_top, snormEssSup_smul_measure hc] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.borel_eq_generateFrom_Ioi_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, {Ioi \u2191a}) ** rw [borel_eq_generateFrom_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 t u : Set \u03b1 \u22a2 MeasurableSpace.generateFrom (range Ioi) = MeasurableSpace.generateFrom (\u22c3 a, {Ioi \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 Ioi \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 (Ioi a) ** have : IsGLB (range ((\u2191) : \u211a \u2192 \u211d) \u2229 Ioi a) a := by\n  simp [isGLB_iff_le_iff, mem_lowerBounds, \u2190 le_iff_forall_lt_rat_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 : IsGLB (range Rat.cast \u2229 Ioi a) a \u22a2 MeasurableSet (Ioi a) ** rw [\u2190 this.biUnion_Ioi_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 : IsGLB (range Rat.cast \u2229 Ioi a) a \u22a2 MeasurableSet (\u22c3 y \u2208 Rat.cast \u207b\u00b9' Ioi a, Ioi \u2191y) ** exact MeasurableSet.biUnion (to_countable _)\n  fun b _ => GenerateMeasurable.basic (Ioi (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 IsGLB (range Rat.cast \u2229 Ioi a) a ** simp [isGLB_iff_le_iff, mem_lowerBounds, \u2190 le_iff_forall_lt_rat_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 : IsGLB (range Rat.cast \u2229 Ioi a) a b : \u211a x\u271d : b \u2208 Rat.cast \u207b\u00b9' Ioi a \u22a2 Ioi \u2191b \u2208 \u22c3 a, {Ioi \u2191a} ** simp ** 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\u271d : \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 p : \u211d hp_pos : 0 < p 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 ** rw [snorm'_lim_eq_lintegral_liminf hp_pos.le h_lim] ** \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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 (\u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc) ^ (1 / p) \u2264 liminf (fun n => snorm' (f n) p \u03bc) atTop ** rw [\u2190 ENNReal.le_rpow_one_div_iff (by simp [hp_pos] : 0 < 1 / p), one_div_one_div] ** \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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 \u222b\u207b (a : \u03b1), liminf (fun m => \u2191\u2016f m a\u2016\u208a ^ p) atTop \u2202\u03bc \u2264 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p ** refine (lintegral_liminf_le' fun m => (hf m).ennnorm.pow_const _).trans_eq ?_ ** \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ p \u2202\u03bc) atTop = liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p ** have h_pow_liminf :\n  (atTop.liminf fun n => snorm' (f n) p \u03bc) ^ p = atTop.liminf fun n => snorm' (f n) p \u03bc ^ p := by\n  have h_rpow_mono := ENNReal.strictMono_rpow_of_pos hp_pos\n  have h_rpow_surj := (ENNReal.rpow_left_bijective hp_pos.ne.symm).2\n  refine' (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).liminf_apply _ _ _ _\n  all_goals isBoundedDefault ** \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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)) h_pow_liminf : liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop \u22a2 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ p \u2202\u03bc) atTop = liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p ** rw [h_pow_liminf] ** \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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)) h_pow_liminf : liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop \u22a2 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ p \u2202\u03bc) atTop = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop ** simp_rw [snorm', \u2190 ENNReal.rpow_mul, one_div, inv_mul_cancel hp_pos.ne.symm, ENNReal.rpow_one] ** \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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 0 < 1 / p ** simp [hp_pos] ** \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop ** have h_rpow_mono := ENNReal.strictMono_rpow_of_pos hp_pos ** \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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)) h_rpow_mono : StrictMono fun x => x ^ p \u22a2 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop ** have h_rpow_surj := (ENNReal.rpow_left_bijective hp_pos.ne.symm).2 ** \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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)) h_rpow_mono : StrictMono fun x => x ^ p h_rpow_surj : Function.Surjective fun y => y ^ p \u22a2 liminf (fun n => snorm' (f n) p \u03bc) atTop ^ p = liminf (fun n => snorm' (f n) p \u03bc ^ p) atTop ** refine' (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).liminf_apply _ _ _ _ ** case refine'_1 \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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)) h_rpow_mono : StrictMono fun x => x ^ p h_rpow_surj : Function.Surjective fun y => y ^ p \u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => snorm' (f n) p \u03bc  case refine'_2 \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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)) h_rpow_mono : StrictMono fun x => x ^ p h_rpow_surj : Function.Surjective fun y => y ^ p \u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => snorm' (f n) p \u03bc  case refine'_3 \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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)) h_rpow_mono : StrictMono fun x => x ^ p h_rpow_surj : Function.Surjective fun y => y ^ p \u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x =>     \u2191(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj) (snorm' (f x) p \u03bc)  case refine'_4 \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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)) h_rpow_mono : StrictMono fun x => x ^ p h_rpow_surj : Function.Surjective fun y => y ^ p \u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x =>     \u2191(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj) (snorm' (f x) p \u03bc) ** all_goals isBoundedDefault ** case refine'_4 \u03b1 : Type u_1 E\u271d : 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\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 p : \u211d hp_pos : 0 < p 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)) h_rpow_mono : StrictMono fun x => x ^ p h_rpow_surj : Function.Surjective fun y => y ^ p \u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x =>     \u2191(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj) (snorm' (f x) p \u03bc) ** isBoundedDefault ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_le_lintegral ** \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\u22650\u221e inst\u271d : IsProbabilityMeasure \u03bc hf : AEMeasurable f \u22a2 \u2203 x, f x \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc ** simpa only [laverage_eq_lintegral] using exists_le_laverage (IsProbabilityMeasure.ne_zero \u03bc) hf ** Qed",
        "informal": ""
    },
    {
        "formal": "AEMeasurable.isLUB ** \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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u22a2 AEMeasurable g ** nontriviality \u03b1 ** \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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 \u22a2 AEMeasurable g ** haveI h\u03b1 : Nonempty \u03b1 := inferInstance ** \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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 \u22a2 AEMeasurable g ** cases' 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 \u22a2 AEMeasurable g ** let p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB { a | \u2203 i, f' i = a } (g x) ** 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) \u22a2 AEMeasurable g ** let g_seq := (aeSeqSet hf p).piecewise g fun _ => h\u03b1.some ** 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 hg_seq : \u2200 (b : \u03b4), IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq b) \u22a2 AEMeasurable g ** refine' \u27e8g_seq, Measurable.isLUB (aeSeq.measurable hf p) hg_seq, _\u27e9 ** 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 hg_seq : \u2200 (b : \u03b4), IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq b) \u22a2 g =\u1d50[\u03bc] g_seq ** exact\n  (ite_ae_eq_of_measure_compl_zero g (fun _ => h\u03b1.some) (aeSeqSet hf p)\n      (aeSeq.measure_compl_aeSeqSet_eq_zero hf hg)).symm ** 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : IsEmpty \u03b9 \u22a2 AEMeasurable g ** simp only [IsEmpty.exists_iff, setOf_false, isLUB_empty_iff] at hg ** 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : IsEmpty \u03b9 hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsBot (g b) \u22a2 AEMeasurable g ** exact aemeasurable_const' (hg.mono fun a ha => hg.mono fun b hb => (ha _).antisymm (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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 \u22a2 \u2200 (b : \u03b4), IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq b) ** intro b ** \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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 b : \u03b4 \u22a2 IsLUB {a | \u2203 i, aeSeq hf p i b = a} (g_seq b) ** simp only [aeSeq, Set.piecewise] ** \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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 b : \u03b4 \u22a2 IsLUB     {a |       \u2203 i,         (if b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) then mk (f i) (_ : AEMeasurable (f i)) b           else Nonempty.some (_ : Nonempty \u03b1)) =           a}     (if b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) then g b else Nonempty.some h\u03b1) ** split_ifs with 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 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 b : \u03b4 h : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) \u22a2 IsLUB {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g b) ** have h_set_eq : { a : \u03b1 | \u2203 i : \u03b9, (hf i).mk (f i) b = a } =\n  { a : \u03b1 | \u2203 i : \u03b9, f i b = a } := by\n  ext x\n  simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf 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 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 b : \u03b4 h : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) h_set_eq : {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | \u2203 i, f i b = a} \u22a2 IsLUB {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} (g b) ** rw [h_set_eq] ** 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 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 b : \u03b4 h : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) h_set_eq : {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | \u2203 i, f i b = a} \u22a2 IsLUB {a | \u2203 i, f i b = a} (g b) ** exact aeSeq.fun_prop_of_mem_aeSeqSet hf 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 b : \u03b4 h : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) \u22a2 {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} = {a | \u2203 i, f i b = a} ** ext x ** case 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 b : \u03b4 h : b \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) x : \u03b1 \u22a2 x \u2208 {a | \u2203 i, mk (f i) (_ : AEMeasurable (f i)) b = a} \u2194 x \u2208 {a | \u2203 i, f i b = a} ** simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf 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 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 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 \u03bc : Measure \u03b4 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 g : \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), AEMeasurable (f i) hg : \u2200\u1d50 (b : \u03b4) \u2202\u03bc, IsLUB {a | \u2203 i, f i b = a} (g b) \u271d : Nontrivial \u03b1 h\u03b1 : Nonempty \u03b1 h\u03b9 : Nonempty \u03b9 p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) g_seq : \u03b4 \u2192 \u03b1 := piecewise (aeSeqSet hf p) g fun x => Nonempty.some h\u03b1 b : \u03b4 h : \u00acb \u2208 aeSeqSet hf fun x f' => IsLUB {a | \u2203 i, f' i = a} (g x) \u22a2 IsLUB {a | \u2203 i, Nonempty.some (_ : Nonempty \u03b1) = a} (Nonempty.some h\u03b1) ** exact IsGreatest.isLUB \u27e8(@exists_const (h\u03b1.some = h\u03b1.some) \u03b9 _).2 rfl, fun x \u27e8i, hi\u27e9 => hi.ge\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_congr_measure_of_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\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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f \u22a2 setToFun \u03bc T hT f = setToFun \u03bc' T hT' f ** have h_int : \u2200 g : \u03b1 \u2192 E, Integrable g \u03bc \u2192 Integrable g \u03bc' := fun g hg =>\n  Integrable.of_measure_le_smul c' hc' h\u03bc'_le hg ** \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g \u22a2 setToFun \u03bc T hT f = setToFun \u03bc' T hT' f ** apply hf\u03bc.induction (P := fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) ** case h_ind \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g \u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) (indicator s fun x => c) ** intro c s hs h\u03bcs ** case h_ind \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g c : E s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 setToFun \u03bc T hT (indicator s fun x => c) = setToFun \u03bc' T hT' (indicator s fun x => c) ** have h\u03bc's : \u03bc' s \u2260 \u221e := by\n  refine' ((h\u03bc'_le s hs).trans_lt _).ne\n  rw [Measure.smul_apply, smul_eq_mul]\n  exact ENNReal.mul_lt_top hc' h\u03bcs.ne ** case h_ind \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g c : E s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 h\u03bc's : \u2191\u2191\u03bc' s \u2260 \u22a4 \u22a2 setToFun \u03bc T hT (indicator s fun x => c) = setToFun \u03bc' T hT' (indicator s fun x => c) ** rw [setToFun_indicator_const hT hs h\u03bcs.ne, setToFun_indicator_const hT' hs h\u03bc's] ** \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g c : E s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2191\u2191\u03bc' s \u2260 \u22a4 ** refine' ((h\u03bc'_le s hs).trans_lt _).ne ** \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g c : E s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2191\u2191(c' \u2022 \u03bc) s < \u22a4 ** rw [Measure.smul_apply, smul_eq_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 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g c : E s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 c' * \u2191\u2191\u03bc s < \u22a4 ** exact ENNReal.mul_lt_top hc' h\u03bcs.ne ** case 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\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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g \u22a2 \u2200 \u2983f g : \u03b1 \u2192 E\u2984,     Disjoint (Function.support f) (Function.support g) \u2192       Integrable f \u2192         Integrable g \u2192           (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) f \u2192             (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) g \u2192               (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) (f + g) ** intro f\u2082 g\u2082 _ hf\u2082 hg\u2082 h_eq_f h_eq_g ** case 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\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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g f\u2082 g\u2082 : \u03b1 \u2192 E a\u271d : Disjoint (Function.support f\u2082) (Function.support g\u2082) hf\u2082 : Integrable f\u2082 hg\u2082 : Integrable g\u2082 h_eq_f : setToFun \u03bc T hT f\u2082 = setToFun \u03bc' T hT' f\u2082 h_eq_g : setToFun \u03bc T hT g\u2082 = setToFun \u03bc' T hT' g\u2082 \u22a2 setToFun \u03bc T hT (f\u2082 + g\u2082) = setToFun \u03bc' T hT' (f\u2082 + g\u2082) ** rw [setToFun_add hT hf\u2082 hg\u2082, setToFun_add hT' (h_int f\u2082 hf\u2082) (h_int g\u2082 hg\u2082), h_eq_f, h_eq_g] ** case h_closed \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g \u22a2 IsClosed {f | (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) \u2191\u2191f} ** refine' isClosed_eq (continuous_setToFun hT) _ ** case h_closed \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g \u22a2 Continuous fun f => setToFun \u03bc' T hT' \u2191\u2191f ** have :\n  (fun f : \u03b1 \u2192\u2081[\u03bc] E => setToFun \u03bc' T hT' f) = fun f : \u03b1 \u2192\u2081[\u03bc] E =>\n    setToFun \u03bc' T hT' ((h_int f (L1.integrable_coeFn f)).toL1 f) := by\n  ext1 f; exact setToFun_congr_ae hT' (Integrable.coeFn_toL1 _).symm ** case h_closed \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g this : (fun f => setToFun \u03bc' T hT' \u2191\u2191f) = fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) \u22a2 Continuous fun f => setToFun \u03bc' T hT' \u2191\u2191f ** rw [this] ** case h_closed \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g this : (fun f => setToFun \u03bc' T hT' \u2191\u2191f) = fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) \u22a2 Continuous fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) ** exact (continuous_setToFun hT').comp (continuous_L1_toL1 c' hc' h\u03bc'_le) ** \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g \u22a2 (fun f => setToFun \u03bc' T hT' \u2191\u2191f) = fun f => setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) ** 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\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\u00b9 g : \u03b1 \u2192 E \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f\u271d : \u03b1 \u2192 E hf\u03bc : Integrable f\u271d h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g f : { x // x \u2208 Lp E 1 } \u22a2 setToFun \u03bc' T hT' \u2191\u2191f = setToFun \u03bc' T hT' \u2191\u2191(Integrable.toL1 \u2191\u2191f (_ : Integrable \u2191\u2191f)) ** exact setToFun_congr_ae hT' (Integrable.coeFn_toL1 _).symm ** case h_ae \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g \u22a2 \u2200 \u2983f g : \u03b1 \u2192 E\u2984,     f =\u1d50[\u03bc] g \u2192       Integrable f \u2192         (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) f \u2192 (fun f => setToFun \u03bc T hT f = setToFun \u03bc' T hT' f) g ** intro f\u2082 g\u2082 hfg _ hf_eq ** case h_ae \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g f\u2082 g\u2082 : \u03b1 \u2192 E hfg : f\u2082 =\u1d50[\u03bc] g\u2082 a\u271d : Integrable f\u2082 hf_eq : setToFun \u03bc T hT f\u2082 = setToFun \u03bc' T hT' f\u2082 \u22a2 setToFun \u03bc T hT g\u2082 = setToFun \u03bc' T hT' g\u2082 ** have hfg' : f\u2082 =\u1d50[\u03bc'] g\u2082 := (Measure.absolutelyContinuous_of_le_smul h\u03bc'_le).ae_eq hfg ** case h_ae \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 \u03bc' : Measure \u03b1 c' : \u211d\u22650\u221e hc' : c' \u2260 \u22a4 h\u03bc'_le : \u03bc' \u2264 c' \u2022 \u03bc hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc' T C' f : \u03b1 \u2192 E hf\u03bc : Integrable f h_int : \u2200 (g : \u03b1 \u2192 E), Integrable g \u2192 Integrable g f\u2082 g\u2082 : \u03b1 \u2192 E hfg : f\u2082 =\u1d50[\u03bc] g\u2082 a\u271d : Integrable f\u2082 hf_eq : setToFun \u03bc T hT f\u2082 = setToFun \u03bc' T hT' f\u2082 hfg' : f\u2082 =\u1d50[\u03bc'] g\u2082 \u22a2 setToFun \u03bc T hT g\u2082 = setToFun \u03bc' T hT' g\u2082 ** rw [\u2190 setToFun_congr_ae hT hfg, hf_eq, setToFun_congr_ae hT' hfg'] ** Qed",
        "informal": ""
    },
    {
        "formal": "PosNum.cast_sub' ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a : PosNum \u22a2 \u2191(sub' a 1) = \u2191a - \u21911 ** rw [sub'_one, Num.cast_toZNum, \u2190 Num.cast_to_nat, pred'_to_nat, \u2190 Nat.sub_one] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a : PosNum \u22a2 \u2191(\u2191a - 1) = \u2191a - \u21911 ** simp [PosNum.cast_pos] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 b : PosNum \u22a2 \u2191(sub' 1 b) = \u21911 - \u2191b ** rw [one_sub', Num.cast_toZNumNeg, \u2190 neg_sub, neg_inj, \u2190 Num.cast_to_nat, pred'_to_nat,\n    \u2190 Nat.sub_one] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 b : PosNum \u22a2 \u2191(\u2191b - 1) = \u2191b - \u21911 ** simp [PosNum.cast_pos] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 \u2191(sub' (bit0 a) (bit0 b)) = \u2191(bit0 a) - \u2191(bit0 b) ** rw [sub', ZNum.cast_bit0, cast_sub' a b] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit0 a) - \u2191(bit0 b) ** have : ((a + -b + (a + -b) : \u2124) : \u03b1) = a + a + (-b + -b) := by simp [add_left_comm] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum this : \u2191(\u2191a + -\u2191b + (\u2191a + -\u2191b)) = \u2191a + \u2191a + (-\u2191b + -\u2191b) \u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit0 a) - \u2191(bit0 b) ** simpa [_root_.bit0, sub_eq_add_neg] using this ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 \u2191(\u2191a + -\u2191b + (\u2191a + -\u2191b)) = \u2191a + \u2191a + (-\u2191b + -\u2191b) ** simp [add_left_comm] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 \u2191(sub' (bit0 a) (bit1 b)) = \u2191(bit0 a) - \u2191(bit1 b) ** rw [sub', ZNum.cast_bitm1, cast_sub' a b] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 _root_.bit0 (\u2191a - \u2191b) - 1 = \u2191(bit0 a) - \u2191(bit1 b) ** have : ((-b + (a + (-b + -1)) : \u2124) : \u03b1) = (a + -1 + (-b + -b) : \u2124) := by\n  simp [add_comm, add_left_comm] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum this : \u2191(-\u2191b + (\u2191a + (-\u2191b + -1))) = \u2191(\u2191a + -1 + (-\u2191b + -\u2191b)) \u22a2 _root_.bit0 (\u2191a - \u2191b) - 1 = \u2191(bit0 a) - \u2191(bit1 b) ** simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 \u2191(-\u2191b + (\u2191a + (-\u2191b + -1))) = \u2191(\u2191a + -1 + (-\u2191b + -\u2191b)) ** simp [add_comm, add_left_comm] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 \u2191(sub' (bit1 a) (bit0 b)) = \u2191(bit1 a) - \u2191(bit0 b) ** rw [sub', ZNum.cast_bit1, cast_sub' a b] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 _root_.bit1 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit0 b) ** have : ((-b + (a + (-b + 1)) : \u2124) : \u03b1) = (a + 1 + (-b + -b) : \u2124) := by\n  simp [add_comm, add_left_comm] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum this : \u2191(-\u2191b + (\u2191a + (-\u2191b + 1))) = \u2191(\u2191a + 1 + (-\u2191b + -\u2191b)) \u22a2 _root_.bit1 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit0 b) ** simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 \u2191(-\u2191b + (\u2191a + (-\u2191b + 1))) = \u2191(\u2191a + 1 + (-\u2191b + -\u2191b)) ** simp [add_comm, add_left_comm] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 \u2191(sub' (bit1 a) (bit1 b)) = \u2191(bit1 a) - \u2191(bit1 b) ** rw [sub', ZNum.cast_bit0, cast_sub' a b] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit1 b) ** have : ((-b + (a + -b) : \u2124) : \u03b1) = a + (-b + -b) := by simp [add_left_comm] ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum this : \u2191(-\u2191b + (\u2191a + -\u2191b)) = \u2191a + (-\u2191b + -\u2191b) \u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit1 b) ** simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 a b : PosNum \u22a2 \u2191(-\u2191b + (\u2191a + -\u2191b)) = \u2191a + (-\u2191b + -\u2191b) ** simp [add_left_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.map_eq ** f : Char \u2192 Char s : String \u22a2 map f s = { data := List.map f s.data } ** simpa using mapAux_of_valid f [] s.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "MulAction.orbitZpowersEquiv_symm_apply' ** n : \u2115 A : Type u_1 R : Type u_2 inst\u271d\u00b3 : AddGroup A inst\u271d\u00b2 : Ring R \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d\u00b9 : Group \u03b1 a : \u03b1 inst\u271d : MulAction \u03b1 \u03b2 b : \u03b2 k : \u2124 \u22a2 \u2191(orbitZpowersEquiv a b).symm \u2191k =     { val := a, property := (_ : a \u2208 zpowers a) } ^ k \u2022       { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) } ** rw [orbitZpowersEquiv_symm_apply, ZMod.coe_int_cast] ** n : \u2115 A : Type u_1 R : Type u_2 inst\u271d\u00b3 : AddGroup A inst\u271d\u00b2 : Ring R \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d\u00b9 : Group \u03b1 a : \u03b1 inst\u271d : MulAction \u03b1 \u03b2 b : \u03b2 k : \u2124 \u22a2 { val := a, property := (_ : a \u2208 zpowers a) } ^ (k % \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b)) \u2022       { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) } =     { val := a, property := (_ : a \u2208 zpowers a) } ^ k \u2022       { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) } ** exact Subtype.ext (zpow_smul_mod_minimalPeriod _ _ k) ** Qed",
        "informal": ""
    },
    {
        "formal": "WithTop.image_coe_Ioo ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ioo a b = Ioo \u2191a \u2191b ** rw [\u2190 preimage_coe_Ioo, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Subset.trans Ioo_subset_Iio_self <| Iio_subset_Iio le_top)] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.diff_erase ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u2081 l\u2082 : List \u03b1 a : \u03b1 \u22a2 List.erase (List.diff l\u2081 l\u2082) a = List.diff (List.erase l\u2081 a) l\u2082 ** rw [\u2190 diff_cons_right, diff_cons] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_prod_of_measurable ** \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 \u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e), Measurable f \u2192 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc ** have m := @measurable_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 m :   \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},     Measurable (Prod.mk x) \u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e), Measurable f \u2192 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc ** refine' Measurable.ennreal_induction\n  (P := fun f => \u222b\u207b z, f z \u2202\u03bc.prod \u03bd = \u222b\u207b x, \u222b\u207b y, f (x, y) \u2202\u03bd \u2202\u03bc) _ _ _ ** case refine'_1 \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 m :   \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},     Measurable (Prod.mk x) \u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set (\u03b1 \u00d7 \u03b2)\u2984,     MeasurableSet s \u2192       (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (indicator s fun x => c) ** intro c s hs ** case refine'_1 \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 m :   \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},     Measurable (Prod.mk x) c : \u211d\u22650\u221e s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =     \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc ** conv_rhs =>\n  enter [2, x, 2, y]\n  rw [\u2190 indicator_comp_right, const_def, const_comp, \u2190 const_def] ** case refine'_1 \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 m :   \u2200 {\u03b1 : Type ?u.166340} {\u03b2 : Type ?u.166341} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},     Measurable (Prod.mk x) c : \u211d\u22650\u221e s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =     \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator (Prod.mk x \u207b\u00b9' s) (fun x => c) y \u2202\u03bd \u2202\u03bc ** conv_rhs =>\n  enter [2, x]\n  rw [lintegral_indicator _ (m (x := x) hs), lintegral_const,\n    Measure.restrict_apply MeasurableSet.univ, univ_inter] ** case refine'_1 \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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) c : \u211d\u22650\u221e s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), c * \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc ** simp [hs, lintegral_const_mul, measurable_measure_prod_mk_left (\u03bd := \u03bd) hs, prod_apply] ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) \u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,     Disjoint (support f) (support g) \u2192       Measurable f \u2192         Measurable g \u2192           (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) f \u2192             (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) g \u2192               (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f + g) ** rintro f g - hf _ h2f h2g ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) f g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable f a\u271d : Measurable g h2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc h2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc ** simp only [Pi.add_apply] ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) f g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable f a\u271d : Measurable g h2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc h2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z + g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc ** conv_lhs => rw [lintegral_add_left hf] ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) f g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable f a\u271d : Measurable g h2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc h2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1 \u00d7 \u03b2), f a \u2202Measure.prod \u03bc \u03bd + \u222b\u207b (a : \u03b1 \u00d7 \u03b2), g a \u2202Measure.prod \u03bc \u03bd =     \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc ** conv_rhs => enter [2, x]; erw [lintegral_add_left (hf.comp (m (x := x)))] ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) f g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable f a\u271d : Measurable g h2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc h2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1 \u00d7 \u03b2), f a \u2202Measure.prod \u03bc \u03bd + \u222b\u207b (a : \u03b1 \u00d7 \u03b2), g a \u2202Measure.prod \u03bc \u03bd =     \u222b\u207b (x : \u03b1), \u222b\u207b (a : \u03b2), (f \u2218 Prod.mk x) a \u2202\u03bd + \u222b\u207b (a : \u03b2), g (x, a) \u2202\u03bd \u2202\u03bc ** simp [lintegral_add_left, Measurable.lintegral_prod_right', hf, h2f, h2g] ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) \u22a2 \u2200 \u2983f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,     (\u2200 (n : \u2115), Measurable (f n)) \u2192       Monotone f \u2192         (\u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)) \u2192           (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) fun x => \u2a06 n, f n x ** intro f hf h2f h3f ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h2f : Monotone f h3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n) \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc ** have kf : \u2200 x n, Measurable fun y => f n (x, y) := fun x n => (hf n).comp m ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h2f : Monotone f h3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n) kf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y) \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc ** have k2f : \u2200 x, Monotone fun n y => f n (x, y) := fun x i j hij y => h2f hij (x, y) ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h2f : Monotone f h3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n) kf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y) k2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y) \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc ** have lf : \u2200 n, Measurable fun x => \u222b\u207b y, f n (x, y) \u2202\u03bd := fun n => (hf n).lintegral_prod_right' ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h2f : Monotone f h3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n) kf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y) k2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y) lf : \u2200 (n : \u2115), Measurable fun x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc ** have l2f : Monotone fun n x => \u222b\u207b y, f n (x, y) \u2202\u03bd := fun i j hij x =>\n  lintegral_mono (k2f x hij) ** case refine'_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\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 m : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x) f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h2f : Monotone f h3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n) kf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y) k2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y) lf : \u2200 (n : \u2115), Measurable fun x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd l2f : Monotone fun n x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 n, f n x) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 n, f n x) (x, y) \u2202\u03bd \u2202\u03bc ** simp only [lintegral_iSup hf h2f, lintegral_iSup (kf _), k2f, lintegral_iSup lf l2f, h3f] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.boundedBy_eq ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 m_empty : m \u2205 = 0 m_mono : \u2200 \u2983t : Set \u03b1\u2984, s \u2286 t \u2192 m s \u2264 m t m_subadd : \u2200 (s : \u2115 \u2192 Set \u03b1), m (\u22c3 i, s i) \u2264 \u2211' (i : \u2115), m (s i) \u22a2 \u2191(boundedBy m) s = m s ** rw [boundedBy_eq_ofFunction m_empty, ofFunction_eq s m_mono m_subadd] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.AssocList.find?_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : BEq \u03b1 a : \u03b1 l : AssocList \u03b1 \u03b2 \u22a2 find? a l = Option.map (fun x => x.snd) (List.find? (fun x => x.fst == a) (toList l)) ** simp [find?_eq_findEntry?] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FinMeasAdditive.map_empty_eq_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\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 \u03b2\u271d : Type u_7 inst\u271d\u00b9 : AddCommMonoid \u03b2\u271d T\u271d T' : Set \u03b1 \u2192 \u03b2\u271d \u03b2 : Type u_8 inst\u271d : AddCancelMonoid \u03b2 T : Set \u03b1 \u2192 \u03b2 hT : FinMeasAdditive \u03bc T \u22a2 T \u2205 = 0 ** have h_empty : \u03bc \u2205 \u2260 \u221e := (measure_empty.le.trans_lt ENNReal.coe_lt_top).ne ** \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 \u03b2\u271d : Type u_7 inst\u271d\u00b9 : AddCommMonoid \u03b2\u271d T\u271d T' : Set \u03b1 \u2192 \u03b2\u271d \u03b2 : Type u_8 inst\u271d : AddCancelMonoid \u03b2 T : Set \u03b1 \u2192 \u03b2 hT : FinMeasAdditive \u03bc T h_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4 \u22a2 T \u2205 = 0 ** specialize hT \u2205 \u2205 MeasurableSet.empty MeasurableSet.empty h_empty h_empty (Set.inter_empty \u2205) ** \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 \u03b2\u271d : Type u_7 inst\u271d\u00b9 : AddCommMonoid \u03b2\u271d T\u271d T' : Set \u03b1 \u2192 \u03b2\u271d \u03b2 : Type u_8 inst\u271d : AddCancelMonoid \u03b2 T : Set \u03b1 \u2192 \u03b2 h_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4 hT : T (\u2205 \u222a \u2205) = T \u2205 + T \u2205 \u22a2 T \u2205 = 0 ** rw [Set.union_empty] at 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\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 \u03b2\u271d : Type u_7 inst\u271d\u00b9 : AddCommMonoid \u03b2\u271d T\u271d T' : Set \u03b1 \u2192 \u03b2\u271d \u03b2 : Type u_8 inst\u271d : AddCancelMonoid \u03b2 T : Set \u03b1 \u2192 \u03b2 h_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4 hT : T \u2205 = T \u2205 + T \u2205 \u22a2 T \u2205 = 0 ** nth_rw 1 [\u2190 add_zero (T \u2205)] at 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\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 \u03b2\u271d : Type u_7 inst\u271d\u00b9 : AddCommMonoid \u03b2\u271d T\u271d T' : Set \u03b1 \u2192 \u03b2\u271d \u03b2 : Type u_8 inst\u271d : AddCancelMonoid \u03b2 T : Set \u03b1 \u2192 \u03b2 h_empty : \u2191\u2191\u03bc \u2205 \u2260 \u22a4 hT : T \u2205 + 0 = T \u2205 + T \u2205 \u22a2 T \u2205 = 0 ** exact (add_left_cancel hT).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "AntilipschitzWith.hausdorffMeasure_preimage_le ** \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y \u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s ** rcases eq_or_ne K 0 with (rfl | h0) ** case inr \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 \u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s ** have hKd0 : (K : \u211d\u22650\u221e) ^ d \u2260 0 := by simp [h0] ** case inr \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 \u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s ** have hKd : (K : \u211d\u22650\u221e) ^ d \u2260 \u221e := by simp [hd] ** case inr \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s ** simp only [hausdorffMeasure_apply, ENNReal.mul_iSup, ENNReal.mul_iInf_of_ne hKd0 hKd,\n  \u2190 ENNReal.tsum_mul_left] ** case inr \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u22a2 \u2a06 r,       \u2a06 (_ : 0 < r),         \u2a05 t,           \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),             \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264     \u2a06 i,       \u2a06 (_ : 0 < i),         \u2a05 i_1,           \u2a05 (_ : s \u2286 \u22c3 n, i_1 n),             \u2a05 (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i), \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191K ^ d * diam (i_1 i) ^ d ** refine' iSup\u2082_le fun \u03b5 \u03b50 => _ ** case inr \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 \u22a2 \u2a05 t,       \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),         \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264     \u2a06 i,       \u2a06 (_ : 0 < i),         \u2a05 i_1,           \u2a05 (_ : s \u2286 \u22c3 n, i_1 n),             \u2a05 (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i), \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191K ^ d * diam (i_1 i) ^ d ** refine' le_iSup\u2082_of_le (\u03b5 / K) (by simp [\u03b50.ne']) _ ** case inr \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 \u22a2 \u2a05 t,       \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),         \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264     \u2a05 i,       \u2a05 (_ : s \u2286 \u22c3 n, i n),         \u2a05 (_ : \u2200 (n : \u2115), diam (i n) \u2264 \u03b5 / \u2191K), \u2211' (i_1 : \u2115), \u2a06 (_ : Set.Nonempty (i i_1)), \u2191K ^ d * diam (i i_1) ^ d ** refine' le_iInf\u2082 fun t hst => le_iInf fun ht\u03b5 => _ ** case inr \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 t : \u2115 \u2192 Set Y hst : s \u2286 \u22c3 n, t n ht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K \u22a2 \u2a05 t,       \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),         \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264     \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d ** replace hst : f \u207b\u00b9' s \u2286 _ := preimage_mono hst ** case inr \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 t : \u2115 \u2192 Set Y ht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K hst : f \u207b\u00b9' s \u2286 f \u207b\u00b9' \u22c3 n, t n \u22a2 \u2a05 t,       \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),         \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264     \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d ** rw [preimage_iUnion] at hst ** case inr \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 t : \u2115 \u2192 Set Y ht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K hst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i \u22a2 \u2a05 t,       \u2a05 (_ : f \u207b\u00b9' s \u2286 \u22c3 n, t n),         \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264     \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d ** refine' iInf\u2082_le_of_le _ hst (iInf_le_of_le (fun n => _) _) ** case inl \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f \u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s ** rcases eq_empty_or_nonempty (f \u207b\u00b9' s) with (hs | \u27e8x, hx\u27e9) ** case inl.inr.intro \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s \u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s ** have : f \u207b\u00b9' s = {x} := by\n  haveI : Subsingleton X := hf.subsingleton\n  have : (f \u207b\u00b9' s).Subsingleton := subsingleton_univ.anti (subset_univ _)\n  exact (subsingleton_iff_singleton hx).1 this ** case inl.inr.intro \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s this : f \u207b\u00b9' s = {x} \u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s ** rw [this] ** case inl.inr.intro \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s this : f \u207b\u00b9' s = {x} \u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s ** rcases eq_or_lt_of_le hd with (rfl | h'd) ** case inl.inl \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f hs : f \u207b\u00b9' s = \u2205 \u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s ** simp only [hs, measure_empty, zero_le] ** \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s \u22a2 f \u207b\u00b9' s = {x} ** haveI : Subsingleton X := hf.subsingleton ** \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s this : Subsingleton X \u22a2 f \u207b\u00b9' s = {x} ** have : (f \u207b\u00b9' s).Subsingleton := subsingleton_univ.anti (subset_univ _) ** \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s this\u271d : Subsingleton X this : Set.Subsingleton (f \u207b\u00b9' s) \u22a2 f \u207b\u00b9' s = {x} ** exact (subsingleton_iff_singleton hx).1 this ** case inl.inr.intro.inl \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 f : X \u2192 Y s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s this : f \u207b\u00b9' s = {x} hd : 0 \u2264 0 \u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 \u21910 ^ 0 * \u2191\u2191\u03bcH[0] s ** simp only [ENNReal.rpow_zero, one_mul, mul_zero] ** case inl.inr.intro.inl \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 f : X \u2192 Y s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s this : f \u207b\u00b9' s = {x} hd : 0 \u2264 0 \u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 \u2191\u2191\u03bcH[0] s ** rw [hausdorffMeasure_zero_singleton] ** case inl.inr.intro.inl \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 f : X \u2192 Y s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s this : f \u207b\u00b9' s = {x} hd : 0 \u2264 0 \u22a2 1 \u2264 \u2191\u2191\u03bcH[0] s ** exact one_le_hausdorffMeasure_zero_of_nonempty \u27e8f x, hx\u27e9 ** case inl.inr.intro.inr \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s this : f \u207b\u00b9' s = {x} h'd : 0 < d \u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s ** haveI := noAtoms_hausdorff X h'd ** case inl.inr.intro.inr \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 f : X \u2192 Y d : \u211d hd : 0 \u2264 d s : Set Y hf : AntilipschitzWith 0 f x : X hx : x \u2208 f \u207b\u00b9' s this\u271d : f \u207b\u00b9' s = {x} h'd : 0 < d this : NoAtoms \u03bcH[d] \u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s ** simp only [zero_le, measure_singleton] ** \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 \u22a2 \u2191K ^ d \u2260 0 ** simp [h0] ** \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 \u22a2 \u2191K ^ d \u2260 \u22a4 ** simp [hd] ** \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 \u22a2 0 < \u03b5 / \u2191K ** simp [\u03b50.ne'] ** case inr.refine'_1 \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 t : \u2115 \u2192 Set Y ht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K hst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i n : \u2115 \u22a2 diam (f \u207b\u00b9' t n) \u2264 \u03b5 ** exact (hf.ediam_preimage_le _).trans (ENNReal.mul_le_of_le_div' <| ht\u03b5 n) ** case inr.refine'_2 \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 t : \u2115 \u2192 Set Y ht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K hst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i \u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (f \u207b\u00b9' t n)), diam (f \u207b\u00b9' t n) ^ d \u2264     \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d ** refine' ENNReal.tsum_le_tsum fun n => iSup_le_iff.2 fun hft => _ ** case inr.refine'_2 \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 t : \u2115 \u2192 Set Y ht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K hst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i n : \u2115 hft : Set.Nonempty (f \u207b\u00b9' t n) \u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 \u2a06 (_ : Set.Nonempty (t n)), \u2191K ^ d * diam (t n) ^ d ** simp only [nonempty_of_nonempty_preimage hft, ciSup_pos] ** case inr.refine'_2 \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 t : \u2115 \u2192 Set Y ht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K hst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i n : \u2115 hft : Set.Nonempty (f \u207b\u00b9' t n) \u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 \u2191K ^ d * diam (t n) ^ d ** rw [\u2190 ENNReal.mul_rpow_of_nonneg _ _ hd] ** case inr.refine'_2 \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 f : X \u2192 Y K : \u211d\u22650 d : \u211d hf : AntilipschitzWith K f hd : 0 \u2264 d s : Set Y h0 : K \u2260 0 hKd0 : \u2191K ^ d \u2260 0 hKd : \u2191K ^ d \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : 0 < \u03b5 t : \u2115 \u2192 Set Y ht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K hst : f \u207b\u00b9' s \u2286 \u22c3 i, f \u207b\u00b9' t i n : \u2115 hft : Set.Nonempty (f \u207b\u00b9' t n) \u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 (\u2191K * diam (t n)) ^ d ** exact ENNReal.rpow_le_rpow (hf.ediam_preimage_le _) hd ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.Tape.map_mk\u2082 ** \u0393 : Type u_1 \u0393' : Type u_2 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Inhabited \u0393' f : PointedMap \u0393 \u0393' L R : List \u0393 \u22a2 map f (mk\u2082 L R) = mk\u2082 (List.map f.f L) (List.map f.f R) ** simp only [Tape.mk\u2082, Tape.map_mk', ListBlank.map_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.cardinal_generateMeasurableRec_le ** \u03b1 : Type u s : Set (Set \u03b1) i : (Quotient.out (ord (aleph 1))).\u03b1 \u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** apply (aleph 1).ord.out.wo.wf.induction i ** \u03b1 : Type u s : Set (Set \u03b1) i : (Quotient.out (ord (aleph 1))).\u03b1 \u22a2 \u2200 (x : (Quotient.out (ord (aleph 1))).\u03b1),     (\u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),         WellOrder.r (Quotient.out (ord (aleph 1))) y x \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080) \u2192       #\u2191(generateMeasurableRec s x) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** intro i IH ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** have A := aleph0_le_aleph 1 ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 \u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** have B : aleph 1 \u2264 max #s 2 ^ aleph0.{u} :=\n  aleph_one_le_continuum.trans (power_le_power_right (le_max_right _ _)) ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** have C : \u2135\u2080 \u2264 max #s 2 ^ aleph0.{u} := A.trans B ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** have J : #(\u22c3 j : Iio i, generateMeasurableRec s j.1) \u2264 max #s 2 ^ aleph0.{u} := by\n  refine (mk_iUnion_le _).trans ?_\n  have D : \u2a06 j : Iio i, #(generateMeasurableRec s j) \u2264 _ := ciSup_le' fun \u27e8j, hj\u27e9 => IH j hj\n  apply (mul_le_mul' ((mk_subtype_le _).trans (aleph 1).mk_ord_out.le) D).trans\n  rw [mul_eq_max A C]\n  exact max_le B le_rfl ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 J : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191(generateMeasurableRec s i) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** rw [generateMeasurableRec] ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 J : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191((s \u222a {\u2205} \u222a compl '' \u22c3 j, generateMeasurableRec s \u2191j) \u222a range fun f => \u22c3 n, \u2191(f n)) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** apply_rules [(mk_union_le _ _).trans, add_le_of_le C, mk_image_le.trans] ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** refine (mk_iUnion_le _).trans ?_ ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191(Iio i) * \u2a06 i_1, #\u2191(generateMeasurableRec s \u2191i_1) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** have D : \u2a06 j : Iio i, #(generateMeasurableRec s j) \u2264 _ := ciSup_le' fun \u27e8j, hj\u27e9 => IH j hj ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 D : \u2a06 j, #\u2191(generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191(Iio i) * \u2a06 i_1, #\u2191(generateMeasurableRec s \u2191i_1) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** apply (mul_le_mul' ((mk_subtype_le _).trans (aleph 1).mk_ord_out.le) D).trans ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 D : \u2a06 j, #\u2191(generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 aleph 1 * max (#\u2191s) 2 ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** rw [mul_eq_max A C] ** \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 D : \u2a06 j, #\u2191(generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 max (aleph 1) (max (#\u2191s) 2 ^ \u2135\u2080) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** exact max_le B le_rfl ** case h1.h1.h1 \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 J : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191s \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** exact (le_max_left _ _).trans (self_le_power _ one_lt_aleph0.le) ** case h1.h1.h2 \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 J : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191{\u2205} \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** rw [mk_singleton] ** case h1.h1.h2 \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 J : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** exact one_lt_aleph0.le.trans C ** case h2 \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 J : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191(range fun f => \u22c3 n, \u2191(f n)) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** apply mk_range_le.trans ** case h2 \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 J : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #(\u2115 \u2192 \u2191(\u22c3 j, generateMeasurableRec s \u2191j)) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** simp only [mk_pi, prod_const, lift_uzero, mk_denumerable, lift_aleph0] ** case h2 \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 J : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 \u22a2 #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** have := @power_le_power_right _ _ \u2135\u2080 J ** case h2 \u03b1 : Type u s : Set (Set \u03b1) i\u271d i : (Quotient.out (ord (aleph 1))).\u03b1 IH :   \u2200 (y : (Quotient.out (ord (aleph 1))).\u03b1),     WellOrder.r (Quotient.out (ord (aleph 1))) y i \u2192 #\u2191(generateMeasurableRec s y) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 A : \u2135\u2080 \u2264 aleph 1 B : aleph 1 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 C : \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 J : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) \u2264 max (#\u2191s) 2 ^ \u2135\u2080 this : #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) ^ \u2135\u2080 \u2264 (max (#\u2191s) 2 ^ \u2135\u2080) ^ \u2135\u2080 \u22a2 #\u2191(\u22c3 j, generateMeasurableRec s \u2191j) ^ \u2135\u2080 \u2264 max (#\u2191s) 2 ^ \u2135\u2080 ** rwa [\u2190 power_mul, aleph0_mul_aleph0] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Content.innerContent_iSup_nat ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) \u22a2 innerContent \u03bc (\u2a06 i, U i) \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i) ** refine' iSup\u2082_le fun K hK => _ ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) \u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i) ** obtain \u27e8t, ht\u27e9 :=\n  K.isCompact.elim_finite_subcover _ (fun i => (U i).isOpen) (by rwa [\u2190 Opens.coe_iSup]) ** case intro G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) \u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i) ** rcases K.isCompact.finite_compact_cover t (SetLike.coe \u2218 U) (fun i _ => (U i).isOpen) ht with\n  \u27e8K', h1K', h2K', h3K'\u27e9 ** case intro.intro.intro.intro G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i \u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i) ** let L : \u2115 \u2192 Compacts G := fun n => \u27e8K' n, h1K' n\u27e9 ** case intro.intro.intro.intro G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i L : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) } \u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i) ** convert le_trans (h3 t L) _ ** case intro.intro.intro.intro.convert_2 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i L : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) } \u22a2 (Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (L i)) \u2264 \u2211' (i : \u2115), innerContent \u03bc (U i) ** refine' le_trans (Finset.sum_le_sum _) (ENNReal.sum_le_tsum t) ** case intro.intro.intro.intro.convert_2 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i L : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) } \u22a2 \u2200 (i : \u2115), i \u2208 t \u2192 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 innerContent \u03bc (U i) ** intro i _ ** case intro.intro.intro.intro.convert_2 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i L : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) } i : \u2115 a\u271d : i \u2208 t \u22a2 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 innerContent \u03bc (U i) ** refine' le_trans _ (le_iSup _ (L i)) ** case intro.intro.intro.intro.convert_2 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i L : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) } i : \u2115 a\u271d : i \u2208 t \u22a2 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 \u2a06 (_ : \u2191(L i) \u2286 \u2191(U i)), (fun s => \u2191(toFun \u03bc s)) (L i) ** refine' le_trans _ (le_iSup _ (h2K' i)) ** case intro.intro.intro.intro.convert_2 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i L : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) } i : \u2115 a\u271d : i \u2208 t \u22a2 (fun s => \u2191(toFun \u03bc s)) (L i) \u2264 (fun s => \u2191(toFun \u03bc s)) (L i) ** rfl ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G \u22a2 \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) ** intro t K ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G t : Finset \u2115 K : \u2115 \u2192 Compacts G \u22a2 (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) ** refine' Finset.induction_on t _ _ ** case refine'_1 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G t : Finset \u2115 K : \u2115 \u2192 Compacts G \u22a2 (fun s => \u2191(toFun \u03bc s)) (Finset.sup \u2205 K) \u2264 Finset.sum \u2205 fun i => (fun s => \u2191(toFun \u03bc s)) (K i) ** simp only [\u03bc.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty] ** case refine'_2 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G t : Finset \u2115 K : \u2115 \u2192 Compacts G \u22a2 \u2200 \u2983a : \u2115\u2984 {s : Finset \u2115},     \u00aca \u2208 s \u2192       ((fun s => \u2191(toFun \u03bc s)) (Finset.sup s K) \u2264 Finset.sum s fun i => (fun s => \u2191(toFun \u03bc s)) (K i)) \u2192         (fun s => \u2191(toFun \u03bc s)) (Finset.sup (insert a s) K) \u2264           Finset.sum (insert a s) fun i => (fun s => \u2191(toFun \u03bc s)) (K i) ** intro n s hn ih ** case refine'_2 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G t : Finset \u2115 K : \u2115 \u2192 Compacts G n : \u2115 s : Finset \u2115 hn : \u00acn \u2208 s ih : (fun s => \u2191(toFun \u03bc s)) (Finset.sup s K) \u2264 Finset.sum s fun i => (fun s => \u2191(toFun \u03bc s)) (K i) \u22a2 (fun s => \u2191(toFun \u03bc s)) (Finset.sup (insert n s) K) \u2264 Finset.sum (insert n s) fun i => (fun s => \u2191(toFun \u03bc s)) (K i) ** rw [Finset.sup_insert, Finset.sum_insert hn] ** case refine'_2 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G t : Finset \u2115 K : \u2115 \u2192 Compacts G n : \u2115 s : Finset \u2115 hn : \u00acn \u2208 s ih : (fun s => \u2191(toFun \u03bc s)) (Finset.sup s K) \u2264 Finset.sum s fun i => (fun s => \u2191(toFun \u03bc s)) (K i) \u22a2 (fun s => \u2191(toFun \u03bc s)) (K n \u2294 Finset.sup s K) \u2264     (fun s => \u2191(toFun \u03bc s)) (K n) + Finset.sum s fun x => (fun s => \u2191(toFun \u03bc s)) (K x) ** exact le_trans (\u03bc.sup_le _ _) (add_le_add_left ih _) ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) \u22a2 \u2191K \u2286 \u22c3 i, \u2191(U i) ** rwa [\u2190 Opens.coe_iSup] ** case h.e'_3.h.e'_1 G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i L : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) } \u22a2 K = Finset.sup t L ** ext1 ** case h.e'_3.h.e'_1.h G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i L : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) } \u22a2 \u2191K = \u2191(Finset.sup t L) ** rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup] ** case h.e'_3.h.e'_1.h G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G U : \u2115 \u2192 Opens G h3 :   \u2200 (t : Finset \u2115) (K : \u2115 \u2192 Compacts G),     (fun s => \u2191(toFun \u03bc s)) (Finset.sup t K) \u2264 Finset.sum t fun i => (fun s => \u2191(toFun \u03bc s)) (K i) K : Compacts G hK : \u2191K \u2286 \u2191(\u2a06 i, U i) t : Finset \u2115 ht : \u2191K \u2286 \u22c3 i \u2208 t, \u2191(U i) K' : \u2115 \u2192 Set G h1K' : \u2200 (i : \u2115), IsCompact (K' i) h2K' : \u2200 (i : \u2115), K' i \u2286 (SetLike.coe \u2218 U) i h3K' : \u2191K = \u22c3 i \u2208 t, K' i L : \u2115 \u2192 Compacts G := fun n => { carrier := K' n, isCompact' := (_ : IsCompact (K' n)) } \u22a2 \u2191K = \u2a06 a \u2208 t, \u2191(L a) ** exact h3K' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.toJordanDecomposition_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 r : \u211d\u22650 \u22a2 toJordanDecomposition (r \u2022 s) = r \u2022 toJordanDecomposition s ** apply toSignedMeasure_injective ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 r : \u211d\u22650 \u22a2 toSignedMeasure (toJordanDecomposition (r \u2022 s)) = toSignedMeasure (r \u2022 toJordanDecomposition s) ** simp [toSignedMeasure_smul] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d c : E \u22a2 \u222b (x : \u211d) in a..b, c = (b - a) \u2022 c ** simp only [integral_const', Real.volume_Ioc, ENNReal.toReal_ofReal', \u2190 neg_sub b,\n  max_zero_sub_eq_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_coe_eq_coe_finsetCard ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 s : Finset \u03b1 \u22a2 encard \u2191s = \u2191(Finset.card s) ** rw [Finite.encard_eq_coe_toFinset_card (Finset.finite_toSet s)] ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 s : Finset \u03b1 \u22a2 \u2191(Finset.card (Finite.toFinset (_ : Set.Finite \u2191s))) = \u2191(Finset.card s) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snormEssSup_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 F \u22a2 snormEssSup (f + g) \u03bc \u2264 snormEssSup f \u03bc + snormEssSup g \u03bc ** refine' le_trans (essSup_mono_ae (eventually_of_forall fun x => _)) (ENNReal.essSup_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 F x : \u03b1 \u22a2 (fun x => \u2191\u2016(f + g) x\u2016\u208a) x \u2264 ((fun x => \u2191\u2016f x\u2016\u208a) + fun x => \u2191\u2016g x\u2016\u208a) x ** simp_rw [Pi.add_apply, \u2190 ENNReal.coe_add, 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 : \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 F x : \u03b1 \u22a2 \u2016f x + g x\u2016\u208a \u2264 \u2016f x\u2016\u208a + \u2016g x\u2016\u208a ** exact nnnorm_add_le _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.range_list_get? ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : List \u03b1 \u22a2 range (get? l) = insert none (some '' {x | x \u2208 l}) ** rw [\u2190 range_list_nthLe, \u2190 range_comp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : List \u03b1 \u22a2 range (get? l) = insert none (range (some \u2218 fun k => nthLe l \u2191k (_ : \u2191k < length l))) ** refine' (range_subset_iff.2 fun n => _).antisymm (insert_subset_iff.2 \u27e8_, _\u27e9) ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 l : List \u03b1 n : \u2115 \u22a2 get? l n \u2208 insert none (range (some \u2218 fun k => nthLe l \u2191k (_ : \u2191k < length l)))  case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 l : List \u03b1 \u22a2 none \u2208 range (get? l)  case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 l : List \u03b1 \u22a2 range (some \u2218 fun k => nthLe l \u2191k (_ : \u2191k < length l)) \u2286 range (get? l) ** exacts [(le_or_lt l.length n).imp get?_eq_none.2 (fun hlt => \u27e8\u27e8_, hlt\u27e9, (get?_eq_get hlt).symm\u27e9),\n  \u27e8_, get?_eq_none.2 le_rfl\u27e9, range_subset_iff.2 <| fun k => \u27e8_, get?_eq_get _\u27e9] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_closedBall ** 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 x : E r : \u211d hr : 0 \u2264 r \u22a2 \u2191\u2191\u03bc (closedBall x r) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 1) ** rw [addHaar_closedBall' \u03bc x hr, addHaar_closed_unit_ball_eq_addHaar_unit_ball] ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.castNum_shiftLeft ** \u03b1 : Type u_1 m : Num n : \u2115 \u22a2 \u2191(m <<< n) = \u2191m <<< n ** cases m <;> dsimp only [\u2190shiftl_eq_shiftLeft, shiftl] ** case pos \u03b1 : Type u_1 n : \u2115 a\u271d : PosNum \u22a2 \u2191(pos (a\u271d <<< n)) = \u2191(pos a\u271d) <<< n ** simp only [cast_pos] ** case pos \u03b1 : Type u_1 n : \u2115 a\u271d : PosNum \u22a2 \u2191(a\u271d <<< n) = \u2191a\u271d <<< n ** induction' n with n IH ** case pos.succ \u03b1 : Type u_1 a\u271d : PosNum n : \u2115 IH : \u2191(a\u271d <<< n) = \u2191a\u271d <<< n \u22a2 \u2191(a\u271d <<< Nat.succ n) = \u2191a\u271d <<< Nat.succ n ** simp [PosNum.shiftl_succ_eq_bit0_shiftl, Nat.shiftLeft_succ, IH,\n      Nat.bit0_val, pow_succ, \u2190 mul_assoc, mul_comm,\n      -shiftl_eq_shiftLeft, -PosNum.shiftl_eq_shiftLeft, shiftl] ** 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_shiftLeft ** case pos.zero \u03b1 : Type u_1 a\u271d : PosNum \u22a2 \u2191(a\u271d <<< Nat.zero) = \u2191a\u271d <<< Nat.zero ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image_add_right_Icc ** \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) (Icc a b) = Icc (a + c) (b + c) ** rw [\u2190 map_add_right_Icc, map_eq_image, addRightEmbedding, Embedding.coeFn_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.countP_eq_length ** \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool l : List \u03b1 \u22a2 countP p l = length l \u2194 \u2200 (a : \u03b1), a \u2208 l \u2192 p a = true ** rw [countP_eq_length_filter, filter_length_eq_length] ** Qed",
        "informal": ""
    },
    {
        "formal": "PosNum.to_int_eq_succ_pred ** \u03b1 : Type u_1 n : PosNum \u22a2 \u2191n = \u2191\u2191(pred' n) + 1 ** rw [\u2190 n.to_nat_to_int, to_nat_eq_succ_pred] ** \u03b1 : Type u_1 n : PosNum \u22a2 \u2191(\u2191(pred' n) + 1) = \u2191\u2191(pred' n) + 1 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.val_neg_of_ne_zero ** n : \u2115 nz : NeZero n a : ZMod n na : NeZero a \u22a2 val (-a) = n - val a ** simp_all [neg_val a, na.out] ** Qed",
        "informal": ""
    },
    {
        "formal": "Measurable.set_lintegral_kernel ** \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 : \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b2 hs : MeasurableSet s \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2) in s, f b \u2202\u2191\u03ba a ** refine Measurable.set_lintegral_kernel_prod_right ?_ hs ** \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 : \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b2 hs : MeasurableSet s \u22a2 Measurable (uncurry fun a b => f b) ** convert (hf.comp measurable_snd) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.disjSups_disjSups_disjSups_comm ** 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 : DistribLattice \u03b1 inst\u271d\u00b9 : OrderBot \u03b1 inst\u271d : DecidableRel Disjoint s t u v : Finset \u03b1 \u22a2 s \u25cb t \u25cb (u \u25cb v) = s \u25cb u \u25cb (t \u25cb v) ** simp_rw [\u2190 disjSups_assoc, disjSups_right_comm] ** 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' f : \u03b1 \u2192 E \u22a2 setToFun \u03bc (T + T') (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) 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' f : \u03b1 \u2192 E hf : Integrable f \u22a2 setToFun \u03bc (T + T') (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f ** simp_rw [setToFun_eq _ hf, L1.setToL1_add_left hT 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\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 setToFun \u03bc (T + T') (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f ** simp_rw [setToFun_undef _ hf, add_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_comp_smul_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' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntegrableOn g (f '' [[a, b]]) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** rw [hf.image_uIcc, \u2190 intervalIntegrable_iff'] at hg1 ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** have h_der :\n  \u2200 x \u2208 Ioo (min a b) (max a b),\n    HasDerivWithinAt (fun u => \u222b t in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x := by\n  intro x hx\n  obtain \u27e8c, hc\u27e9 := nonempty_Ioo.mpr hx.1\n  obtain \u27e8d, hd\u27e9 := nonempty_Ioo.mpr hx.2\n  have cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) := by\n    rw [uIcc_of_le (hc.2.trans hd.1).le]\n    exact Icc_subset_Ioo hc.1 hd.2\n  replace hg_cont := hg_cont.mono (image_subset f cdsub)\n  let J := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]\n  have hJ : f '' [[c, d]] = J := (hf.mono (cdsub.trans Ioo_subset_Icc_self)).image_uIcc\n  rw [hJ] at hg_cont\n  have h2x : f x \u2208 J := by rw [\u2190 hJ]; exact mem_image_of_mem _ (mem_uIcc_of_le hc.2.le hd.1.le)\n  have h2g : IntervalIntegrable g volume (f a) (f x) := by\n    refine' hg1.mono_set _\n    rw [\u2190 hf.image_uIcc]\n    exact hf.surjOn_uIcc left_mem_uIcc (Ioo_subset_Icc_self hx)\n  have h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) :=\n    hg_cont.stronglyMeasurableAtFilter_nhdsWithin measurableSet_Icc (f x)\n  haveI : Fact (f x \u2208 J) := \u27e8h2x\u27e9\n  have : HasDerivWithinAt (fun u => \u222b x in f a..u, g x) (g (f x)) J (f x) :=\n    intervalIntegral.integral_hasDerivWithinAt_right h2g h3g (hg_cont (f x) h2x)\n  refine' (this.scomp x ((hff' x hx).Ioo_of_Ioi hd.1) _).Ioi_of_Ioo hd.1\n  rw [\u2190 hJ]\n  refine' (mapsTo_image _ _).mono _ Subset.rfl\n  exact Ioo_subset_Icc_self.trans ((Icc_subset_Icc_left hc.2.le).trans Icc_subset_uIcc) ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] h_der :   \u2200 (x : \u211d),     x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** rw [\u2190 intervalIntegrable_iff'] at hg2 ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntervalIntegrable (fun x => f' x \u2022 (g \u2218 f) x) volume a b h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] h_der :   \u2200 (x : \u211d),     x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** simp_rw [integral_eq_sub_of_hasDeriv_right h_cont h_der hg2, integral_same, sub_zero] ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] \u22a2 ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] ** refine' (continuousOn_primitive_interval' hg1 _).comp hf _ ** case refine'_1 \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] \u22a2 f a \u2208 [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] ** rw [\u2190 hf.image_uIcc] ** case refine'_1 \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] \u22a2 f a \u2208 f '' [[a, b]] ** exact mem_image_of_mem f left_mem_uIcc ** case refine'_2 \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] \u22a2 MapsTo (fun u => f u) [[a, b]] [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] ** rw [\u2190 hf.image_uIcc] ** case refine'_2 \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] \u22a2 MapsTo (fun u => f u) [[a, b]] (f '' [[a, b]]) ** exact mapsTo_image _ _ ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] \u22a2 \u2200 (x : \u211d),     x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** intro 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\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** obtain \u27e8c, hc\u27e9 := nonempty_Ioo.mpr hx.1 ** case 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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** obtain \u27e8d, hd\u27e9 := nonempty_Ioo.mpr hx.2 ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** have cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) := by\n  rw [uIcc_of_le (hc.2.trans hd.1).le]\n  exact Icc_subset_Ioo hc.1 hd.2 ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** replace hg_cont := hg_cont.mono (image_subset f cdsub) ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) hg_cont : ContinuousOn g (f '' [[c, d]]) \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** let J := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) hg_cont : ContinuousOn g (f '' [[c, d]]) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** have hJ : f '' [[c, d]] = J := (hf.mono (cdsub.trans Ioo_subset_Icc_self)).image_uIcc ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) hg_cont : ContinuousOn g (f '' [[c, d]]) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hJ : f '' [[c, d]] = J \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** rw [hJ] at hg_cont ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** have h2x : f x \u2208 J := by rw [\u2190 hJ]; exact mem_image_of_mem _ (mem_uIcc_of_le hc.2.le hd.1.le) ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** have h2g : IntervalIntegrable g volume (f a) (f x) := by\n  refine' hg1.mono_set _\n  rw [\u2190 hf.image_uIcc]\n  exact hf.surjOn_uIcc left_mem_uIcc (Ioo_subset_Icc_self hx) ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J h2g : IntervalIntegrable g volume (f a) (f x) \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** have h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) :=\n  hg_cont.stronglyMeasurableAtFilter_nhdsWithin measurableSet_Icc (f x) ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J h2g : IntervalIntegrable g volume (f a) (f x) h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** haveI : Fact (f x \u2208 J) := \u27e8h2x\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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J h2g : IntervalIntegrable g volume (f a) (f x) h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) this : Fact (f x \u2208 J) \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** have : HasDerivWithinAt (fun u => \u222b x in f a..u, g x) (g (f x)) J (f x) :=\n  intervalIntegral.integral_hasDerivWithinAt_right h2g h3g (hg_cont (f x) h2x) ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J h2g : IntervalIntegrable g volume (f a) (f x) h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) this\u271d : Fact (f x \u2208 J) this : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x) \u22a2 HasDerivWithinAt (fun u => \u222b (t : \u211d) in f a..f u, g t) (f' x \u2022 (g \u2218 f) x) (Ioi x) x ** refine' (this.scomp x ((hff' x hx).Ioo_of_Ioi hd.1) _).Ioi_of_Ioo hd.1 ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J h2g : IntervalIntegrable g volume (f a) (f x) h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) this\u271d : Fact (f x \u2208 J) this : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x) \u22a2 MapsTo f (Ioo x d) J ** rw [\u2190 hJ] ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J h2g : IntervalIntegrable g volume (f a) (f x) h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) this\u271d : Fact (f x \u2208 J) this : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x) \u22a2 MapsTo f (Ioo x d) (f '' [[c, d]]) ** refine' (mapsTo_image _ _).mono _ Subset.rfl ** 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\u271d\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J h2g : IntervalIntegrable g volume (f a) (f x) h3g : StronglyMeasurableAtFilter g (\ud835\udcdd[J] f x) this\u271d : Fact (f x \u2208 J) this : HasDerivWithinAt (fun u => \u222b (x : \u211d) in f a..u, g x) (g (f x)) J (f x) \u22a2 Ioo x d \u2286 [[c, d]] ** exact Ioo_subset_Icc_self.trans ((Icc_subset_Icc_left hc.2.le).trans Icc_subset_uIcc) ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) \u22a2 [[c, d]] \u2286 Ioo (min a b) (max a b) ** rw [uIcc_of_le (hc.2.trans hd.1).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\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) \u22a2 Icc c d \u2286 Ioo (min a b) (max a b) ** exact Icc_subset_Ioo hc.1 hd.2 ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J \u22a2 f x \u2208 J ** rw [\u2190 hJ] ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J \u22a2 f x \u2208 f '' [[c, d]] ** exact mem_image_of_mem _ (mem_uIcc_of_le hc.2.le hd.1.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\u00b9 : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J \u22a2 IntervalIntegrable g volume (f a) (f x) ** refine' hg1.mono_set _ ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J \u22a2 [[f a, f x]] \u2286 [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] ** rw [\u2190 hf.image_uIcc] ** \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\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) [[a, b]] h_cont : ContinuousOn (fun u => \u222b (t : \u211d) in f a..f u, g t) [[a, b]] x : \u211d hx : x \u2208 Ioo (min a b) (max a b) c : \u211d hc : c \u2208 Ioo (min a b) x d : \u211d hd : d \u2208 Ioo x (max a b) cdsub : [[c, d]] \u2286 Ioo (min a b) (max a b) J : Set \u211d := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]] hg_cont : ContinuousOn g J hJ : f '' [[c, d]] = J h2x : f x \u2208 J \u22a2 [[f a, f x]] \u2286 f '' [[a, b]] ** exact hf.surjOn_uIcc left_mem_uIcc (Ioo_subset_Icc_self hx) ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.val_coe_unit_coprime ** n : \u2115 u : (ZMod n)\u02e3 \u22a2 Nat.Coprime (val \u2191u) n ** cases' n with n ** case succ n : \u2115 u : (ZMod (Nat.succ n))\u02e3 \u22a2 Nat.Coprime (val \u2191u) (Nat.succ n) ** apply Nat.coprime_of_mul_modEq_one ((u\u207b\u00b9 : Units (ZMod (n + 1))) : ZMod (n + 1)).val ** case succ n : \u2115 u : (ZMod (Nat.succ n))\u02e3 \u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n] ** have := Units.ext_iff.1 (mul_right_inv u) ** case succ n : \u2115 u : (ZMod (Nat.succ n))\u02e3 this : \u2191(u * u\u207b\u00b9) = \u21911 \u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n] ** rw [Units.val_one] at this ** case succ n : \u2115 u : (ZMod (Nat.succ n))\u02e3 this : \u2191(u * u\u207b\u00b9) = 1 \u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n] ** rw [\u2190 eq_iff_modEq_nat, Nat.cast_one, \u2190 this] ** case succ n : \u2115 u : (ZMod (Nat.succ n))\u02e3 this : \u2191(u * u\u207b\u00b9) = 1 \u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(u * u\u207b\u00b9) ** clear this ** case succ n : \u2115 u : (ZMod (Nat.succ n))\u02e3 \u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(u * u\u207b\u00b9) ** rw [\u2190 nat_cast_zmod_val ((u * u\u207b\u00b9 : Units (ZMod (n + 1))) : ZMod (n + 1))] ** case succ n : \u2115 u : (ZMod (Nat.succ n))\u02e3 \u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(val \u2191(u * u\u207b\u00b9)) ** rw [Units.val_mul, val_mul, nat_cast_mod] ** case zero u : (ZMod Nat.zero)\u02e3 \u22a2 Nat.Coprime (val \u2191u) Nat.zero ** rcases Int.units_eq_one_or u with (rfl | rfl) <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Filtration.filtrationOfSet_eq_natural ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 filtrationOfSet hsm =     natural (fun i => Set.indicator (s i) fun x => 1) (_ : \u2200 (i : \u03b9), StronglyMeasurable (Set.indicator (s i) 1)) ** simp only [filtrationOfSet, natural, measurableSpace_iSup_eq, exists_prop, mk.injEq] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 (fun i => MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t}) = fun i =>     MeasurableSpace.generateFrom {s_1 | \u2203 n, MeasurableSet s_1} ** ext1 i ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 \u22a2 MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t} = MeasurableSpace.generateFrom {s_1 | \u2203 n, MeasurableSet s_1} ** refine' le_antisymm (generateFrom_le _) (generateFrom_le _) ** case h.refine'_1 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 \u22a2 \u2200 (t : Set \u03a9), t \u2208 {t | \u2203 j, j \u2264 i \u2227 s j = t} \u2192 MeasurableSet t ** rintro _ \u27e8j, hij, rfl\u27e9 ** case h.refine'_1.intro.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i j : \u03b9 hij : j \u2264 i \u22a2 MeasurableSet (s j) ** refine' measurableSet_generateFrom \u27e8{1}, measurableSet_singleton 1, _\u27e9 ** case h.refine'_1.intro.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i j : \u03b9 hij : j \u2264 i \u22a2 (Set.indicator (s j) fun x => 1) \u207b\u00b9' {1} = s j ** ext x ** case h.refine'_1.intro.intro.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i j : \u03b9 hij : j \u2264 i x : \u03a9 \u22a2 x \u2208 (Set.indicator (s j) fun x => 1) \u207b\u00b9' {1} \u2194 x \u2208 s j ** simp [Set.indicator_const_preimage_eq_union] ** case h.refine'_2 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 \u22a2 \u2200 (t : Set \u03a9), t \u2208 {s_1 | \u2203 n, MeasurableSet s_1} \u2192 MeasurableSet t ** rintro t \u27e8n, ht\u27e9 ** case h.refine'_2.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t : Set \u03a9 n : \u03b9 ht : MeasurableSet t \u22a2 MeasurableSet t ** suffices MeasurableSpace.generateFrom {t | n \u2264 i \u2227\n  MeasurableSet[MeasurableSpace.comap ((s n).indicator (fun _ => 1 : \u03a9 \u2192 \u03b2)) m\u03b2] t} \u2264\n    MeasurableSpace.generateFrom {t | \u2203 (j : \u03b9), j \u2264 i \u2227 s j = t} by\n  exact this _ ht ** case h.refine'_2.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t : Set \u03a9 n : \u03b9 ht : MeasurableSet t \u22a2 MeasurableSpace.generateFrom {t | n \u2264 i \u2227 MeasurableSet t} \u2264 MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t} ** refine' generateFrom_le _ ** case h.refine'_2.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t : Set \u03a9 n : \u03b9 ht : MeasurableSet t \u22a2 \u2200 (t : Set \u03a9), t \u2208 {t | n \u2264 i \u2227 MeasurableSet t} \u2192 MeasurableSet t ** rintro t \u27e8hn, u, _, hu'\u27e9 ** case h.refine'_2.intro.intro.intro.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t \u22a2 MeasurableSet t ** obtain heq | heq | heq | heq := Set.indicator_const_preimage (s n) u (1 : \u03b2) ** case h.refine'_2.intro.intro.intro.intro.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ \u22a2 MeasurableSet t  case h.refine'_2.intro.intro.intro.intro.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n \u22a2 MeasurableSet t  case h.refine'_2.intro.intro.intro.intro.inr.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c \u22a2 MeasurableSet t  case h.refine'_2.intro.intro.intro.intro.inr.inr.inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u \u2208 {\u2205} \u22a2 MeasurableSet t ** pick_goal 4 ** case h.refine'_2.intro.intro.intro.intro.inr.inr.inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u \u2208 {\u2205} \u22a2 MeasurableSet t  case h.refine'_2.intro.intro.intro.intro.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ \u22a2 MeasurableSet t  case h.refine'_2.intro.intro.intro.intro.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n \u22a2 MeasurableSet t  case h.refine'_2.intro.intro.intro.intro.inr.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c \u22a2 MeasurableSet t ** rw [Set.mem_singleton_iff] at heq ** case h.refine'_2.intro.intro.intro.intro.inr.inr.inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = \u2205 \u22a2 MeasurableSet t  case h.refine'_2.intro.intro.intro.intro.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ \u22a2 MeasurableSet t  case h.refine'_2.intro.intro.intro.intro.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n \u22a2 MeasurableSet t  case h.refine'_2.intro.intro.intro.intro.inr.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c \u22a2 MeasurableSet t ** all_goals rw [heq] at hu'; rw [\u2190 hu'] ** case h.refine'_2.intro.intro.intro.intro.inr.inr.inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : \u2205 = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = \u2205 \u22a2 MeasurableSet \u2205  case h.refine'_2.intro.intro.intro.intro.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : Set.univ = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = Set.univ \u22a2 MeasurableSet Set.univ  case h.refine'_2.intro.intro.intro.intro.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : s n = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = s n \u22a2 MeasurableSet (s n)  case h.refine'_2.intro.intro.intro.intro.inr.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (s n)\u1d9c = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c \u22a2 MeasurableSet (s n)\u1d9c ** exacts [measurableSet_empty _, MeasurableSet.univ, measurableSet_generateFrom \u27e8n, hn, rfl\u27e9,\n  MeasurableSet.compl (measurableSet_generateFrom \u27e8n, hn, rfl\u27e9)] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t : Set \u03a9 n : \u03b9 ht : MeasurableSet t this :   MeasurableSpace.generateFrom {t | n \u2264 i \u2227 MeasurableSet t} \u2264 MeasurableSpace.generateFrom {t | \u2203 j, j \u2264 i \u2227 s j = t} \u22a2 MeasurableSet t ** exact this _ ht ** case h.refine'_2.intro.intro.intro.intro.inr.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c \u22a2 MeasurableSet t ** rw [heq] at hu' ** case h.refine'_2.intro.intro.intro.intro.inr.inr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : MetrizableSpace \u03b2 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b3 : BorelSpace \u03b2 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : MulZeroOneClass \u03b2 inst\u271d : Nontrivial \u03b2 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) i : \u03b9 t\u271d : Set \u03a9 n : \u03b9 ht : MeasurableSet t\u271d t : Set \u03a9 hn : n \u2264 i u : Set \u03b2 left\u271d : MeasurableSet u hu' : (s n)\u1d9c = t heq : (Set.indicator (s n) fun x => 1) \u207b\u00b9' u = (s n)\u1d9c \u22a2 MeasurableSet t ** rw [\u2190 hu'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.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 k : K q : Stmt\u2082 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) f : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3 \u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S =     TM2.stepAux q (stVar v (S k) (StAct.peek f)) (update S k (stWrite v (S k) (StAct.peek f))) ** unfold stWrite ** 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 q : Stmt\u2082 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) f : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3 \u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S =     TM2.stepAux q (stVar v (S k) (StAct.peek f))       (update S k         (match StAct.peek f with         | StAct.push f => f v :: S k         | StAct.peek a => S k         | StAct.pop a => List.tail (S k))) ** rw [Function.update_eq_self] ** 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 q : Stmt\u2082 v : \u03c3 S : (k : K) \u2192 List (\u0393 k) f : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3 \u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S = TM2.stepAux q (stVar v (S k) (StAct.peek f)) S ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.cgf_zero ** \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 \u22a2 cgf X \u03bc 0 = 0 ** simp only [cgf_zero', measure_univ, ENNReal.one_toReal, log_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.LocallyIntegrableOn.smul_continuousOn ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace X inst\u271d\u2079 : TopologicalSpace X inst\u271d\u2078 : MeasurableSpace Y inst\u271d\u2077 : TopologicalSpace Y inst\u271d\u2076 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s\u271d : Set X inst\u271d\u2075 : OpensMeasurableSpace X A K : Set X inst\u271d\u2074 : LocallyCompactSpace X inst\u271d\u00b3 : T2Space X \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NormedField \ud835\udd5c inst\u271d\u00b9 : SecondCountableTopologyEither X E inst\u271d : NormedSpace \ud835\udd5c E f : X \u2192 \ud835\udd5c g : X \u2192 E s : Set X hs : IsOpen s hf : LocallyIntegrableOn f s hg : ContinuousOn g s \u22a2 LocallyIntegrableOn (fun x => f x \u2022 g x) s ** rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf \u22a2 ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace X inst\u271d\u2079 : TopologicalSpace X inst\u271d\u2078 : MeasurableSpace Y inst\u271d\u2077 : TopologicalSpace Y inst\u271d\u2076 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s\u271d : Set X inst\u271d\u2075 : OpensMeasurableSpace X A K : Set X inst\u271d\u2074 : LocallyCompactSpace X inst\u271d\u00b3 : T2Space X \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NormedField \ud835\udd5c inst\u271d\u00b9 : SecondCountableTopologyEither X E inst\u271d : NormedSpace \ud835\udd5c E f : X \u2192 \ud835\udd5c g : X \u2192 E s : Set X hs : IsOpen s hf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k hg : ContinuousOn g s \u22a2 \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn (fun x => f x \u2022 g x) k ** exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_partition_approximatesLinearOn_of_hasFDerivWithinAt ** E : Type u_1 F : Type u_2 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : FiniteDimensional \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f'\u271d : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E \u03bc : Measure E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc inst\u271d : SecondCountableTopology F f : E \u2192 F s : Set E f' : E \u2192 E \u2192L[\u211d] F hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x r : (E \u2192L[\u211d] F) \u2192 \u211d\u22650 rpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0 \u22a2 \u2203 t A,     Pairwise (Disjoint on t) \u2227       (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227         s \u2286 \u22c3 n, t n \u2227           (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227             (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y) ** rcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with\n  \u27e8t, A, t_closed, st, t_approx, ht\u27e9 ** case intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : FiniteDimensional \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f'\u271d : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E \u03bc : Measure E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc inst\u271d : SecondCountableTopology F f : E \u2192 F s : Set E f' : E \u2192 E \u2192L[\u211d] F hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x r : (E \u2192L[\u211d] F) \u2192 \u211d\u22650 rpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0 t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] F t_closed : \u2200 (n : \u2115), IsClosed (t n) st : s \u2286 \u22c3 n, t n t_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n)) ht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2203 t A,     Pairwise (Disjoint on t) \u2227       (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227         s \u2286 \u22c3 n, t n \u2227           (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227             (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y) ** refine'\n  \u27e8disjointed t, A, disjoint_disjointed _,\n    MeasurableSet.disjointed fun n => (t_closed n).measurableSet, _, _, ht\u27e9 ** case intro.intro.intro.intro.intro.refine'_1 E : Type u_1 F : Type u_2 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : FiniteDimensional \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f'\u271d : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E \u03bc : Measure E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc inst\u271d : SecondCountableTopology F f : E \u2192 F s : Set E f' : E \u2192 E \u2192L[\u211d] F hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x r : (E \u2192L[\u211d] F) \u2192 \u211d\u22650 rpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0 t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] F t_closed : \u2200 (n : \u2115), IsClosed (t n) st : s \u2286 \u22c3 n, t n t_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n)) ht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 s \u2286 \u22c3 n, disjointed t n ** rw [iUnion_disjointed] ** case intro.intro.intro.intro.intro.refine'_1 E : Type u_1 F : Type u_2 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : FiniteDimensional \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f'\u271d : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E \u03bc : Measure E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc inst\u271d : SecondCountableTopology F f : E \u2192 F s : Set E f' : E \u2192 E \u2192L[\u211d] F hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x r : (E \u2192L[\u211d] F) \u2192 \u211d\u22650 rpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0 t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] F t_closed : \u2200 (n : \u2115), IsClosed (t n) st : s \u2286 \u22c3 n, t n t_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n)) ht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 s \u2286 \u22c3 n, t n ** exact st ** case intro.intro.intro.intro.intro.refine'_2 E : Type u_1 F : Type u_2 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : FiniteDimensional \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f'\u271d : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E \u03bc : Measure E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc inst\u271d : SecondCountableTopology F f : E \u2192 F s : Set E f' : E \u2192 E \u2192L[\u211d] F hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x r : (E \u2192L[\u211d] F) \u2192 \u211d\u22650 rpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0 t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] F t_closed : \u2200 (n : \u2115), IsClosed (t n) st : s \u2286 \u22c3 n, t n t_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n)) ht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 disjointed t n) (r (A n)) ** intro n ** case intro.intro.intro.intro.intro.refine'_2 E : Type u_1 F : Type u_2 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : FiniteDimensional \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f'\u271d : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E \u03bc : Measure E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc inst\u271d : SecondCountableTopology F f : E \u2192 F s : Set E f' : E \u2192 E \u2192L[\u211d] F hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x r : (E \u2192L[\u211d] F) \u2192 \u211d\u22650 rpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0 t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] F t_closed : \u2200 (n : \u2115), IsClosed (t n) st : s \u2286 \u22c3 n, t n t_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n)) ht : 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 disjointed t n) (r (A n)) ** exact (t_approx n).mono_set (inter_subset_inter_right _ (disjointed_subset _ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.range_extend ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 f : \u03b1 \u2192 \u03b2 hf : Injective f g : \u03b1 \u2192 \u03b3 g' : \u03b2 \u2192 \u03b3 \u22a2 range (extend f g g') = range g \u222a g' '' (range f)\u1d9c ** refine' (range_extend_subset _ _ _).antisymm _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 f : \u03b1 \u2192 \u03b2 hf : Injective f g : \u03b1 \u2192 \u03b3 g' : \u03b2 \u2192 \u03b3 \u22a2 range g \u222a g' '' (range f)\u1d9c \u2286 range (extend f g g') ** rintro z (\u27e8x, rfl\u27e9 | \u27e8y, hy, rfl\u27e9) ** case inl.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 f : \u03b1 \u2192 \u03b2 hf : Injective f g : \u03b1 \u2192 \u03b3 g' : \u03b2 \u2192 \u03b3 x : \u03b1 \u22a2 g x \u2208 range (extend f g g')  case inr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 f : \u03b1 \u2192 \u03b2 hf : Injective f g : \u03b1 \u2192 \u03b3 g' : \u03b2 \u2192 \u03b3 y : \u03b2 hy : y \u2208 (range f)\u1d9c \u22a2 g' y \u2208 range (extend f g g') ** exacts [\u27e8f x, hf.extend_apply _ _ _\u27e9, \u27e8y, extend_apply' _ _ _ hy\u27e9] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.count_apply_finite ** \u03b1 : Type u_1 \u03b2 : Type ?u.8455 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 s\u271d : Set \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 s : Set \u03b1 hs : Set.Finite s \u22a2 \u2191\u2191count s = \u2191(Finset.card (Finite.toFinset hs)) ** rw [\u2190 count_apply_finset, Finite.coe_toFinset] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.set_lintegral_condDistrib_of_measurableSet ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : 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 hX : Measurable X hY : AEMeasurable Y hs : MeasurableSet s t : Set \u03b1 ht : MeasurableSet t \u22a2 \u222b\u207b (a : \u03b1) in t, \u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s \u2202\u03bc = \u2191\u2191\u03bc (t \u2229 Y \u207b\u00b9' s) ** obtain \u27e8t', ht', rfl\u27e9 := ht ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : 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 hX : Measurable X hY : AEMeasurable Y hs : MeasurableSet s t' : Set \u03b2 ht' : MeasurableSet t' \u22a2 \u222b\u207b (a : \u03b1) in X \u207b\u00b9' t', \u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s \u2202\u03bc = \u2191\u2191\u03bc (X \u207b\u00b9' t' \u2229 Y \u207b\u00b9' s) ** rw [set_lintegral_preimage_condDistrib hX hY hs ht'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_positive ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v) ** have hwuv : s ((w \u2229 u) \u2206 (w \u2229 v)) = 0 := by\n  refine'\n    subset_positive_null_set (hu.union hv) ((hw.inter hu).symmDiff (hw.inter hv))\n      (hu.symmDiff hv) (restrict_le_restrict_union _ _ hu hsu hv hsv) hs\n      Set.symmDiff_subset_union _\n  rw [\u2190 Set.inter_symmDiff_distrib_left]\n  exact Set.inter_subset_right _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 hwuv : \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0 \u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v) ** obtain \u27e8huv, hvu\u27e9 :=\n  of_diff_eq_zero_of_symmDiff_eq_zero_positive (hw.inter hu) (hw.inter hv)\n    (restrict_le_restrict_subset _ _ hu hsu (w.inter_subset_right u))\n    (restrict_le_restrict_subset _ _ hv hsv (w.inter_subset_right v)) hwuv ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 hwuv : \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0 huv : \u2191s ((w \u2229 u) \\ (w \u2229 v)) = 0 hvu : \u2191s ((w \u2229 v) \\ (w \u2229 u)) = 0 \u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v) ** rw [\u2190 of_diff_of_diff_eq_zero (hw.inter hu) (hw.inter hv) hvu, huv, zero_add] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0 ** refine'\n  subset_positive_null_set (hu.union hv) ((hw.inter hu).symmDiff (hw.inter hv))\n    (hu.symmDiff hv) (restrict_le_restrict_union _ _ hu hsu hv hsv) hs\n    Set.symmDiff_subset_union _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 (w \u2229 u) \u2206 (w \u2229 v) \u2286 u \u2206 v ** rw [\u2190 Set.inter_symmDiff_distrib_left] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hw : MeasurableSet w hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 w \u2229 u \u2206 v \u2286 u \u2206 v ** exact Set.inter_subset_right _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.eq_one_of_mul_eq_self_left ** a b : Int Hpos : a \u2260 0 H : b * a = a \u22a2 b * a = 1 * a ** rw [Int.one_mul, H] ** Qed",
        "informal": ""
    },
    {
        "formal": "NFA.to\u03b5NFA_evalFrom_match ** \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S : Set \u03c3 x : List \u03b1 s : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start : Set \u03c3 \u22a2 \u03b5NFA.evalFrom (to\u03b5NFA M) start = evalFrom M start ** rw [evalFrom, \u03b5NFA.evalFrom, to\u03b5NFA_\u03b5Closure] ** \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S : Set \u03c3 x : List \u03b1 s : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start : Set \u03c3 \u22a2 List.foldl (\u03b5NFA.stepSet (to\u03b5NFA M)) start = List.foldl (stepSet M) start ** suffices \u03b5NFA.stepSet (to\u03b5NFA M) = stepSet M by rw [this] ** \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S : Set \u03c3 x : List \u03b1 s : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start : Set \u03c3 \u22a2 \u03b5NFA.stepSet (to\u03b5NFA M) = stepSet M ** ext S s ** case h.h.h \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S\u271d : Set \u03c3 x : List \u03b1 s\u271d : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start S : Set \u03c3 s : \u03b1 x\u271d : \u03c3 \u22a2 x\u271d \u2208 \u03b5NFA.stepSet (to\u03b5NFA M) S s \u2194 x\u271d \u2208 stepSet M S s ** simp only [stepSet, \u03b5NFA.stepSet, exists_prop, Set.mem_iUnion] ** case h.h.h \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S\u271d : Set \u03c3 x : List \u03b1 s\u271d : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start S : Set \u03c3 s : \u03b1 x\u271d : \u03c3 \u22a2 (\u2203 i, i \u2208 S \u2227 x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) i (some s))) \u2194 \u2203 i, i \u2208 S \u2227 x\u271d \u2208 step M i s ** apply exists_congr ** case h.h.h.h \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S\u271d : Set \u03c3 x : List \u03b1 s\u271d : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start S : Set \u03c3 s : \u03b1 x\u271d : \u03c3 \u22a2 \u2200 (a : \u03c3), a \u2208 S \u2227 x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) a (some s)) \u2194 a \u2208 S \u2227 x\u271d \u2208 step M a s ** simp only [and_congr_right_iff] ** case h.h.h.h \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S\u271d : Set \u03c3 x : List \u03b1 s\u271d : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start S : Set \u03c3 s : \u03b1 x\u271d : \u03c3 \u22a2 \u2200 (a : \u03c3), a \u2208 S \u2192 (x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) a (some s)) \u2194 x\u271d \u2208 step M a s) ** intro _ _ ** case h.h.h.h \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S\u271d : Set \u03c3 x : List \u03b1 s\u271d : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start S : Set \u03c3 s : \u03b1 x\u271d a\u271d\u00b9 : \u03c3 a\u271d : a\u271d\u00b9 \u2208 S \u22a2 x\u271d \u2208 \u03b5NFA.\u03b5Closure (to\u03b5NFA M) (\u03b5NFA.step (to\u03b5NFA M) a\u271d\u00b9 (some s)) \u2194 x\u271d \u2208 step M a\u271d\u00b9 s ** rw [M.to\u03b5NFA_\u03b5Closure] ** case h.h.h.h \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S\u271d : Set \u03c3 x : List \u03b1 s\u271d : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start S : Set \u03c3 s : \u03b1 x\u271d a\u271d\u00b9 : \u03c3 a\u271d : a\u271d\u00b9 \u2208 S \u22a2 x\u271d \u2208 \u03b5NFA.step (to\u03b5NFA M) a\u271d\u00b9 (some s) \u2194 x\u271d \u2208 step M a\u271d\u00b9 s ** rfl ** \u03b1 : Type u \u03c3 \u03c3' : Type v M\u271d : \u03b5NFA \u03b1 \u03c3 S : Set \u03c3 x : List \u03b1 s : \u03c3 a : \u03b1 M : NFA \u03b1 \u03c3 start : Set \u03c3 this : \u03b5NFA.stepSet (to\u03b5NFA M) = stepSet M \u22a2 List.foldl (\u03b5NFA.stepSet (to\u03b5NFA M)) start = List.foldl (stepSet M) start ** rw [this] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.card_insert ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x a : \u03b1 s : Set \u03b1 inst\u271d : Fintype \u2191s h : \u00aca \u2208 s d : Fintype \u2191(insert a s) \u22a2 Fintype.card \u2191(insert a s) = Fintype.card \u2191s + 1 ** rw [\u2190 card_fintypeInsertOfNotMem s h] ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x a : \u03b1 s : Set \u03b1 inst\u271d : Fintype \u2191s h : \u00aca \u2208 s d : Fintype \u2191(insert a s) \u22a2 Fintype.card \u2191(insert a s) = Fintype.card \u2191(insert a s) ** congr ** case h.e_2.h \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x a : \u03b1 s : Set \u03b1 inst\u271d : Fintype \u2191s h : \u00aca \u2208 s d : Fintype \u2191(insert a s) \u22a2 d = fintypeInsertOfNotMem s h ** exact Subsingleton.elim _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.mod_add_div ** m k : Nat \u22a2 m % k + k * (m / k) = m ** induction m, k using mod.inductionOn with rw [div_eq, mod_eq]\n| base x y h => simp [h]\n| ind x y h IH => simp [h]; rw [Nat.mul_succ, \u2190 Nat.add_assoc, IH, Nat.sub_add_cancel h.2] ** case base x y : Nat h : \u00ac(0 < y \u2227 y \u2264 x) \u22a2 ((if 0 < y \u2227 y \u2264 x then (x - y) % y else x) + y * if 0 < y \u2227 y \u2264 x then (x - y) / y + 1 else 0) = x ** simp [h] ** case ind x y : Nat h : 0 < y \u2227 y \u2264 x IH : (x - y) % y + y * ((x - y) / y) = x - y \u22a2 ((if 0 < y \u2227 y \u2264 x then (x - y) % y else x) + y * if 0 < y \u2227 y \u2264 x then (x - y) / y + 1 else 0) = x ** simp [h] ** case ind x y : Nat h : 0 < y \u2227 y \u2264 x IH : (x - y) % y + y * ((x - y) / y) = x - y \u22a2 (x - y) % y + y * ((x - y) / y + 1) = x ** rw [Nat.mul_succ, \u2190 Nat.add_assoc, IH, Nat.sub_add_cancel h.2] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.image_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 (Int.fract '' s) ** simp only [Int.image_fract, sub_eq_add_neg, image_add_right'] ** \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 m => (measurable_add_const _ hs).inter measurableSet_Ico ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.expand_WF.foldl ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat l : List (\u03b1 \u00d7 \u03b2) i : Nat hl\u2081 : \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) l hl\u2082 : \u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 l \u2192 rank x.fst = i target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 l \u2192 \u00ac(x.fst == k) = true)         bucket \u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target l) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target l).val.data \u2192         AssocList.All (fun k x => rank k \u2264 i) bucket ** induction l generalizing target with\n| nil => exact \u27e8ht\u2081, fun _ h\u2081 _ h\u2082 => (ht\u2082 _ h\u2081 _ h\u2082).1\u27e9\n| cons _ _ ih =>\n  simp at hl\u2081 hl\u2082 ht\u2082\n  refine ih hl\u2081.2 hl\u2082.2\n    (reinsertAux_WF ht\u2081 fun _ h => (ht\u2082 _ (Array.getElem_mem_data ..) _ h).2.1)\n    (fun _ h => ?_)\n  simp [reinsertAux, Buckets.update] at h\n  match List.mem_or_eq_of_mem_set h with\n  | .inl h =>\n    intro _ hf\n    have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ h _ hf\n    exact \u27e8h\u2081, h\u2082.2\u27e9\n  | .inr h => subst h; intro\n    | _, .head .. =>\n      exact \u27e8hl\u2082.1 \u25b8 Nat.le_refl _, fun _ h h' => hl\u2081.1 _ h (PartialEquivBEq.symm h')\u27e9\n    | _, .tail _ h =>\n      have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (Array.getElem_mem_data ..) _ h\n      exact \u27e8h\u2081, h\u2082.2\u27e9 ** case nil \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat hl\u2081 : \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) [] hl\u2082 : \u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 [] \u2192 rank x.fst = i target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 [] \u2192 \u00ac(x.fst == k) = true)         bucket \u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target []) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target []).val.data \u2192         AssocList.All (fun k x => rank k \u2264 i) bucket ** exact \u27e8ht\u2081, fun _ h\u2081 _ h\u2082 => (ht\u2082 _ h\u2081 _ h\u2082).1\u27e9 ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (head\u271d :: tail\u271d) hl\u2082 : \u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 head\u271d :: tail\u271d \u2192 rank x.fst = i target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),               x \u2208 head\u271d :: tail\u271d \u2192 \u00ac(x.fst == k) = true)         bucket \u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)).val.data \u2192         AssocList.All (fun k x => rank k \u2264 i) bucket ** simp at hl\u2081 hl\u2082 ht\u2082 ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket \u22a2 Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)) \u2227     \u2200 (bucket : AssocList \u03b1 \u03b2),       bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (head\u271d :: tail\u271d)).val.data \u2192         AssocList.All (fun k x => rank k \u2264 i) bucket ** refine ih hl\u2081.2 hl\u2082.2\n  (reinsertAux_WF ht\u2081 fun _ h => (ht\u2082 _ (Array.getElem_mem_data ..) _ h).2.1)\n  (fun _ h => ?_) ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket x\u271d : AssocList \u03b1 \u03b2 h : x\u271d \u2208 ((fun d x => reinsertAux d x.fst x.snd) target head\u271d).val.data \u22a2 AssocList.All     (fun k x =>       rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)     x\u271d ** simp [reinsertAux, Buckets.update] at h ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket x\u271d : AssocList \u03b1 \u03b2 h :   x\u271d \u2208     List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) \u22a2 AssocList.All     (fun k x =>       rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)     x\u271d ** match List.mem_or_eq_of_mem_set h with\n| .inl h =>\n  intro _ hf\n  have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ h _ hf\n  exact \u27e8h\u2081, h\u2082.2\u27e9\n| .inr h => subst h; intro\n  | _, .head .. =>\n    exact \u27e8hl\u2082.1 \u25b8 Nat.le_refl _, fun _ h h' => hl\u2081.1 _ h (PartialEquivBEq.symm h')\u27e9\n  | _, .tail _ h =>\n    have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (Array.getElem_mem_data ..) _ h\n    exact \u27e8h\u2081, h\u2082.2\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket x\u271d : AssocList \u03b1 \u03b2 h\u271d :   x\u271d \u2208     List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) h : x\u271d \u2208 target.val.data \u22a2 AssocList.All     (fun k x =>       rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)     x\u271d ** intro _ hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket x\u271d : AssocList \u03b1 \u03b2 h\u271d :   x\u271d \u2208     List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) h : x\u271d \u2208 target.val.data a\u271d : \u03b1 \u00d7 \u03b2 hf : a\u271d \u2208 AssocList.toList x\u271d \u22a2 rank a\u271d.fst \u2264 i \u2227     \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true ** have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ h _ hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket x\u271d : AssocList \u03b1 \u03b2 h\u271d :   x\u271d \u2208     List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) h : x\u271d \u2208 target.val.data a\u271d : \u03b1 \u00d7 \u03b2 hf : a\u271d \u2208 AssocList.toList x\u271d h\u2081 : rank a\u271d.fst \u2264 i h\u2082 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     \u00ac(head\u271d.fst == a\u271d.fst) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == a\u271d.fst) = true \u22a2 rank a\u271d.fst \u2264 i \u2227     \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true ** exact \u27e8h\u2081, h\u2082.2\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket x\u271d : AssocList \u03b1 \u03b2 h\u271d :   x\u271d \u2208     List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) h :   x\u271d =     AssocList.cons head\u271d.fst head\u271d.snd       target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u22a2 AssocList.All     (fun k x =>       rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)     x\u271d ** subst h ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket h :   AssocList.cons head\u271d.fst head\u271d.snd       target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208     List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) \u22a2 AssocList.All     (fun k x =>       rank k \u2264 i \u2227 \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)     (AssocList.cons head\u271d.fst head\u271d.snd       target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) ** intro\n| _, .head .. =>\nexact \u27e8hl\u2082.1 \u25b8 Nat.le_refl _, fun _ h h' => hl\u2081.1 _ h (PartialEquivBEq.symm h')\u27e9\n| _, .tail _ h =>\nhave \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (Array.getElem_mem_data ..) _ h\nexact \u27e8h\u2081, h\u2082.2\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket h :   AssocList.cons head\u271d.fst head\u271d.snd       target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208     List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) x\u271d\u00b9 : \u03b1 \u00d7 \u03b2 x\u271d :   x\u271d\u00b9 \u2208     AssocList.toList       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) \u22a2 rank (head\u271d.fst, head\u271d.snd).fst \u2264 i \u2227     \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),       x \u2208 tail\u271d \u2192 \u00ac(x.fst == (head\u271d.fst, head\u271d.snd).fst) = true ** exact \u27e8hl\u2082.1 \u25b8 Nat.le_refl _, fun _ h h' => hl\u2081.1 _ h (PartialEquivBEq.symm h')\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket h\u271d :   AssocList.cons head\u271d.fst head\u271d.snd       target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208     List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) x\u271d\u00b9 : \u03b1 \u00d7 \u03b2 x\u271d :   x\u271d\u00b9 \u2208     AssocList.toList       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) a\u271d : \u03b1 \u00d7 \u03b2 h :   List.Mem a\u271d     (AssocList.toList       target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) \u22a2 rank a\u271d.fst \u2264 i \u2227     \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true ** have \u27e8h\u2081, h\u2082\u27e9 := ht\u2082 _ (Array.getElem_mem_data ..) _ h ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 rank : \u03b1 \u2192 Nat i : Nat head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) ih :   (\u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1], List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d) \u2192     (\u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 rank x.fst = i) \u2192       \u2200 {target : Buckets \u03b1 \u03b2},         Buckets.WF target \u2192           (\u2200 (bucket : AssocList \u03b1 \u03b2),               bucket \u2208 target.val.data \u2192                 AssocList.All                   (fun k x =>                     rank k \u2264 i \u2227                       \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2),                         x \u2208 tail\u271d \u2192 \u00ac(x.fst == k) = true)                   bucket) \u2192             Buckets.WF (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d) \u2227               \u2200 (bucket : AssocList \u03b1 \u03b2),                 bucket \u2208 (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data \u2192                   AssocList.All (fun k x => rank k \u2264 i) bucket target : Buckets \u03b1 \u03b2 ht\u2081 : Buckets.WF target hl\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     (\u2200 (a' : \u03b1 \u00d7 \u03b2), a' \u2208 tail\u271d \u2192 \u00ac(head\u271d.fst == a'.fst) = true) \u2227       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) tail\u271d hl\u2082 : rank head\u271d.fst = i \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 rank a.fst = i ht\u2082 :   \u2200 (bucket : AssocList \u03b1 \u03b2),     bucket \u2208 target.val.data \u2192       AssocList.All         (fun k x =>           rank k \u2264 i \u2227             \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],               \u00ac(head\u271d.fst == k) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == k) = true)         bucket h\u271d :   AssocList.cons head\u271d.fst head\u271d.snd       target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val] \u2208     List.set target.val.data (USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val)       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) x\u271d\u00b9 : \u03b1 \u00d7 \u03b2 x\u271d :   x\u271d\u00b9 \u2208     AssocList.toList       (AssocList.cons head\u271d.fst head\u271d.snd         target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) a\u271d : \u03b1 \u00d7 \u03b2 h :   List.Mem a\u271d     (AssocList.toList       target.val[USize.toNat (mkIdx (_ : 0 < Array.size target.val) (UInt64.toUSize (hash head\u271d.fst))).val]) h\u2081 : rank a\u271d.fst \u2264 i h\u2082 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     \u00ac(head\u271d.fst == a\u271d.fst) = true \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), a \u2208 tail\u271d \u2192 \u00ac(a.fst == a\u271d.fst) = true \u22a2 rank a\u271d.fst \u2264 i \u2227     \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1] (x : \u03b1 \u00d7 \u03b2), x \u2208 tail\u271d \u2192 \u00ac(x.fst == a\u271d.fst) = true ** exact \u27e8h\u2081, h\u2082.2\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.InnerRegular.weaklyRegular_of_finite ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen \u22a2 WeaklyRegular \u03bc ** have hfin : \u2200 {s}, \u03bc s \u2260 \u22a4 := @(measure_ne_top \u03bc) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 WeaklyRegular \u03bc ** suffices \u2200 s, MeasurableSet s \u2192 \u2200 \u03b5, \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227\n    IsClosed F \u2227 IsOpen U \u2227 \u03bc s \u2264 \u03bc F + \u03b5 \u2227 \u03bc U \u2264 \u03bc s + \u03b5 by\n  refine'\n    { outerRegular := fun s hs r hr => _\n      innerRegular := H }\n  rcases exists_between hr with \u27e8r', hsr', hr'r\u27e9\n  rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with \u27e8-, -, U, hsU, -, hUo, -, H\u27e9\n  refine' \u27e8U, hsU, hUo, _\u27e9\n  rw [add_tsub_cancel_of_le hsr'.le] at H\n  exact H.trans_lt hr'r ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 ** refine' MeasurableSet.induction_on_open _ _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 this :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 \u22a2 WeaklyRegular \u03bc ** refine'\n  { outerRegular := fun s hs r hr => _\n    innerRegular := H } ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 this :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 s : Set \u03b1 hs : MeasurableSet s r : \u211d\u22650\u221e hr : r > \u2191\u2191\u03bc s \u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r ** rcases exists_between hr with \u27e8r', hsr', hr'r\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 this :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 s : Set \u03b1 hs : MeasurableSet s r : \u211d\u22650\u221e hr : r > \u2191\u2191\u03bc s r' : \u211d\u22650\u221e hsr' : \u2191\u2191\u03bc s < r' hr'r : r' < r \u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r ** rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with \u27e8-, -, U, hsU, -, hUo, -, H\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 this :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 s : Set \u03b1 hs : MeasurableSet s r : \u211d\u22650\u221e hr : r > \u2191\u2191\u03bc s r' : \u211d\u22650\u221e hsr' : \u2191\u2191\u03bc s < r' hr'r : r' < r U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U H : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + (r' - \u2191\u2191\u03bc s) \u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r ** refine' \u27e8U, hsU, hUo, _\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 this :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 s : Set \u03b1 hs : MeasurableSet s r : \u211d\u22650\u221e hr : r > \u2191\u2191\u03bc s r' : \u211d\u22650\u221e hsr' : \u2191\u2191\u03bc s < r' hr'r : r' < r U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U H : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + (r' - \u2191\u2191\u03bc s) \u22a2 \u2191\u2191\u03bc U < r ** rw [add_tsub_cancel_of_le hsr'.le] at H ** case intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 this :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 s : Set \u03b1 hs : MeasurableSet s r : \u211d\u22650\u221e hr : r > \u2191\u2191\u03bc s r' : \u211d\u22650\u221e hsr' : \u2191\u2191\u03bc s < r' hr'r : r' < r U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U H : \u2191\u2191\u03bc U \u2264 r' \u22a2 \u2191\u2191\u03bc U < r ** exact H.trans_lt hr'r ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200 (U : Set \u03b1),     IsOpen U \u2192       \u2200 (\u03b5 : \u211d\u22650\u221e),         \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5 ** intro U hU \u03b5 h\u03b5 ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 U : Set \u03b1 hU : IsOpen U \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5 ** rcases H.exists_subset_lt_add isClosed_empty hU hfin h\u03b5 with \u27e8F, hsF, hFc, hF\u27e9 ** case refine'_1.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 U : Set \u03b1 hU : IsOpen U \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 F : Set \u03b1 hsF : F \u2286 U hFc : IsClosed F hF : \u2191\u2191\u03bc U < \u2191\u2191\u03bc F + \u03b5 \u22a2 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5 ** exact \u27e8F, hsF, U, Subset.rfl, hFc, hU, hF.le, le_self_add\u27e9 ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200 (t : Set \u03b1),     MeasurableSet t \u2192       (\u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 t \u2227 \u2203 U, U \u2287 t \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc t + \u03b5) \u2192         \u2200 (\u03b5 : \u211d\u22650\u221e),           \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 t\u1d9c \u2227 \u2203 U, U \u2287 t\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc t\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc t\u1d9c + \u03b5 ** rintro s hs H \u03b5 h\u03b5 ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : Set \u03b1 hs : MeasurableSet s H : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 F, F \u2286 s\u1d9c \u2227 \u2203 U, U \u2287 s\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5 ** rcases H \u03b5 h\u03b5 with \u27e8F, hFs, U, hsU, hFc, hUo, hF, hU\u27e9 ** case refine'_2.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : Set \u03b1 hs : MeasurableSet s H : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 F : Set \u03b1 hFs : F \u2286 s U : Set \u03b1 hsU : U \u2287 s hFc : IsClosed F hUo : IsOpen U hF : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 hU : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 \u22a2 \u2203 F, F \u2286 s\u1d9c \u2227 \u2203 U, U \u2287 s\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5 ** refine'\n  \u27e8U\u1d9c, compl_subset_compl.2 hsU, F\u1d9c, compl_subset_compl.2 hFs, hUo.isClosed_compl,\n    hFc.isOpen_compl, _\u27e9 ** case refine'_2.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : Set \u03b1 hs : MeasurableSet s H : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 F : Set \u03b1 hFs : F \u2286 s U : Set \u03b1 hsU : U \u2287 s hFc : IsClosed F hUo : IsOpen U hF : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 hU : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5 \u22a2 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc U\u1d9c + \u03b5 \u2227 \u2191\u2191\u03bc F\u1d9c \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5 ** simp only [measure_compl_le_add_iff, *, hUo.measurableSet, hFc.measurableSet, true_and_iff] ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),     Pairwise (Disjoint on f) \u2192       (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192         (\u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),             \u03b5 \u2260 0 \u2192               \u2203 F, F \u2286 f i \u2227 \u2203 U, U \u2287 f i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (f i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (f i) + \u03b5) \u2192           \u2200 (\u03b5 : \u211d\u22650\u221e),             \u03b5 \u2260 0 \u2192               \u2203 F,                 F \u2286 \u22c3 i, f i \u2227                   \u2203 U, U \u2287 \u22c3 i, f i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, f i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, f i) + \u03b5 ** intro s hsd hsm H \u03b5 \u03b50 ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u22a2 \u2203 F,     F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5 ** have \u03b50' : \u03b5 / 2 \u2260 0 := (ENNReal.half_pos \u03b50).ne' ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u22a2 \u2203 F,     F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5 ** rcases ENNReal.exists_pos_sum_of_countable' \u03b50' \u2115 with \u27e8\u03b4, \u03b40, h\u03b4\u03b5\u27e9 ** case refine'_3.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 \u22a2 \u2203 F,     F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5 ** choose F hFs U hsU hFc hUo hF hU using fun n => H n (\u03b4 n) (\u03b40 n).ne' ** case refine'_3.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n \u22a2 \u2203 F,     F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5 ** have : Tendsto (fun t => (\u2211 k in t, \u03bc (s k)) + \u03b5 / 2) atTop (\ud835\udcdd <| \u03bc (\u22c3 n, s n) + \u03b5 / 2) := by\n  rw [measure_iUnion hsd hsm]\n  exact Tendsto.add ENNReal.summable.hasSum tendsto_const_nhds ** case refine'_3.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) \u22a2 \u2203 F,     F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5 ** rcases (this.eventually <| lt_mem_nhds <| ENNReal.lt_add_right hfin \u03b50').exists with \u27e8t, ht\u27e9 ** case refine'_3.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) t : Finset \u2115 ht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u22a2 \u2203 F,     F \u2286 \u22c3 i, s i \u2227 \u2203 U, U \u2287 \u22c3 i, s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 i, s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5 ** refine'\n  \u27e8\u22c3 k \u2208 t, F k, iUnion_mono fun k => iUnion_subset fun _ => hFs _, \u22c3 n, U n, iUnion_mono hsU,\n    isClosed_biUnion_finset fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n \u22a2 Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) ** rw [measure_iUnion hsd hsm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n \u22a2 Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2211' (i : \u2115), \u2191\u2191\u03bc (s i) + \u03b5 / 2)) ** exact Tendsto.add ENNReal.summable.hasSum tendsto_const_nhds ** case refine'_3.intro.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) t : Finset \u2115 ht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2191\u2191\u03bc (\u22c3 k \u2208 t, F k) + \u03b5 ** calc\n  (\u2211 k in t, \u03bc (s k)) + \u03b5 / 2 \u2264 ((\u2211 k in t, \u03bc (F k)) + \u2211 k in t, \u03b4 k) + \u03b5 / 2 := by\n    rw [\u2190 sum_add_distrib]\n    exact add_le_add_right (sum_le_sum fun k _ => hF k) _\n  _ \u2264 (\u2211 k in t, \u03bc (F k)) + \u03b5 / 2 + \u03b5 / 2 :=\n    (add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans h\u03b4\u03b5.le) _) _)\n  _ = \u03bc (\u22c3 k \u2208 t, F k) + \u03b5 := by\n    rw [measure_biUnion_finset, add_assoc, ENNReal.add_halves]\n    exacts [fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n),\n      fun k _ => (hFc k).measurableSet] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) t : Finset \u2115 ht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2211 k in t, \u2191\u2191\u03bc (F k) + \u2211 k in t, \u03b4 k + \u03b5 / 2 ** rw [\u2190 sum_add_distrib] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) t : Finset \u2115 ht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2211 x in t, (\u2191\u2191\u03bc (F x) + \u03b4 x) + \u03b5 / 2 ** exact add_le_add_right (sum_le_sum fun k _ => hF k) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) t : Finset \u2115 ht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u22a2 \u2211 k in t, \u2191\u2191\u03bc (F k) + \u03b5 / 2 + \u03b5 / 2 = \u2191\u2191\u03bc (\u22c3 k \u2208 t, F k) + \u03b5 ** rw [measure_biUnion_finset, add_assoc, ENNReal.add_halves] ** case hd \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) t : Finset \u2115 ht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u22a2 PairwiseDisjoint \u2191t fun k => F k  case hm \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) t : Finset \u2115 ht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u22a2 \u2200 (b : \u2115), b \u2208 t \u2192 MeasurableSet (F b) ** exacts [fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n),\n  fun k _ => (hFc k).measurableSet] ** case refine'_3.intro.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) t : Finset \u2115 ht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u22a2 \u2191\u2191\u03bc (\u22c3 n, U n) \u2264 \u2191\u2191\u03bc (\u22c3 i, s i) + \u03b5 ** calc\n  \u03bc (\u22c3 n, U n) \u2264 \u2211' n, \u03bc (U n) := measure_iUnion_le _\n  _ \u2264 \u2211' n, (\u03bc (s n) + \u03b4 n) := (ENNReal.tsum_le_tsum hU)\n  _ = \u03bc (\u22c3 n, s n) + \u2211' n, \u03b4 n := by rw [measure_iUnion hsd hsm, ENNReal.tsum_add]\n  _ \u2264 \u03bc (\u22c3 n, s n) + \u03b5 := add_le_add_left (h\u03b4\u03b5.le.trans ENNReal.half_le_self) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc H\u271d : InnerRegular \u03bc IsClosed IsOpen hfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4 s : \u2115 \u2192 Set \u03b1 hsd : Pairwise (Disjoint on s) hsm : \u2200 (i : \u2115), MeasurableSet (s i) H :   \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),     \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u03b50' : \u03b5 / 2 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2 F : \u2115 \u2192 Set \u03b1 hFs : \u2200 (n : \u2115), F n \u2286 s n U : \u2115 \u2192 Set \u03b1 hsU : \u2200 (n : \u2115), U n \u2287 s n hFc : \u2200 (n : \u2115), IsClosed (F n) hUo : \u2200 (n : \u2115), IsOpen (U n) hF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n hU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n this : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 n, s n) + \u03b5 / 2)) t : Finset \u2115 ht : \u2191\u2191\u03bc (\u22c3 n, s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (s n) + \u03b4 n) = \u2191\u2191\u03bc (\u22c3 n, s n) + \u2211' (n : \u2115), \u03b4 n ** rw [measure_iUnion hsd hsm, ENNReal.tsum_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.eventually_nonempty_inter_smul_of_density_one ** 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 \u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 t)) ** obtain \u27e8t', t'_meas, t't, t'pos, t'top\u27e9 : \u2203 t', MeasurableSet t' \u2227 t' \u2286 t \u2227 0 < \u03bc t' \u2227 \u03bc t' < \u22a4 :=\n  exists_subset_measure_lt_top ht h't.bot_lt ** case intro.intro.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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 t' : Set E t'_meas : MeasurableSet t' t't : t' \u2286 t t'pos : 0 < \u2191\u2191\u03bc t' t'top : \u2191\u2191\u03bc t' < \u22a4 \u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 t)) ** filter_upwards [(tendsto_order.1\n        (tendsto_addHaar_inter_smul_one_of_density_one \u03bc s x h t' t'_meas t'pos.ne' t'top.ne)).1\n    0 zero_lt_one] ** 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 t' : Set E t'_meas : MeasurableSet t' t't : t' \u2286 t t'pos : 0 < \u2191\u2191\u03bc t' t'top : \u2191\u2191\u03bc t' < \u22a4 \u22a2 \u2200 (a : \u211d), 0 < \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t')) / \u2191\u2191\u03bc ({x} + a \u2022 t') \u2192 Set.Nonempty (s \u2229 ({x} + a \u2022 t)) ** intro r hr ** 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 t' : Set E t'_meas : MeasurableSet t' t't : t' \u2286 t t'pos : 0 < \u2191\u2191\u03bc t' t'top : \u2191\u2191\u03bc t' < \u22a4 r : \u211d hr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t') \u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t)) ** have : \u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0 := fun h' => by\n  simp only [ENNReal.not_lt_zero, ENNReal.zero_div, h'] at hr ** 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 t' : Set E t'_meas : MeasurableSet t' t't : t' \u2286 t t'pos : 0 < \u2191\u2191\u03bc t' t'top : \u2191\u2191\u03bc t' < \u22a4 r : \u211d hr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t') this : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0 \u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t)) ** have : (s \u2229 ({x} + r \u2022 t')).Nonempty := nonempty_of_measure_ne_zero this ** 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 t' : Set E t'_meas : MeasurableSet t' t't : t' \u2286 t t'pos : 0 < \u2191\u2191\u03bc t' t'top : \u2191\u2191\u03bc t' < \u22a4 r : \u211d hr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t') this\u271d : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0 this : Set.Nonempty (s \u2229 ({x} + r \u2022 t')) \u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t)) ** apply this.mono (inter_subset_inter Subset.rfl _) ** 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 t' : Set E t'_meas : MeasurableSet t' t't : t' \u2286 t t'pos : 0 < \u2191\u2191\u03bc t' t'top : \u2191\u2191\u03bc t' < \u22a4 r : \u211d hr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t') this\u271d : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0 this : Set.Nonempty (s \u2229 ({x} + r \u2022 t')) \u22a2 {x} + r \u2022 t' \u2286 {x} + r \u2022 t ** exact add_subset_add Subset.rfl (smul_set_mono t'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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 t' : Set E t'_meas : MeasurableSet t' t't : t' \u2286 t t'pos : 0 < \u2191\u2191\u03bc t' t'top : \u2191\u2191\u03bc t' < \u22a4 r : \u211d hr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t') h' : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) = 0 \u22a2 False ** simp only [ENNReal.not_lt_zero, ENNReal.zero_div, h'] at hr ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.exists_ae_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) \u2202\u03bc, \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** filter_upwards [hf.upcrossings_ae_lt_top hbdd, ae_bdd_liminf_atTop_of_snorm_bdd one_ne_zero\n  (fun n => (hf.stronglyMeasurable n).measurable.mono (\u2131.le n) le_rfl) hbdd] with \u03c9 h\u2081 h\u2082 ** case h \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 \u03c9 : \u03a9 h\u2081 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4 h\u2082 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4 \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** exact tendsto_of_uncrossing_lt_top h\u2082 h\u2081 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.IccExtend_range ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : LinearOrder \u03b1 a b : \u03b1 h : a \u2264 b x : \u03b1 f : \u2191(Icc a b) \u2192 \u03b2 \u22a2 range (IccExtend h f) = range f ** simp only [IccExtend, range_comp f, range_projIcc, image_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.All.append ** \u03b1\u271d : Type u_1 l r : RBNode \u03b1\u271d p : \u03b1\u271d \u2192 Prop hl : All p l hr : All p r \u22a2 All p (append l r) ** unfold append ** \u03b1\u271d : Type u_1 l r : RBNode \u03b1\u271d p : \u03b1\u271d \u2192 Prop hl : All p l hr : All p r \u22a2 All p     (match l, r with     | nil, x => x     | x, nil => x     | node red a x b, node red c y d =>       match append b c with       | node red b' z c' => node red (node red a x b') z (node red c' y d)       | bc => node red a x (node red bc y d)     | node black a x b, node black c y d =>       match append b c with       | node red b' z c' => node red (node black a x b') z (node black c' y d)       | bc => balLeft a x (node black bc y d)     | a@h:(node black l v r), node red b x c => node red (append a b) x c     | node red a x b, c@h:(node black l v r) => node red a x (append b c)) ** split <;> try simp [*] ** case h_6 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d l\u271d : RBNode \u03b1\u271d v\u271d : \u03b1\u271d r\u271d : RBNode \u03b1\u271d hl : All p (node red a\u271d x\u271d b\u271d) hr : All p (node black l\u271d v\u271d r\u271d) \u22a2 All p (node red a\u271d x\u271d (append b\u271d (node black l\u271d v\u271d r\u271d))) ** simp [*] ** case h_3 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d c\u271d : RBNode \u03b1\u271d y\u271d : \u03b1\u271d d\u271d : RBNode \u03b1\u271d hl : All p (node red a\u271d x\u271d b\u271d) hr : All p (node red c\u271d y\u271d d\u271d) \u22a2 All p     (match append b\u271d c\u271d with     | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)     | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d)) ** have \u27e8hx, ha, hb\u27e9 := hl ** case h_3 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d c\u271d : RBNode \u03b1\u271d y\u271d : \u03b1\u271d d\u271d : RBNode \u03b1\u271d hl : All p (node red a\u271d x\u271d b\u271d) hr : All p (node red c\u271d y\u271d d\u271d) hx : p x\u271d ha : All p a\u271d hb : All p b\u271d \u22a2 All p     (match append b\u271d c\u271d with     | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)     | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d)) ** have \u27e8hy, hc, hd\u27e9 := hr ** case h_3 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d c\u271d : RBNode \u03b1\u271d y\u271d : \u03b1\u271d d\u271d : RBNode \u03b1\u271d hl : All p (node red a\u271d x\u271d b\u271d) hr : All p (node red c\u271d y\u271d d\u271d) hx : p x\u271d ha : All p a\u271d hb : All p b\u271d hy : p y\u271d hc : All p c\u271d hd : All p d\u271d \u22a2 All p     (match append b\u271d c\u271d with     | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)     | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d)) ** have := hb.append hc ** case h_3 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d c\u271d : RBNode \u03b1\u271d y\u271d : \u03b1\u271d d\u271d : RBNode \u03b1\u271d hl : All p (node red a\u271d x\u271d b\u271d) hr : All p (node red c\u271d y\u271d d\u271d) hx : p x\u271d ha : All p a\u271d hb : All p b\u271d hy : p y\u271d hc : All p c\u271d hd : All p d\u271d this : All p (append b\u271d c\u271d) \u22a2 All p     (match append b\u271d c\u271d with     | node red b' z c' => node red (node red a\u271d x\u271d b') z (node red c' y\u271d d\u271d)     | bc => node red a\u271d x\u271d (node red bc y\u271d d\u271d)) ** split <;> simp_all ** case h_4 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d c\u271d : RBNode \u03b1\u271d y\u271d : \u03b1\u271d d\u271d : RBNode \u03b1\u271d hl : All p (node black a\u271d x\u271d b\u271d) hr : All p (node black c\u271d y\u271d d\u271d) \u22a2 All p     (match append b\u271d c\u271d with     | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)     | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d)) ** have \u27e8hx, ha, hb\u27e9 := hl ** case h_4 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d c\u271d : RBNode \u03b1\u271d y\u271d : \u03b1\u271d d\u271d : RBNode \u03b1\u271d hl : All p (node black a\u271d x\u271d b\u271d) hr : All p (node black c\u271d y\u271d d\u271d) hx : p x\u271d ha : All p a\u271d hb : All p b\u271d \u22a2 All p     (match append b\u271d c\u271d with     | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)     | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d)) ** have \u27e8hy, hc, hd\u27e9 := hr ** case h_4 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d c\u271d : RBNode \u03b1\u271d y\u271d : \u03b1\u271d d\u271d : RBNode \u03b1\u271d hl : All p (node black a\u271d x\u271d b\u271d) hr : All p (node black c\u271d y\u271d d\u271d) hx : p x\u271d ha : All p a\u271d hb : All p b\u271d hy : p y\u271d hc : All p c\u271d hd : All p d\u271d \u22a2 All p     (match append b\u271d c\u271d with     | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)     | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d)) ** have := hb.append hc ** case h_4 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d c\u271d : RBNode \u03b1\u271d y\u271d : \u03b1\u271d d\u271d : RBNode \u03b1\u271d hl : All p (node black a\u271d x\u271d b\u271d) hr : All p (node black c\u271d y\u271d d\u271d) hx : p x\u271d ha : All p a\u271d hb : All p b\u271d hy : p y\u271d hc : All p c\u271d hd : All p d\u271d this : All p (append b\u271d c\u271d) \u22a2 All p     (match append b\u271d c\u271d with     | node red b' z c' => node red (node black a\u271d x\u271d b') z (node black c' y\u271d d\u271d)     | bc => balLeft a\u271d x\u271d (node black bc y\u271d d\u271d)) ** split <;> simp_all [All.balLeft] ** case h_5 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 l\u271d : RBNode \u03b1\u271d v\u271d : \u03b1\u271d r\u271d b\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d c\u271d : RBNode \u03b1\u271d hl : All p (node black l\u271d v\u271d r\u271d) hr : All p (node red b\u271d x\u271d c\u271d) \u22a2 p x\u271d \u2227 All p (append (node black l\u271d v\u271d r\u271d) b\u271d) \u2227 All p c\u271d ** simp_all [hl.append hr.2.1] ** case h_6 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x\u271d\u00b2 x\u271d\u00b9 a\u271d : RBNode \u03b1\u271d x\u271d : \u03b1\u271d b\u271d l\u271d : RBNode \u03b1\u271d v\u271d : \u03b1\u271d r\u271d : RBNode \u03b1\u271d hl : All p (node red a\u271d x\u271d b\u271d) hr : All p (node black l\u271d v\u271d r\u271d) \u22a2 p x\u271d \u2227 All p a\u271d \u2227 All p (append b\u271d (node black l\u271d v\u271d r\u271d)) ** simp_all [hl.2.2.append hr] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.measurable_injection_nat_bool_of_countablyGenerated ** \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ \u22a2 \u2203 f, Measurable f \u2227 Injective f ** rcases exists_seq_separating \u03b1 MeasurableSet.empty univ with \u27e8e, hem, he\u27e9 ** case intro.intro.refine_1 \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ e : \u2115 \u2192 Set \u03b1 hem : \u2200 (n : \u2115), MeasurableSet (e n) he : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y \u22a2 Measurable fun x x_1 => decide (x \u2208 e x_1) ** rw [measurable_pi_iff] ** case intro.intro.refine_1 \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ e : \u2115 \u2192 Set \u03b1 hem : \u2200 (n : \u2115), MeasurableSet (e n) he : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y \u22a2 \u2200 (a : \u2115), Measurable fun x => decide (x \u2208 e a) ** refine fun n \u21a6 measurable_to_bool ?_ ** case intro.intro.refine_1 \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ e : \u2115 \u2192 Set \u03b1 hem : \u2200 (n : \u2115), MeasurableSet (e n) he : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y n : \u2115 \u22a2 MeasurableSet ((fun x => decide (x \u2208 e n)) \u207b\u00b9' {true}) ** simpa only [preimage, mem_singleton_iff, Bool.decide_iff, setOf_mem_eq] using hem n ** case intro.intro.refine_2 \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 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : HasCountableSeparatingOn \u03b1 MeasurableSet univ e : \u2115 \u2192 Set \u03b1 hem : \u2200 (n : \u2115), MeasurableSet (e n) he : \u2200 (x : \u03b1), x \u2208 univ \u2192 \u2200 (y : \u03b1), y \u2208 univ \u2192 (\u2200 (n : \u2115), x \u2208 e n \u2194 y \u2208 e n) \u2192 x = y \u22a2 Injective fun x x_1 => decide (x \u2208 e x_1) ** exact fun x y h \u21a6 he x trivial y trivial fun n \u21a6 decide_eq_decide.1 <| congr_fun h _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.measurable_floor ** \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 (floor \u207b\u00b9' {\u230an\u230b\u208a}) ** cases' eq_or_ne \u230an\u230b\u208a 0 with h h <;> simp_all [h, Nat.preimage_floor_of_ne_zero, -floor_eq_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "torusIntegral_succ ** 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 \u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =     \u222e (x : \u2102) in C(c 0, R 0), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succ, R \u2218 Fin.succ), f (Fin.cons x y) ** simpa using torusIntegral_succAbove hf 0 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsFundamentalDomain.preimage_of_equiv ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Group G inst\u271d\u2075 : Group H inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MulAction H \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 h : IsFundamentalDomain G s f : \u03b2 \u2192 \u03b1 hf : QuasiMeasurePreserving f e : G \u2192 H he : Bijective e hef : \u2200 (g : G), Semiconj f (fun x => e g \u2022 x) fun x => g \u2022 x x : \u03b2 x\u271d : \u2203 g, g \u2022 f x \u2208 s g : G hg : g \u2022 f x \u2208 s \u22a2 e g \u2022 x \u2208 f \u207b\u00b9' s ** rwa [mem_preimage, hef g x] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Group G inst\u271d\u2075 : Group H inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MulAction H \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 h : IsFundamentalDomain G s f : \u03b2 \u2192 \u03b1 hf : QuasiMeasurePreserving f e : G \u2192 H he : Bijective e hef : \u2200 (g : G), Semiconj f (fun x => e g \u2022 x) fun x => g \u2022 x a b : H hab : a \u2260 b \u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b ** lift e to G \u2243 H using he ** case intro G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Group G inst\u271d\u2075 : Group H inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MulAction H \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 h : IsFundamentalDomain G s f : \u03b2 \u2192 \u03b1 hf : QuasiMeasurePreserving f a b : H hab : a \u2260 b e : G \u2243 H hef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x \u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b ** have : (e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (e.symm b\u207b\u00b9)\u207b\u00b9 := by simp [hab] ** case intro G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Group G inst\u271d\u2075 : Group H inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MulAction H \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 h : IsFundamentalDomain G s f : \u03b2 \u2192 \u03b1 hf : QuasiMeasurePreserving f a b : H hab : a \u2260 b e : G \u2243 H hef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x this : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9 \u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b ** have := (h.aedisjoint this).preimage hf ** case intro G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Group G inst\u271d\u2075 : Group H inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MulAction H \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 h : IsFundamentalDomain G s f : \u03b2 \u2192 \u03b1 hf : QuasiMeasurePreserving f a b : H hab : a \u2260 b e : G \u2243 H hef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x this\u271d : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9 this : AEDisjoint \u03bd (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm a\u207b\u00b9)\u207b\u00b9) (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm b\u207b\u00b9)\u207b\u00b9) \u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b ** simp only [Semiconj] at hef ** case intro G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Group G inst\u271d\u2075 : Group H inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MulAction H \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 h : IsFundamentalDomain G s f : \u03b2 \u2192 \u03b1 hf : QuasiMeasurePreserving f a b : H hab : a \u2260 b e : G \u2243 H hef : \u2200 (g : G) (x : \u03b2), f (\u2191e g \u2022 x) = g \u2022 f x this\u271d : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9 this : AEDisjoint \u03bd (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm a\u207b\u00b9)\u207b\u00b9) (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm b\u207b\u00b9)\u207b\u00b9) \u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b ** simpa only [onFun, \u2190 preimage_smul_inv, preimage_preimage, \u2190 hef, e.apply_symm_apply, inv_inv]\n  using this ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Group G inst\u271d\u2075 : Group H inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MulAction H \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 h : IsFundamentalDomain G s f : \u03b2 \u2192 \u03b1 hf : QuasiMeasurePreserving f a b : H hab : a \u2260 b e : G \u2243 H hef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x \u22a2 (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9 ** simp [hab] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSet_self ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 t : Set \u03a9 h_indep : IndepSet t t \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4 ** rw [IndepSet_iff] at h_indep ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 t : Set \u03a9 h_indep : \u2200 (t1 t2 : Set \u03a9), MeasurableSet t1 \u2192 MeasurableSet t2 \u2192 \u2191\u2191\u03bc (t1 \u2229 t2) = \u2191\u2191\u03bc t1 * \u2191\u2191\u03bc t2 \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4 ** specialize h_indep t t (measurableSet_generateFrom (Set.mem_singleton t))\n  (measurableSet_generateFrom (Set.mem_singleton t)) ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 t : Set \u03a9 h_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4 ** by_cases h0 : \u03bc t = 0 ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 t : Set \u03a9 h_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t h0 : \u00ac\u2191\u2191\u03bc t = 0 \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4 ** by_cases h_top : \u03bc t = \u221e ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 t : Set \u03a9 h_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t h0 : \u00ac\u2191\u2191\u03bc t = 0 h_top : \u00ac\u2191\u2191\u03bc t = \u22a4 \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4 ** rw [\u2190 one_mul (\u03bc (t \u2229 t)), Set.inter_self, ENNReal.mul_eq_mul_right h0 h_top] at h_indep ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 t : Set \u03a9 h_indep : 1 = \u2191\u2191\u03bc t h0 : \u00ac\u2191\u2191\u03bc t = 0 h_top : \u00ac\u2191\u2191\u03bc t = \u22a4 \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4 ** exact Or.inr (Or.inl h_indep.symm) ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 t : Set \u03a9 h_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t h0 : \u2191\u2191\u03bc t = 0 \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4 ** exact Or.inl h0 ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 t : Set \u03a9 h_indep : \u2191\u2191\u03bc (t \u2229 t) = \u2191\u2191\u03bc t * \u2191\u2191\u03bc t h0 : \u00ac\u2191\u2191\u03bc t = 0 h_top : \u2191\u2191\u03bc t = \u22a4 \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4 ** exact Or.inr (Or.inr h_top) ** Qed",
        "informal": ""
    },
    {
        "formal": "isAddFundamentalDomain_Ioc' ** T : \u211d hT : 0 < T t : \u211d \u03bc : autoParam (Measure \u211d) _auto\u271d \u22a2 IsAddFundamentalDomain { x // x \u2208 AddSubgroup.op (zmultiples T) } (Ioc t (t + T)) ** refine' IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => _ ** T : \u211d hT : 0 < T t : \u211d \u03bc : autoParam (Measure \u211d) _auto\u271d x : \u211d \u22a2 \u2203! g, g +\u1d65 x \u2208 Ioc t (t + T) ** have : Bijective (codRestrict (fun n : \u2124 => n \u2022 T) (AddSubgroup.zmultiples T) _) :=\n  (Equiv.ofInjective (fun n : \u2124 => n \u2022 T) (zsmul_strictMono_left hT).injective).bijective ** T : \u211d hT : 0 < T t : \u211d \u03bc : autoParam (Measure \u211d) _auto\u271d x : \u211d this : Bijective (codRestrict (fun n => n \u2022 T) \u2191(zmultiples T) (_ : \u2200 (x : \u2124), \u2203 y, y \u2022 T = x \u2022 T)) \u22a2 \u2203! g, g +\u1d65 x \u2208 Ioc t (t + T) ** refine' (AddSubgroup.equivOp _).bijective.comp this |>.existsUnique_iff.2 _ ** T : \u211d hT : 0 < T t : \u211d \u03bc : autoParam (Measure \u211d) _auto\u271d x : \u211d this : Bijective (codRestrict (fun n => n \u2022 T) \u2191(zmultiples T) (_ : \u2200 (x : \u2124), \u2203 y, y \u2022 T = x \u2022 T)) \u22a2 \u2203! x_1,     (\u2191(equivOp (zmultiples T)) \u2218 codRestrict (fun n => n \u2022 T) \u2191(zmultiples T) (_ : \u2200 (x : \u2124), \u2203 y, y \u2022 T = x \u2022 T))           x_1 +\u1d65         x \u2208       Ioc t (t + T) ** simpa using existsUnique_add_zsmul_mem_Ioc hT x t ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.measurable_condCdfRat ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) q : \u211a \u22a2 Measurable fun a => condCdfRat \u03c1 a q ** simp_rw [condCdfRat, ite_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) q : \u211a \u22a2 Measurable fun a => if a \u2208 condCdfSet \u03c1 then ENNReal.toReal (preCdf \u03c1 q a) else if q < 0 then 0 else 1 ** exact\n  Measurable.ite (measurableSet_condCdfSet \u03c1) measurable_preCdf.ennreal_toReal\n    measurable_const ** Qed",
        "informal": ""
    },
    {
        "formal": "QPF.Cofix.bisim' ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u_1 Q : \u03b1 \u2192 Prop u v : \u03b1 \u2192 Cofix F h :   \u2200 (x : \u03b1),     Q x \u2192       \u2203 a f f',         dest (u x) = abs { fst := a, snd := f } \u2227           dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x' x\u271d\u00b9 : \u03b1 Qx : Q x\u271d\u00b9 R : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x' x y : Cofix F x\u271d : R x y x' : \u03b1 Qx' : Q x' xeq : x = u x' yeq : y = v x' \u22a2 Liftr R (dest x) (dest y) ** rcases h x' Qx' with \u27e8a, f, f', ux'eq, vx'eq, h'\u27e9 ** case intro.intro.intro.intro.intro F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u_1 Q : \u03b1 \u2192 Prop u v : \u03b1 \u2192 Cofix F h :   \u2200 (x : \u03b1),     Q x \u2192       \u2203 a f f',         dest (u x) = abs { fst := a, snd := f } \u2227           dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x' x\u271d\u00b9 : \u03b1 Qx : Q x\u271d\u00b9 R : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x' x y : Cofix F x\u271d : R x y x' : \u03b1 Qx' : Q x' xeq : x = u x' yeq : y = v x' a : (P F).A f f' : PFunctor.B (P F) a \u2192 Cofix F ux'eq : dest (u x') = abs { fst := a, snd := f } vx'eq : dest (v x') = abs { fst := a, snd := f' } h' : \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x' \u22a2 Liftr R (dest x) (dest y) ** rw [liftr_iff] ** case intro.intro.intro.intro.intro F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u_1 Q : \u03b1 \u2192 Prop u v : \u03b1 \u2192 Cofix F h :   \u2200 (x : \u03b1),     Q x \u2192       \u2203 a f f',         dest (u x) = abs { fst := a, snd := f } \u2227           dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x' x\u271d\u00b9 : \u03b1 Qx : Q x\u271d\u00b9 R : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x' x y : Cofix F x\u271d : R x y x' : \u03b1 Qx' : Q x' xeq : x = u x' yeq : y = v x' a : (P F).A f f' : PFunctor.B (P F) a \u2192 Cofix F ux'eq : dest (u x') = abs { fst := a, snd := f } vx'eq : dest (v x') = abs { fst := a, snd := f' } h' : \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x' \u22a2 \u2203 a f\u2080 f\u2081,     dest x = abs { fst := a, snd := f\u2080 } \u2227       dest y = abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : PFunctor.B (P F) a), R (f\u2080 i) (f\u2081 i) ** refine' \u27e8a, f, f', xeq.symm \u25b8 ux'eq, yeq.symm \u25b8 vx'eq, h'\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.norm_le_mul_norm_of_ae_le_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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : \u211d f : { x // x \u2208 Lp E p } g : { x // x \u2208 Lp F p } h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016 \u22a2 \u2016f\u2016 \u2264 c * \u2016g\u2016 ** cases' le_or_lt 0 c with hc hc ** case inl \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 : \u211d f : { x // x \u2208 Lp E p } g : { x // x \u2208 Lp F p } h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016 hc : 0 \u2264 c \u22a2 \u2016f\u2016 \u2264 c * \u2016g\u2016 ** lift c to \u211d\u22650 using hc ** case inl.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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : { x // x \u2208 Lp E p } g : { x // x \u2208 Lp F p } c : \u211d\u22650 h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 \u2191c * \u2016\u2191\u2191g x\u2016 \u22a2 \u2016f\u2016 \u2264 \u2191c * \u2016g\u2016 ** exact NNReal.coe_le_coe.mpr (nnnorm_le_mul_nnnorm_of_ae_le_mul h) ** case inr \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 : \u211d f : { x // x \u2208 Lp E p } g : { x // x \u2208 Lp F p } h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016 hc : c < 0 \u22a2 \u2016f\u2016 \u2264 c * \u2016g\u2016 ** simp only [norm_def] ** case inr \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 : \u211d f : { x // x \u2208 Lp E p } g : { x // x \u2208 Lp F p } h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016 hc : c < 0 \u22a2 ENNReal.toReal (snorm (\u2191\u2191f) p \u03bc) \u2264 c * ENNReal.toReal (snorm (\u2191\u2191g) p \u03bc) ** have := snorm_eq_zero_and_zero_of_ae_le_mul_neg h hc p ** case inr \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 : \u211d f : { x // x \u2208 Lp E p } g : { x // x \u2208 Lp F p } h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 c * \u2016\u2191\u2191g x\u2016 hc : c < 0 this : snorm (fun x => \u2191\u2191f x) p \u03bc = 0 \u2227 snorm (fun x => \u2191\u2191g x) p \u03bc = 0 \u22a2 ENNReal.toReal (snorm (\u2191\u2191f) p \u03bc) \u2264 c * ENNReal.toReal (snorm (\u2191\u2191g) p \u03bc) ** simp [this] ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_vector_succ ** \u03b1 : Type u_1 m n : \u2115 f : Vector3 \u03b1 (succ n) \u2192 Prop x\u271d : Exists f v : Vector3 \u03b1 (succ n) fv : f v \u22a2 f (?m.41426 f x\u271d v fv :: ?m.41427 f x\u271d v fv) ** rw [cons_head_tail v] ** \u03b1 : Type u_1 m n : \u2115 f : Vector3 \u03b1 (succ n) \u2192 Prop x\u271d : Exists f v : Vector3 \u03b1 (succ n) fv : f v \u22a2 f v ** exact fv ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.drop ** 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.drop 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.drop 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.drop s n) ** simp [Substring.drop, 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 + { byteIdx := utf8Len (List.take n m) }, stopPos := s.stopPos } ** 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 } + { byteIdx := utf8Len (List.take n m) },       stopPos := { byteIdx := utf8Len l + utf8Len 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 } + { byteIdx := utf8Len (List.take n m) },       stopPos := { byteIdx := utf8Len l + utf8Len 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 } + { byteIdx := utf8Len (List.take n m) },           stopPos := { byteIdx := utf8Len l + utf8Len m } }.stopPos.byteIdx =     utf8Len (l ++ List.take n m) + utf8Len (List.drop n m) ** 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 } + { byteIdx := utf8Len (List.take n (List.take n m ++ List.drop n m)) },           stopPos := { byteIdx := utf8Len l + utf8Len (List.take n m ++ List.drop n m) } }.stopPos.byteIdx =     utf8Len (l ++ List.take n m) + utf8Len (List.drop n m) ** 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 } + { byteIdx := utf8Len (List.take n m) },           stopPos := { byteIdx := utf8Len l + utf8Len m } }.str.data =     l ++ List.take n m ++ List.drop n m ++ r ** 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 } + { byteIdx := utf8Len (List.take n m) },           stopPos := { byteIdx := utf8Len l + utf8Len m } }.startPos.byteIdx =     utf8Len (l ++ List.take n m) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Holor.sum_unitVec_mul_slice ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) \u22a2 \u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d) = x ** apply slice_eq _ _ _ ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) \u22a2 slice (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d)) = slice x ** ext i hid ** case h.h \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d \u22a2 slice (\u2211 i in Finset.attach (Finset.range d), unitVec d \u2191i \u2297 slice x \u2191i (_ : Nat.succ \u2191i \u2264 d)) i hid = slice x i hid ** rw [\u2190 slice_sum] ** case h.h \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d \u22a2 \u2211 x_1 in Finset.attach (Finset.range d), slice (unitVec d \u2191x_1 \u2297 slice x \u2191x_1 (_ : Nat.succ \u2191x_1 \u2264 d)) i hid =     slice x i hid ** simp only [slice_unitVec_mul hid] ** case h.h \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d \u22a2 (\u2211 x_1 in Finset.attach (Finset.range d), if i = \u2191x_1 then slice x \u2191x_1 (_ : Nat.succ \u2191x_1 \u2264 d) else 0) =     slice x i hid ** rw [Finset.sum_eq_single (Subtype.mk i <| Finset.mem_range.2 hid)] ** case h.h \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d \u22a2 (if i = \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } then       slice x \u2191{ val := i, property := (_ : i \u2208 Finset.range d) }         (_ : Nat.succ \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } \u2264 d)     else 0) =     slice x i hid ** simp ** case h.h.h\u2080 \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d \u22a2 \u2200 (b : { x // x \u2208 Finset.range d }),     b \u2208 Finset.attach (Finset.range d) \u2192       b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) } \u2192         (if i = \u2191b then slice x \u2191b (_ : Nat.succ \u2191b \u2264 d) else 0) = 0 ** intro (b : { x // x \u2208 Finset.range d }) (_ : b \u2208 (Finset.range d).attach) (hbi : b \u2260 \u27e8i, _\u27e9) ** case h.h.h\u2080 \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d b : { x // x \u2208 Finset.range d } x\u271d : b \u2208 Finset.attach (Finset.range d) hbi : b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) } \u22a2 (if i = \u2191b then slice x \u2191b (_ : Nat.succ \u2191b \u2264 d) else 0) = 0 ** have hbi' : i \u2260 b := by simpa only [Ne.def, Subtype.ext_iff, Subtype.coe_mk] using hbi.symm ** case h.h.h\u2080 \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d b : { x // x \u2208 Finset.range d } x\u271d : b \u2208 Finset.attach (Finset.range d) hbi : b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) } hbi' : i \u2260 \u2191b \u22a2 (if i = \u2191b then slice x \u2191b (_ : Nat.succ \u2191b \u2264 d) else 0) = 0 ** simp [hbi'] ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d b : { x // x \u2208 Finset.range d } x\u271d : b \u2208 Finset.attach (Finset.range d) hbi : b \u2260 { val := i, property := (_ : i \u2208 Finset.range d) } \u22a2 i \u2260 \u2191b ** simpa only [Ne.def, Subtype.ext_iff, Subtype.coe_mk] using hbi.symm ** case h.h.h\u2081 \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d \u22a2 \u00ac{ val := i, property := (_ : i \u2208 Finset.range d) } \u2208 Finset.attach (Finset.range d) \u2192     (if i = \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } then         slice x \u2191{ val := i, property := (_ : i \u2208 Finset.range d) }           (_ : Nat.succ \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } \u2264 d)       else 0) =       0 ** intro (hid' : Subtype.mk i _ \u2209 Finset.attach (Finset.range d)) ** case h.h.h\u2081 \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d hid' : \u00ac{ val := i, property := (_ : i \u2208 Finset.range d) } \u2208 Finset.attach (Finset.range d) \u22a2 (if i = \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } then       slice x \u2191{ val := i, property := (_ : i \u2208 Finset.range d) }         (_ : Nat.succ \u2191{ val := i, property := (_ : i \u2208 Finset.range d) } \u2264 d)     else 0) =     0 ** exfalso ** case h.h.h\u2081.h \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 (d :: ds) i : \u2115 hid : i < d hid' : \u00ac{ val := i, property := (_ : i \u2208 Finset.range d) } \u2208 Finset.attach (Finset.range d) \u22a2 False ** exact absurd (Finset.mem_attach _ _) hid' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.stronglyMeasurable_const' ** \u03b1\u271d : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_5 \u03b2 : Type u_6 m : MeasurableSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : \u2200 (x y : \u03b1), f x = f y \u22a2 StronglyMeasurable f ** cases' isEmpty_or_nonempty \u03b1 with _ h ** case inl \u03b1\u271d : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_5 \u03b2 : Type u_6 m : MeasurableSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : \u2200 (x y : \u03b1), f x = f y h\u271d : IsEmpty \u03b1 \u22a2 StronglyMeasurable f ** exact stronglyMeasurable_of_isEmpty f ** case inr \u03b1\u271d : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_5 \u03b2 : Type u_6 m : MeasurableSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : \u2200 (x y : \u03b1), f x = f y h : Nonempty \u03b1 \u22a2 StronglyMeasurable f ** convert stronglyMeasurable_const (\u03b2 := \u03b2) using 1 ** case h.e'_5 \u03b1\u271d : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_5 \u03b2 : Type u_6 m : MeasurableSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : \u2200 (x y : \u03b1), f x = f y h : Nonempty \u03b1 \u22a2 f = fun x => ?inr.convert_3  case inr.convert_3 \u03b1\u271d : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 \u03b1 : Type u_5 \u03b2 : Type u_6 m : MeasurableSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : \u2200 (x y : \u03b1), f x = f y h : Nonempty \u03b1 \u22a2 \u03b2 ** exact funext fun x => hf x h.some ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.abs_toReal_measure_sub_le_measure_symmDiff' ** \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 hs : MeasurableSet s ht : MeasurableSet t hs' : \u2191\u2191\u03bc s \u2260 \u22a4 ht' : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t)) ** have hst : \u03bc (s \\ t) \u2260 \u221e := (measure_lt_top_of_subset (diff_subset s t) hs').ne ** \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 hs : MeasurableSet s ht : MeasurableSet t hs' : \u2191\u2191\u03bc s \u2260 \u22a4 ht' : \u2191\u2191\u03bc t \u2260 \u22a4 hst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4 \u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t)) ** have hts : \u03bc (t \\ s) \u2260 \u221e := (measure_lt_top_of_subset (diff_subset t s) ht').ne ** \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 hs : MeasurableSet s ht : MeasurableSet t hs' : \u2191\u2191\u03bc s \u2260 \u22a4 ht' : \u2191\u2191\u03bc t \u2260 \u22a4 hst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4 hts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4 \u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t)) ** suffices : (\u03bc s).toReal - (\u03bc t).toReal = (\u03bc (s \\ t)).toReal - (\u03bc (t \\ s)).toReal ** case this \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 hs : MeasurableSet s ht : MeasurableSet t hs' : \u2191\u2191\u03bc s \u2260 \u22a4 ht' : \u2191\u2191\u03bc t \u2260 \u22a4 hst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4 hts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s)) ** rw [measure_diff' s ht ht', measure_diff' t hs hs',\n  ENNReal.toReal_sub_of_le measure_le_measure_union_right (measure_union_ne_top hs' ht'),\n  ENNReal.toReal_sub_of_le measure_le_measure_union_right (measure_union_ne_top ht' hs'),\n  union_comm t s] ** case this \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 hs : MeasurableSet s ht : MeasurableSet t hs' : \u2191\u2191\u03bc s \u2260 \u22a4 ht' : \u2191\u2191\u03bc t \u2260 \u22a4 hst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4 hts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) =     ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc t) - (ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc s)) ** abel ** \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 hs : MeasurableSet s ht : MeasurableSet t hs' : \u2191\u2191\u03bc s \u2260 \u22a4 ht' : \u2191\u2191\u03bc t \u2260 \u22a4 hst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4 hts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4 this : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s)) \u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t)) ** rw [this, measure_symmDiff_eq hs ht, ENNReal.toReal_add hst hts] ** \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 hs : MeasurableSet s ht : MeasurableSet t hs' : \u2191\u2191\u03bc s \u2260 \u22a4 ht' : \u2191\u2191\u03bc t \u2260 \u22a4 hst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4 hts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4 this : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s)) \u22a2 |ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))| \u2264     ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) + ENNReal.toReal (\u2191\u2191\u03bc (t \\ s)) ** convert abs_sub (\u03bc (s \\ t)).toReal (\u03bc (t \\ s)).toReal <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.mem_finsupp_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f : \u03b9 \u2192\u2080 \u03b1 t : \u03b9 \u2192 Finset \u03b1 \u22a2 f \u2208 Finset.finsupp s t \u2194 f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i ** refine' mem_map.trans \u27e8_, _\u27e9 ** case refine'_1 \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f : \u03b9 \u2192\u2080 \u03b1 t : \u03b9 \u2192 Finset \u03b1 \u22a2 (\u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } a = f) \u2192     f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i ** rintro \u27e8f, hf, rfl\u27e9 ** case refine'_1.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 t : \u03b9 \u2192 Finset \u03b1 f : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1 hf : f \u2208 pi s t \u22a2 (\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f).support \u2286 s \u2227     \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i \u2208 t i ** refine' \u27e8support_indicator_subset _ _, fun i hi => _\u27e9 ** case refine'_1.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 t : \u03b9 \u2192 Finset \u03b1 f : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1 hf : f \u2208 pi s t i : \u03b9 hi : i \u2208 s \u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i \u2208 t i ** convert mem_pi.1 hf i hi ** case h.e'_4 \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 t : \u03b9 \u2192 Finset \u03b1 f : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1 hf : f \u2208 pi s t i : \u03b9 hi : i \u2208 s \u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i = f i hi ** exact indicator_of_mem hi _ ** case refine'_2 \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f : \u03b9 \u2192\u2080 \u03b1 t : \u03b9 \u2192 Finset \u03b1 \u22a2 (f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i) \u2192     \u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } a = f ** refine' fun h => \u27e8fun i _ => f i, mem_pi.2 h.2, _\u27e9 ** case refine'_2 \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f : \u03b9 \u2192\u2080 \u03b1 t : \u03b9 \u2192 Finset \u03b1 h : f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i \u22a2 (\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } fun i x => \u2191f i) = f ** ext i ** case refine'_2.h \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f : \u03b9 \u2192\u2080 \u03b1 t : \u03b9 \u2192 Finset \u03b1 h : f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i i : \u03b9 \u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } fun i x => \u2191f i) i = \u2191f i ** exact ite_eq_left_iff.2 fun hi => (not_mem_support_iff.1 fun H => hi <| h.1 H).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.withDensity\u1d65_absolutelyContinuous ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d \u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : Integrable f \u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc ** refine' VectorMeasure.AbsolutelyContinuous.mk fun i hi\u2081 hi\u2082 => _ ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : Integrable f i : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : \u2191(toENNRealVectorMeasure \u03bc) i = 0 \u22a2 \u2191(withDensity\u1d65 \u03bc f) i = 0 ** rw [toENNRealVectorMeasure_apply_measurable hi\u2081] at hi\u2082 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : Integrable f i : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : \u2191\u2191\u03bc i = 0 \u22a2 \u2191(withDensity\u1d65 \u03bc f) i = 0 ** rw [withDensity\u1d65_apply hf hi\u2081, Measure.restrict_zero_set hi\u2082, integral_zero_measure] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : \u00acIntegrable f \u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc ** rw [withDensity\u1d65, dif_neg hf] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d hf : \u00acIntegrable f \u22a2 0 \u226a\u1d65 toENNRealVectorMeasure \u03bc ** exact VectorMeasure.AbsolutelyContinuous.zero _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv ** E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL \u22a2 \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b ** ext1 x ** case h E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL x : F \u22a2 x \u2208 \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) \u2194 x \u2208 Set.Icc a b ** simp only [Set.mem_preimage, Set.mem_Icc, he_ord] ** E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b \u22a2 Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b ** rw [\u2190 hIcc, eL.symm_preimage_preimage] ** E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b \u22a2 \u222b (x : F) in Set.Icc a b, DF x =     \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) ** simp only [hDF] ** E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b \u22a2 \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =     \u222b (x : (fun x => Fin (n + 1) \u2192 \u211d) a) in Set.Icc (\u2191eL a) (\u2191eL b),       \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) ** rw [\u2190 he_vol.set_integral_preimage_emb he_emb] ** E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b \u22a2 \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =     \u222b (x : F) in \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b),       \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) (\u2191eL x))) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) ** simp only [hIcc, eL.symm_apply_apply] ** E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b \u22a2 \u222b (x : (fun x => Fin (n + 1) \u2192 \u211d) a) in Set.Icc (\u2191eL a) (\u2191eL b),       \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (\u2191eL a \u2218 Fin.succAbove i) (\u2191eL b \u2218 Fin.succAbove i),           f i (\u2191(ContinuousLinearEquiv.symm eL) (Fin.insertNth i (\u2191eL b i) x))) -         \u222b (x : Fin n \u2192 \u211d) in Set.Icc (\u2191eL a \u2218 Fin.succAbove i) (\u2191eL b \u2218 Fin.succAbove i),           f i (\u2191(ContinuousLinearEquiv.symm eL) (Fin.insertNth i (\u2191eL a i) x))) ** refine integral_divergence_of_hasFDerivWithinAt_off_countable' (eL a) (eL b)\n  ((he_ord _ _).2 hle) (fun i x => f i (eL.symm x))\n  (fun i x => f' i (eL.symm x) \u2218L (eL.symm : \u211d\u207f\u207a\u00b9 \u2192L[\u211d] F)) (eL.symm \u207b\u00b9' s)\n  (hs.preimage eL.symm.injective) ?_ ?_ ?_ ** case refine_1 E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b \u22a2 \u2200 (i : Fin (n + 1)), ContinuousOn ((fun i x => f i (\u2191(ContinuousLinearEquiv.symm eL) x)) i) (Set.Icc (\u2191eL a) (\u2191eL b)) ** exact fun i => (Hc i).comp eL.symm.continuousOn hIcc'.subset ** case refine_2 E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b \u22a2 \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s \u2192       \u2200 (i : Fin (n + 1)),         HasFDerivAt ((fun i x => f i (\u2191(ContinuousLinearEquiv.symm eL) x)) i)           ((fun i x =>               ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))             i x)           x ** refine' fun x hx i => (Hd (eL.symm x) \u27e8_, hx.2\u27e9 i).comp x eL.symm.hasFDerivAt ** case refine_2 E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b x : Fin (n + 1) \u2192 \u211d hx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s i : Fin (n + 1) \u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 interior (Set.Icc a b) ** rw [\u2190 hIcc] ** case refine_2 E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b x : Fin (n + 1) \u2192 \u211d hx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s i : Fin (n + 1) \u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 interior (\u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b)) ** refine' preimage_interior_subset_interior_preimage eL.continuous _ ** case refine_2 E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b x : Fin (n + 1) \u2192 \u211d hx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s i : Fin (n + 1) \u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 \u2191eL \u207b\u00b9' interior (Set.Icc (\u2191eL a) (\u2191eL b)) ** simpa only [Set.mem_preimage, eL.apply_symm_apply, \u2190 pi_univ_Icc,\n  interior_pi_set (@finite_univ (Fin _) _), interior_Icc] using hx.1 ** case refine_3 E : Type u inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : CompleteSpace E n : \u2115 F : Type u_1 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : PartialOrder F inst\u271d\u00b9 : MeasureSpace F inst\u271d : BorelSpace F eL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d he_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y he_vol : MeasurePreserving \u2191eL f : Fin (n + 1) \u2192 F \u2192 E f' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E s : Set F hs : Set.Countable s a b : F hle : a \u2264 b Hc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b) Hd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x DF : F \u2192 E hDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) Hi : IntegrableOn DF (Set.Icc a b) he_emb : MeasurableEmbedding \u2191eL hIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b hIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b \u22a2 IntegrableOn     (fun x =>       \u2211 i : Fin (n + 1),         \u2191((fun i x =>                 ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))               i x)           (e i))     (Set.Icc (\u2191eL a) (\u2191eL b)) ** rw [\u2190 he_vol.integrableOn_comp_preimage he_emb, hIcc] ** Qed",
        "informal": ""
    },
    {
        "formal": "WithTop.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 (Subset.trans Ico_subset_Iio_self <| Iio_subset_Iio le_top)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Continuous.isOpenPosMeasure_map ** X : Type u_1 Y : Type u_2 inst\u271d\u2077 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace Y inst\u271d\u2075 : T2Space Y \u03bc \u03bd : Measure X inst\u271d\u2074 : IsOpenPosMeasure \u03bc s U F : Set X x : X inst\u271d\u00b3 : OpensMeasurableSpace X Z : Type u_3 inst\u271d\u00b2 : TopologicalSpace Z inst\u271d\u00b9 : MeasurableSpace Z inst\u271d : BorelSpace Z f : X \u2192 Z hf : Continuous f hf_surj : Surjective f \u22a2 IsOpenPosMeasure (map f \u03bc) ** refine' \u27e8fun U hUo hUne => _\u27e9 ** X : Type u_1 Y : Type u_2 inst\u271d\u2077 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace Y inst\u271d\u2075 : T2Space Y \u03bc \u03bd : Measure X inst\u271d\u2074 : IsOpenPosMeasure \u03bc s U\u271d F : Set X x : X inst\u271d\u00b3 : OpensMeasurableSpace X Z : Type u_3 inst\u271d\u00b2 : TopologicalSpace Z inst\u271d\u00b9 : MeasurableSpace Z inst\u271d : BorelSpace Z f : X \u2192 Z hf : Continuous f hf_surj : Surjective f U : Set Z hUo : IsOpen U hUne : Set.Nonempty U \u22a2 \u2191\u2191(map f \u03bc) U \u2260 0 ** rw [Measure.map_apply hf.measurable hUo.measurableSet] ** X : Type u_1 Y : Type u_2 inst\u271d\u2077 : TopologicalSpace X m : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace Y inst\u271d\u2075 : T2Space Y \u03bc \u03bd : Measure X inst\u271d\u2074 : IsOpenPosMeasure \u03bc s U\u271d F : Set X x : X inst\u271d\u00b3 : OpensMeasurableSpace X Z : Type u_3 inst\u271d\u00b2 : TopologicalSpace Z inst\u271d\u00b9 : MeasurableSpace Z inst\u271d : BorelSpace Z f : X \u2192 Z hf : Continuous f hf_surj : Surjective f U : Set Z hUo : IsOpen U hUne : Set.Nonempty U \u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' U) \u2260 0 ** exact (hUo.preimage hf).measure_ne_zero \u03bc (hf_surj.nonempty_preimage.mpr hUne) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.tsum_eapproxDiff ** \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 \u2211' (n : \u2115), \u2191(\u2191(eapproxDiff f n) a) = f a ** simp_rw [ENNReal.tsum_eq_iSup_nat' (tendsto_add_atTop_nat 1), sum_eapproxDiff,\n  iSup_eapprox_apply f hf a] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.martingale_const ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f g : \u03b9 \u2192 \u03a9 \u2192 E \u2131\u271d \u2131 : Filtration \u03b9 m0 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc x : E i j : \u03b9 x\u271d : i \u2264 j \u22a2 \u03bc[(fun x_1 x_2 => x) j|\u2191\u2131 i] =\u1d50[\u03bc] (fun x_1 x_2 => x) i ** rw [condexp_const (\u2131.le _)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.exists_insert_toList_zoom_nil ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering p : Path \u03b1 v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n e : zoom (cmp v) t Path.root = (nil, p) \u22a2 toList t = Path.listL p ++ Path.listR p \u2227 toList (insert cmp t v) = Path.listL p ++ v :: Path.listR p ** simp [\u2190 zoom_toList e, insert_toList_zoom_nil ht e] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.Range.forIn_eq_forIn_range' ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) \u22a2 forIn r init f = forIn (List.range' r.start (numElems r) r.step) init f ** refine Eq.trans ?_ <| (forIn'_eq_forIn_range' r init (fun x _ => f x)).trans ?_ ** case refine_1 m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) \u22a2 forIn r init f = forIn' r init fun x x_1 => f x ** simp [forIn, forIn', Range.forIn, Range.forIn'] ** case refine_1 m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) \u22a2 forIn.loop f r.stop r.start r.stop r.step init =     forIn'.loop r.start r.stop r.step (fun x x_1 => f x) r.stop r.start (_ : r.start \u2264 r.start) init ** suffices \u2200 fuel i hl b, forIn'.loop r.start r.stop r.step (fun x _ => f x) fuel i hl b =\n    forIn.loop f fuel i r.stop r.step b from (this _ ..).symm ** case refine_1 m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) \u22a2 \u2200 (fuel i : Nat) (hl : r.start \u2264 i) (b : \u03b2),     forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b ** intro fuel ** case refine_1 m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) fuel : Nat \u22a2 \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),     forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b ** induction fuel <;> intro i hl b <;>\nunfold forIn.loop forIn'.loop <;> simp [*] <;> split <;> try simp ** case refine_1.zero.inr m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) i : Nat hl : r.start \u2264 i b : \u03b2 h\u271d : \u00aci < r.stop \u22a2 pure b = pure b ** simp ** case refine_1.succ.inl m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) n\u271d : Nat n_ih\u271d :   \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),     forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b i : Nat hl : r.start \u2264 i b : \u03b2 h\u271d : i < r.stop \u22a2 (do       let __do_lift \u2190 f i b       match __do_lift with         | ForInStep.done b => pure b         | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b) =     if i \u2265 r.stop then pure b     else do       let __do_lift \u2190 f i b       match __do_lift with         | ForInStep.done b => pure b         | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b ** simp [if_neg (Nat.not_le.2 \u2039_\u203a)] ** case refine_1.succ.inr m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) n\u271d : Nat n_ih\u271d :   \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),     forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b i : Nat hl : r.start \u2264 i b : \u03b2 h\u271d : \u00aci < r.stop \u22a2 pure b =     if i \u2265 r.stop then pure b     else do       let __do_lift \u2190 f i b       match __do_lift with         | ForInStep.done b => pure b         | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b ** simp [if_pos (Nat.not_lt.1 \u2039_\u203a)] ** case refine_2 m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) \u22a2 \u2200 (L : List Nat) (H : \u2200 (a : Nat), a \u2208 L \u2192 a \u2208 r),     (forIn (List.pmap Subtype.mk L H) init fun x => f x.val) = forIn L init f ** intro L ** case refine_2 m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 inst\u271d : Monad m r : Range init : \u03b2 f : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2) L : List Nat \u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 L \u2192 a \u2208 r), (forIn (List.pmap Subtype.mk L H) init fun x => f x.val) = forIn L init f ** induction L generalizing init <;> intro H <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.toNat_lt ** n : Nat z : Int h : 0 \u2264 z \u22a2 toNat z < n \u2194 z < \u2191n ** rw [\u2190 Int.not_le, \u2190 Nat.not_le, Int.le_toNat h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_congr_ae ** \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 : f =\u1d50[\u03bc] g \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g ** by_cases hfi : 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 : f =\u1d50[\u03bc] g hfi : Integrable f \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g ** have hgi : Integrable g \u03bc := hfi.congr h ** 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 : f =\u1d50[\u03bc] g hfi : Integrable f hgi : Integrable g \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g ** rw [setToFun_eq hT hfi, setToFun_eq hT hgi, (Integrable.toL1_eq_toL1_iff f g hfi hgi).2 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 g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C h : f =\u1d50[\u03bc] g hfi : \u00acIntegrable f \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g ** have hgi : \u00acIntegrable g \u03bc := by rw [integrable_congr h] at hfi; exact hfi ** 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 : f =\u1d50[\u03bc] g hfi : \u00acIntegrable f hgi : \u00acIntegrable g \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g ** rw [setToFun_undef hT hfi, setToFun_undef hT hgi] ** \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 : f =\u1d50[\u03bc] g hfi : \u00acIntegrable f \u22a2 \u00acIntegrable g ** rw [integrable_congr h] at hfi ** \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 : f =\u1d50[\u03bc] g hfi : \u00acIntegrable g \u22a2 \u00acIntegrable g ** exact hfi ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.is_left_invariant_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) g : G K : Compacts G \u22a2 prehaar (\u2191K\u2080) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (\u2191K\u2080) U K ** simp only [prehaar, Compacts.coe_map, is_left_invariant_index K.isCompact _ hU] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.singleton_product_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s s' : Finset \u03b1 t t' : Finset \u03b2 a\u271d : \u03b1 b\u271d : \u03b2 a : \u03b1 b : \u03b2 \u22a2 {a} \u00d7\u02e2 {b} = {(a, b)} ** simp only [product_singleton, Function.Embedding.coeFn_mk, map_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.measure_limsup_eq_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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 \u22a2 \u2191\u2191\u03bc (limsup s atTop) = 1 ** rw [measure_congr (eventuallyEq_set.2 (ae_mem_limsup_atTop_iff \u03bc <|\n  measurableSet_filtrationOfSet' hsm) : (limsup s atTop : Set \u03a9) =\u1d50[\u03bc]\n    {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n,\n      (\u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm k]) \u03c9) atTop atTop})] ** \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 \u22a2 \u2191\u2191\u03bc       {\u03c9 |         Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop           atTop} =     1 ** suffices {\u03c9 | Tendsto (fun n => \u2211 k in Finset.range n,\n    (\u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm k]) \u03c9) atTop atTop} =\u1d50[\u03bc] Set.univ by\n  rw [measure_congr this, measure_univ] ** \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 \u22a2 {\u03c9 |       Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop         atTop} =\u1d50[\u03bc]     Set.univ ** have : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, (\u03bc[(s (n + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm n]) \u03c9 = _ :=\n  ae_all_iff.2 fun n => hs.condexp_indicator_filtrationOfSet_ae_eq hsm n.lt_succ_self ** \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u22a2 {\u03c9 |       Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop         atTop} =\u1d50[\u03bc]     Set.univ ** filter_upwards [this] with \u03c9 h\u03c9 ** case h \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) \u22a2 setOf       (fun \u03c9 =>         Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop           atTop)       \u03c9 =     Set.univ \u03c9 ** refine' eq_true (_ : Tendsto _ _ _) ** case h \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) \u22a2 Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop ** simp_rw [h\u03c9] ** case h \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) \u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop ** have htends : Tendsto (fun n => \u2211 k in Finset.range n, \u03bc (s (k + 1))) atTop (\ud835\udcdd \u221e) := by\n  rw [\u2190 ENNReal.tsum_add_one_eq_top hs' (measure_ne_top _ _)]\n  exact ENNReal.tendsto_nat_tsum _ ** case h \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4) \u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop ** rw [ENNReal.tendsto_nhds_top_iff_nnreal] at htends ** case h \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) \u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop ** refine' tendsto_atTop_atTop_of_monotone' _ _ ** \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   {\u03c9 |       Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop         atTop} =\u1d50[\u03bc]     Set.univ \u22a2 \u2191\u2191\u03bc       {\u03c9 |         Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop           atTop} =     1 ** rw [measure_congr this, measure_univ] ** \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) \u22a2 Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4) ** rw [\u2190 ENNReal.tsum_add_one_eq_top hs' (measure_ne_top _ _)] ** \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) \u22a2 Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd (\u2211' (n : \u2115), \u2191\u2191\u03bc (s (n + 1)))) ** exact ENNReal.tendsto_nat_tsum _ ** case h.refine'_1 \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) \u22a2 Monotone fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))) ** refine' monotone_nat_of_le_succ fun n => _ ** case h.refine'_1 \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) n : \u2115 \u22a2 \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))) \u2264     \u2211 x in Finset.range (n + 1), ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))) ** rw [\u2190 sub_nonneg, Finset.sum_range_succ_sub_sum] ** case h.refine'_1 \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) n : \u2115 \u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) ** exact ENNReal.toReal_nonneg ** case h.refine'_2 \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) \u22a2 \u00acBddAbove (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) ** rintro \u27e8B, hB\u27e9 ** case h.refine'_2.intro \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d hB : B \u2208 upperBounds (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u22a2 False ** refine' not_eventually.2 (frequently_of_forall fun n => _) (htends B.toNNReal) ** case h.refine'_2.intro \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d hB : B \u2208 upperBounds (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) n : \u2115 \u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1)) ** rw [mem_upperBounds] at hB ** case h.refine'_2.intro \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d hB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B n : \u2115 \u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1)) ** specialize hB (\u2211 k : \u2115 in Finset.range n, \u03bc (s (k + 1))).toReal _ ** case h.refine'_2.intro \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d hB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B n : \u2115 \u22a2 ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2208     Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))) ** refine' \u27e8n, _\u27e9 ** case h.refine'_2.intro \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d hB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B n : \u2115 \u22a2 (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) n =     ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) ** rw [ENNReal.toReal_sum] ** case h.refine'_2.intro \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d hB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B n : \u2115 \u22a2 \u2200 (a : \u2115), a \u2208 Finset.range n \u2192 \u2191\u2191\u03bc (s (a + 1)) \u2260 \u22a4 ** exact fun _ _ => measure_ne_top _ _ ** case h.refine'_2.intro \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d n : \u2115 hB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B \u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1)) ** rw [not_lt, \u2190 ENNReal.toReal_le_toReal (ENNReal.sum_lt_top _).ne ENNReal.coe_ne_top] ** case h.refine'_2.intro \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d n : \u2115 hB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B \u22a2 ENNReal.toReal (\u2211 a in Finset.range n, \u2191\u2191\u03bc (s (a + 1))) \u2264 ENNReal.toReal \u2191(Real.toNNReal B) ** exact hB.trans (by simp) ** \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d n : \u2115 hB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B \u22a2 B \u2264 ENNReal.toReal \u2191(Real.toNNReal B) ** simp ** \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\u271d : \u03b9 \u2192 Set \u03a9 s : \u2115 \u2192 Set \u03a9 hsm : \u2200 (n : \u2115), MeasurableSet (s n) hs : iIndepSet s hs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4 this :   \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,     \u2200 (n : \u2115),       (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9 \u03c9 : \u03a9 h\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1))) htends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1)) B : \u211d n : \u2115 hB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B \u22a2 \u2200 (a : \u2115), a \u2208 Finset.range n \u2192 \u2191\u2191\u03bc (s (a + 1)) \u2260 \u22a4 ** exact fun _ _ => measure_ne_top _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_pos ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 0 < encard s \u2194 Set.Nonempty s ** rw [pos_iff_ne_zero, encard_ne_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "ack_three ** n : \u2115 \u22a2 ack 3 n = 2 ^ (n + 3) - 3 ** induction' n with n IH ** case zero  \u22a2 ack 3 zero = 2 ^ (zero + 3) - 3 ** rfl ** case succ n : \u2115 IH : ack 3 n = 2 ^ (n + 3) - 3 \u22a2 ack 3 (succ n) = 2 ^ (succ n + 3) - 3 ** rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,\n    Nat.mul_sub_left_distrib, \u2190 Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right] ** case succ n : \u2115 IH : ack 3 n = 2 ^ (n + 3) - 3 \u22a2 2 * 3 \u2264 2 * 2 ^ (n + 3) ** have H : 2 * 3 \u2264 2 * 2 ^ 3 := by norm_num ** case succ n : \u2115 IH : ack 3 n = 2 ^ (n + 3) - 3 H : 2 * 3 \u2264 2 * 2 ^ 3 \u22a2 2 * 3 \u2264 2 * 2 ^ (n + 3) ** apply H.trans ** case succ n : \u2115 IH : ack 3 n = 2 ^ (n + 3) - 3 H : 2 * 3 \u2264 2 * 2 ^ 3 \u22a2 2 * 2 ^ 3 \u2264 2 * 2 ^ (n + 3) ** simp [pow_le_pow] ** n : \u2115 IH : ack 3 n = 2 ^ (n + 3) - 3 \u22a2 2 * 3 \u2264 2 * 2 ^ 3 ** norm_num ** Qed",
        "informal": ""
    },
    {
        "formal": "measurableSet_pi ** \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 s : Set \u03b4 t : (i : \u03b4) \u2192 Set (\u03c0 i) hs : Set.Countable s \u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205 ** cases' (pi s t).eq_empty_or_nonempty with h h ** case inl \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 s : Set \u03b4 t : (i : \u03b4) \u2192 Set (\u03c0 i) hs : Set.Countable s h : Set.pi s t = \u2205 \u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205 ** simp [h] ** case inr \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 s : Set \u03b4 t : (i : \u03b4) \u2192 Set (\u03c0 i) hs : Set.Countable s h : Set.Nonempty (Set.pi s t) \u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205 ** simp [measurableSet_pi_of_nonempty hs, h, \u2190 not_nonempty_iff_eq_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.DominatedFinMeasAdditive.eq_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\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\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2\u271d : Type u_7 inst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\u271d T\u271d T' : Set \u03b1 \u2192 \u03b2\u271d C\u271d C' : \u211d \u03b2 : Type u_8 inst\u271d : NormedAddCommGroup \u03b2 T : Set \u03b1 \u2192 \u03b2 C : \u211d m : MeasurableSpace \u03b1 hT : DominatedFinMeasAdditive 0 T C s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u21910 s = 0 ** simp only [Measure.coe_zero, Pi.zero_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.totalDegree_add_eq_right_of_totalDegree_lt ** 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 : totalDegree q < totalDegree p \u22a2 totalDegree (q + p) = totalDegree p ** rw [add_comm, totalDegree_add_eq_left_of_totalDegree_lt h] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.swapRight_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 \u00d7 \u03b2) hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(swapRight \u03ba) a) s = \u2191\u2191(\u2191\u03ba a) {p | Prod.swap p \u2208 s} ** rw [swapRight_apply, Measure.map_apply measurable_swap 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 \u00d7 \u03b2) hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191\u03ba a) (Prod.swap \u207b\u00b9' s) = \u2191\u2191(\u2191\u03ba a) {p | Prod.swap p \u2208 s} ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.mem_uIcc' ** \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 a \u2208 [[b, c]] \u2194 b \u2264 a \u2227 a \u2264 c \u2228 c \u2264 a \u2227 a \u2264 b ** simp [uIcc_eq_union] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal ** \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\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 ** have fmeas : AEMeasurable f \u03bc := by\n  convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable\n  simp only [Real.toNNReal_coe] ** \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\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 fmeas : AEMeasurable f \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 ** lift \u03b5 to \u211d\u22650 using \u03b5pos.le ** case 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u03b5\u27e9 : \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227 \u03b4 < \u03b5 ** \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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 < \u03b5  case 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** exact exists_between \u03b5pos ** case 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** have int_f_ne_top : (\u222b\u207b a : \u03b1, f a \u2202\u03bc) \u2260 \u221e :=\n  (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne ** case 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable \u03bc f fmeas\n    (ENNReal.coe_ne_zero.2 \u03b4pos.ne') with\n  \u27e8g, f_lt_g, gcont, gint\u27e9 ** case 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** have gint_ne : (\u222b\u207b x : \u03b1, g x \u2202\u03bc) \u2260 \u221e := ne_top_of_le_ne_top (by simpa) gint ** case 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** have g_lt_top : \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u221e := ae_lt_top gcont.measurable gint_ne ** case 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** have Ig : (\u222b\u207b a : \u03b1, ENNReal.ofReal (g a).toReal \u2202\u03bc) = \u222b\u207b a : \u03b1, g a \u2202\u03bc := by\n  apply lintegral_congr_ae\n  filter_upwards [g_lt_top] with _ hx\n  simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff] ** case 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227           (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** refine' \u27e8g, f_lt_g, gcont, g_lt_top, _, _\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\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 AEMeasurable f ** convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable ** 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\u22650 fint : Integrable fun x => \u2191(f x) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 x\u271d : \u03b1 \u22a2 f x\u271d = Real.toNNReal \u2191(f x\u271d) ** simp only [Real.toNNReal_coe] ** \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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4 ** simpa ** \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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc ** apply lintegral_congr_ae ** 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 \u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (g a))) =\u1da0[ae \u03bc] fun a => g a ** filter_upwards [g_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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 a\u271d : \u03b1 hx : g a\u271d < \u22a4 \u22a2 ENNReal.ofReal (ENNReal.toReal (g a\u271d)) = g a\u271d ** simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff] ** case intro.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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 Integrable fun x => ENNReal.toReal (g x) ** refine' \u27e8gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _\u27e9 ** case intro.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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 HasFiniteIntegral fun x => ENNReal.toReal (g x) ** simp only [hasFiniteIntegral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg] ** case intro.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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc < \u22a4 ** convert gint_ne.lt_top using 1 ** case intro.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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5 ** rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae] ** case intro.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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) <     ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5 ** calc\n  ENNReal.toReal (\u222b\u207b a : \u03b1, ENNReal.ofReal (g a).toReal \u2202\u03bc) =\n      ENNReal.toReal (\u222b\u207b a : \u03b1, g a \u2202\u03bc) :=\n    by congr 1\n  _ \u2264 ENNReal.toReal ((\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4) := by\n    apply ENNReal.toReal_mono _ gint\n    simpa using int_f_ne_top\n  _ = ENNReal.toReal (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4 := by\n    rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]\n  _ < ENNReal.toReal (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b5 := (add_lt_add_left h\u03b4\u03b5 _)\n  _ = (\u222b\u207b a : \u03b1, ENNReal.ofReal \u2191(f a) \u2202\u03bc).toReal + \u03b5 := 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), g a \u2202\u03bc) ** congr 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), g a \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc + \u2191\u03b4) ** apply ENNReal.toReal_mono _ gint ** \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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4 ** simpa using int_f_ne_top ** \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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc + \u2191\u03b4) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc) + \u2191\u03b4 ** rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc) + \u2191\u03b5 = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5 ** simp ** case intro.intro.intro.intro.intro.intro.refine'_2.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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x) ** apply Filter.eventually_of_forall fun 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x ** simp ** case intro.intro.intro.intro.intro.intro.refine'_2.hfm \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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc ** exact fmeas.coe_nnreal_real.aestronglyMeasurable ** case intro.intro.intro.intro.intro.intro.refine'_2.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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 0 \u2264\u1da0[ae \u03bc] fun x => ENNReal.toReal (g x) ** apply Filter.eventually_of_forall fun 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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => ENNReal.toReal (g x)) x ** simp ** case intro.intro.intro.intro.intro.intro.refine'_2.hfm \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\u22650 fint : Integrable fun x => \u2191(f x) fmeas : AEMeasurable f \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u2191\u03b5 \u03b4 : \u211d\u22650 \u03b4pos : 0 < \u03b4 h\u03b4\u03b5 : \u03b4 < \u03b5 int_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x gcont : LowerSemicontinuous g gint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 gint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 Ig : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) \u03bc ** apply gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.natDegree_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 \u22a2 natDegree (\u2191(finSuccEquiv R n) f) = degreeOf 0 f ** by_cases c : f = 0 ** case pos 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 c : f = 0 \u22a2 natDegree (\u2191(finSuccEquiv R n) f) = degreeOf 0 f ** rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero] ** case neg 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 c : \u00acf = 0 \u22a2 natDegree (\u2191(finSuccEquiv R n) f) = degreeOf 0 f ** rw [Polynomial.natDegree, degree_finSuccEquiv (by simpa only [Ne.def] )] ** case neg 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 c : \u00acf = 0 \u22a2 WithBot.unbot' 0 \u2191(degreeOf 0 f) = degreeOf 0 f ** erw [WithBot.unbot'_coe] ** case neg 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 c : \u00acf = 0 \u22a2 \u2191(degreeOf 0 f) = degreeOf 0 f ** simp ** 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 c : \u00acf = 0 \u22a2 f \u2260 0 ** simpa only [Ne.def] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.norm_indicatorConstLp_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\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 h\u03bcs_ne_zero : \u2191\u2191\u03bc s \u2260 0 \u22a2 \u2016indicatorConstLp \u22a4 hs h\u03bcs c\u2016 = \u2016c\u2016 ** rw [Lp.norm_def, snorm_congr_ae indicatorConstLp_coeFn,\n  snorm_indicator_const' hs h\u03bcs_ne_zero ENNReal.top_ne_zero, ENNReal.top_toReal, _root_.div_zero,\n  ENNReal.rpow_zero, mul_one, ENNReal.coe_toReal, coe_nnnorm] ** 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 \u22a2 \u2203 k, Partrec k \u2227 \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) ** let \u27e8k, hk, H\u27e9 := Nat.Partrec.merge' (bind_decode\u2082_iff.1 hf) (bind_decode\u2082_iff.1 hg) ** \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) \u22a2 \u2203 k, Partrec k \u2227 \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) ** let k' (a : \u03b1) := (k (encode a)).bind fun n => (decode (\u03b1 := \u03c3) n : Part \u03c3) ** \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) \u22a2 \u2203 k, Partrec k \u2227 \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) ** refine'\n  \u27e8k', ((nat_iff.2 hk).comp Computable.encode).bind (Computable.decode.ofOption.comp snd).to\u2082,\n    fun a => _\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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 \u22a2 (\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) ** suffices ** \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 this : ?m.108648 \u22a2 (\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)  case this \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 \u22a2 ?m.108648 ** refine' \u27e8this, \u27e8fun h => (this _ \u27e8h, rfl\u27e9).imp Exists.fst Exists.fst, _\u27e9\u27e9 ** case this \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 \u22a2 \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a ** intro x h' ** case this \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 x : \u03c3 h' : x \u2208 k' a \u22a2 x \u2208 f a \u2228 x \u2208 g a ** simp only [exists_prop, mem_coe, mem_bind_iff, Option.mem_def] at h' ** case this \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 x : \u03c3 h' : \u2203 a_1, a_1 \u2208 k (encode a) \u2227 decode a_1 = Option.some x \u22a2 x \u2208 f a \u2228 x \u2208 g a ** obtain \u27e8n, hn, hx\u27e9 := h' ** case this.intro.intro \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 x : \u03c3 n : \u2115 hn : n \u2208 k (encode a) hx : decode n = Option.some x \u22a2 x \u2208 f a \u2228 x \u2208 g a ** have := (H _).1 _ hn ** case this.intro.intro \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 x : \u03c3 n : \u2115 hn : n \u2208 k (encode a) hx : decode n = Option.some x this :   (n \u2208 Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (f a)) \u2228     n \u2208 Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (g a) \u22a2 x \u2208 f a \u2228 x \u2208 g a ** simp [mem_decode\u2082, encode_injective.eq_iff] at this ** case this.intro.intro \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 x : \u03c3 n : \u2115 hn : n \u2208 k (encode a) hx : decode n = Option.some x this : (\u2203 a_1, a_1 \u2208 f a \u2227 encode a_1 = n) \u2228 \u2203 a_1, a_1 \u2208 g a \u2227 encode a_1 = n \u22a2 x \u2208 f a \u2228 x \u2208 g a ** obtain \u27e8a', ha, rfl\u27e9 | \u27e8a', ha, rfl\u27e9 := this <;> simp only [encodek, Option.some_inj] at hx <;>\n  rw [hx] at ha ** case this.intro.intro.inr.intro.intro \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 x a' : \u03c3 ha : x \u2208 g a hn : encode a' \u2208 k (encode a) hx : a' = x \u22a2 x \u2208 f a \u2228 x \u2208 g a ** exact Or.inr ha ** \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 this : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a \u22a2 (f a).Dom \u2228 (g a).Dom \u2192 (k' a).Dom ** intro h ** \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 this : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a h : (f a).Dom \u2228 (g a).Dom \u22a2 (k' a).Dom ** rw [bind_dom] ** \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 this : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a h : (f a).Dom \u2228 (g a).Dom \u22a2 \u2203 h, (\u2191(decode (Part.get (k (encode a)) h))).Dom ** have hk : (k (encode a)).Dom :=\n  (H _).2.2 (by simpa only [encodek\u2082, bind_some, coe_some] using h) ** \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 k : \u2115 \u2192. \u2115 hk\u271d : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 this : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a h : (f a).Dom \u2228 (g a).Dom hk : (k (encode a)).Dom \u22a2 \u2203 h, (\u2191(decode (Part.get (k (encode a)) h))).Dom ** exists hk ** \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 k : \u2115 \u2192. \u2115 hk\u271d : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 this : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a h : (f a).Dom \u2228 (g a).Dom hk : (k (encode a)).Dom \u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom ** simp only [exists_prop, mem_map_iff, mem_coe, mem_bind_iff, Option.mem_def] at H ** \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 k : \u2115 \u2192. \u2115 hk\u271d : Nat.Partrec k k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 this : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a h : (f a).Dom \u2228 (g a).Dom hk : (k (encode a)).Dom H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (\u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 f a_2 \u2227 encode a = x) \u2228             \u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 g a_2 \u2227 encode a = x) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) \u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom ** obtain \u27e8a', _, y, _, e\u27e9 | \u27e8a', _, y, _, e\u27e9 := (H _).1 _ \u27e8hk, rfl\u27e9 <;>\n  simp only [e.symm, encodek, coe_some, some_dom] ** \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 this : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a h : (f a).Dom \u2228 (g a).Dom \u22a2 (Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (f a)).Dom \u2228     (Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (g a)).Dom ** simpa only [encodek\u2082, bind_some, coe_some] using h ** case this.intro.intro.inl.intro.intro \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 k : \u2115 \u2192. \u2115 hk : Nat.Partrec k H :   \u2200 (a : \u2115),     (\u2200 (x : \u2115),         x \u2208 k a \u2192           (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228             x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227       ((k a).Dom \u2194         (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228           (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom) k' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n) a : \u03b1 x a' : \u03c3 ha : x \u2208 f a hn : encode a' \u2208 k (encode a) hx : a' = x \u22a2 x \u2208 f a \u2228 x \u2208 g a ** exact Or.inl ha ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.product_biUnion ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s\u271d s' : Finset \u03b1 t\u271d t' : Finset \u03b2 a : \u03b1 b : \u03b2 inst\u271d : DecidableEq \u03b3 s : Finset \u03b1 t : Finset \u03b2 f : \u03b1 \u00d7 \u03b2 \u2192 Finset \u03b3 \u22a2 Finset.biUnion (s \u00d7\u02e2 t) f = Finset.biUnion s fun a => Finset.biUnion t fun b => f (a, b) ** classical simp_rw [product_eq_biUnion, biUnion_biUnion, image_biUnion] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s\u271d s' : Finset \u03b1 t\u271d t' : Finset \u03b2 a : \u03b1 b : \u03b2 inst\u271d : DecidableEq \u03b3 s : Finset \u03b1 t : Finset \u03b2 f : \u03b1 \u00d7 \u03b2 \u2192 Finset \u03b3 \u22a2 Finset.biUnion (s \u00d7\u02e2 t) f = Finset.biUnion s fun a => Finset.biUnion t fun b => f (a, b) ** simp_rw [product_eq_biUnion, biUnion_biUnion, image_biUnion] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.abs_sub_le_of_uIcc_subset_uIcc ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 a b c d : \u03b1 h : [[c, d]] \u2286 [[a, b]] \u22a2 |d - c| \u2264 |b - a| ** rw [\u2190 max_sub_min_eq_abs, \u2190 max_sub_min_eq_abs] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 a b c d : \u03b1 h : [[c, d]] \u2286 [[a, b]] \u22a2 max c d - min c d \u2264 max a b - min a b ** rw [uIcc_subset_uIcc_iff_le] at h ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 a b c d : \u03b1 h : min a b \u2264 min c d \u2227 max c d \u2264 max a b \u22a2 max c d - min c d \u2264 max a b - min a b ** exact sub_le_sub h.2 h.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_sphere ** 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 \u22a2 \u2191\u2191\u03bc (sphere x r) = 0 ** rcases eq_or_ne r 0 with (rfl | h) ** case inl 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 \u22a2 \u2191\u2191\u03bc (sphere x 0) = 0 ** rw [sphere_zero, measure_singleton] ** case inr 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 h : r \u2260 0 \u22a2 \u2191\u2191\u03bc (sphere x r) = 0 ** exact addHaar_sphere_of_ne_zero \u03bc x h ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.empty_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 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": "Finset.monotone_iff ** \u03b1 : Type u_1 s t : Finset \u03b1 \u03b2 : Type u_2 inst\u271d : Preorder \u03b2 f : Finset \u03b1 \u2192 \u03b2 \u22a2 Monotone f \u2194 \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s \u2264 f (cons i s hi) ** classical\nsimp only [monotone_iff_forall_covby, covby_iff, forall_exists_index, and_imp]\naesop ** \u03b1 : Type u_1 s t : Finset \u03b1 \u03b2 : Type u_2 inst\u271d : Preorder \u03b2 f : Finset \u03b1 \u2192 \u03b2 \u22a2 Monotone f \u2194 \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s \u2264 f (cons i s hi) ** simp only [monotone_iff_forall_covby, covby_iff, forall_exists_index, and_imp] ** \u03b1 : Type u_1 s t : Finset \u03b1 \u03b2 : Type u_2 inst\u271d : Preorder \u03b2 f : Finset \u03b1 \u2192 \u03b2 \u22a2 (\u2200 (a b : Finset \u03b1) (x : \u03b1) (x_1 : \u00acx \u2208 a), b = cons x a x_1 \u2192 f a \u2264 f b) \u2194     \u2200 (s : Finset \u03b1) {i : \u03b1} (hi : \u00aci \u2208 s), f s \u2264 f (cons i s hi) ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_le_of_forall_fin_meas_le' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : AEMeasurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C ** let f' := hf_meas.mk f ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : AEMeasurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C ** have hf' : \u2200 s, MeasurableSet[m] s \u2192 \u03bc s \u2260 \u221e \u2192 \u222b\u207b x in s, f' x \u2202\u03bc \u2264 C := by\n  refine' fun s hs h\u03bcs => (le_of_eq _).trans (hf s hs h\u03bcs)\n  refine' lintegral_congr_ae (ae_restrict_of_ae (hf_meas.ae_eq_mk.mono fun x hx => _))\n  dsimp only\n  rw [hx] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : AEMeasurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas hf' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C ** rw [lintegral_congr_ae hf_meas.ae_eq_mk] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : AEMeasurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas hf' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C \u22a2 \u222b\u207b (a : \u03b1), AEMeasurable.mk f hf_meas a \u2202\u03bc \u2264 C ** exact lintegral_le_of_forall_fin_meas_le_of_measurable hm C hf_meas.measurable_mk hf' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : AEMeasurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C ** refine' fun s hs h\u03bcs => (le_of_eq _).trans (hf s hs h\u03bcs) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : AEMeasurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc ** refine' lintegral_congr_ae (ae_restrict_of_ae (hf_meas.ae_eq_mk.mono fun x hx => _)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : AEMeasurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : \u03b1 hx : f x = AEMeasurable.mk f hf_meas x \u22a2 (fun x => f' x) x = (fun x => f x) x ** dsimp only ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : OpensMeasurableSpace E \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : AEMeasurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : \u03b1 hx : f x = AEMeasurable.mk f hf_meas x \u22a2 AEMeasurable.mk f hf_meas x = f x ** rw [hx] ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.aemeasurable_of_exist_almost_disjoint_supersets ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h :   \u2200 (p q : \u211d\u22650),     p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 \u22a2 AEMeasurable f ** obtain \u27e8s, s_count, s_dense, _, s_top\u27e9 :\n  \u2203 s : Set \u211d\u22650\u221e, s.Countable \u2227 Dense s \u2227 0 \u2209 s \u2227 \u221e \u2209 s :=\n  ENNReal.exists_countable_dense_no_zero_top ** case intro.intro.intro.intro \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h :   \u2200 (p q : \u211d\u22650),     p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 s : Set \u211d\u22650\u221e s_count : Set.Countable s s_dense : Dense s left\u271d : \u00ac0 \u2208 s s_top : \u00ac\u22a4 \u2208 s \u22a2 AEMeasurable f ** have I : \u2200 x \u2208 s, x \u2260 \u221e := fun x xs hx => s_top (hx \u25b8 xs) ** case intro.intro.intro.intro \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h :   \u2200 (p q : \u211d\u22650),     p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 s : Set \u211d\u22650\u221e s_count : Set.Countable s s_dense : Dense s left\u271d : \u00ac0 \u2208 s s_top : \u00ac\u22a4 \u2208 s I : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4 \u22a2 AEMeasurable f ** apply MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets \u03bc s s_count s_dense _ ** case intro.intro.intro.intro \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h :   \u2200 (p q : \u211d\u22650),     p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 s : Set \u211d\u22650\u221e s_count : Set.Countable s s_dense : Dense s left\u271d : \u00ac0 \u2208 s s_top : \u00ac\u22a4 \u2208 s I : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4 \u22a2 \u2200 (p : \u211d\u22650\u221e),     p \u2208 s \u2192       \u2200 (q : \u211d\u22650\u221e),         q \u2208 s \u2192           p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 ** rintro p hp q hq hpq ** case intro.intro.intro.intro \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h :   \u2200 (p q : \u211d\u22650),     p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 s : Set \u211d\u22650\u221e s_count : Set.Countable s s_dense : Dense s left\u271d : \u00ac0 \u2208 s s_top : \u00ac\u22a4 \u2208 s I : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4 p : \u211d\u22650\u221e hp : p \u2208 s q : \u211d\u22650\u221e hq : q \u2208 s hpq : p < q \u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 ** lift p to \u211d\u22650 using I p hp ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h :   \u2200 (p q : \u211d\u22650),     p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 s : Set \u211d\u22650\u221e s_count : Set.Countable s s_dense : Dense s left\u271d : \u00ac0 \u2208 s s_top : \u00ac\u22a4 \u2208 s I : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4 q : \u211d\u22650\u221e hq : q \u2208 s p : \u211d\u22650 hp : \u2191p \u2208 s hpq : \u2191p < q \u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 ** lift q to \u211d\u22650 using I q hq ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h :   \u2200 (p q : \u211d\u22650),     p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 s : Set \u211d\u22650\u221e s_count : Set.Countable s s_dense : Dense s left\u271d : \u00ac0 \u2208 s s_top : \u00ac\u22a4 \u2208 s I : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4 p : \u211d\u22650 hp : \u2191p \u2208 s q : \u211d\u22650 hq : \u2191q \u2208 s hpq : \u2191p < \u2191q \u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0 ** exact h p q (ENNReal.coe_lt_coe.1 hpq) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.mem_powerset ** \u03b1 : Type u_1 s\u271d t\u271d s t : Finset \u03b1 \u22a2 s \u2208 powerset t \u2194 s \u2286 t ** cases s ** case mk \u03b1 : Type u_1 s t\u271d t : Finset \u03b1 val\u271d : Multiset \u03b1 nodup\u271d : Nodup val\u271d \u22a2 { val := val\u271d, nodup := nodup\u271d } \u2208 powerset t \u2194 { val := val\u271d, nodup := nodup\u271d } \u2286 t ** simp [powerset, mem_mk, mem_pmap, mk.injEq, mem_powerset, exists_prop, exists_eq_right,\n  \u2190 val_le_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_deriv_comp_smul_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' : \u211d \u2192 \u211d g g' : \u211d \u2192 E 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, f' x \u2022 (g' \u2218 f) x = (g \u2218 f) b - (g \u2218 f) a ** rw [integral_comp_smul_deriv'' hf hff' hf' hg',\n  integral_eq_sub_of_hasDeriv_right hg hgg' (hg'.mono _).intervalIntegrable] ** \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' : \u211d \u2192 \u211d g g' : \u211d \u2192 E 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 g (f b) - g (f a) = (g \u2218 f) b - (g \u2218 f) a  \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' : \u211d \u2192 \u211d g g' : \u211d \u2192 E 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 [[f a, f b]] \u2286 f '' [[a, b]] ** exacts [rfl, intermediate_value_uIcc hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "PEquiv.symm_trans_self ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 \u22a2 PEquiv.symm (PEquiv.trans (PEquiv.symm f) f) = PEquiv.symm (ofSet {b | isSome (\u2191(PEquiv.symm f) b) = true}) ** simp [symm_trans_rev, self_trans_symm, -symm_symm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.leastGE_eq_min ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n \u22a2 leastGE f r (\u03c0 \u03c9) \u03c9 = min (\u03c0 \u03c9) (leastGE f r n \u03c9) ** refine' le_antisymm (le_min (leastGE_le _) (leastGE_mono (h\u03c0n \u03c9) r \u03c9)) _ ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n \u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9 ** by_cases hle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9 ** 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 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n hle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9 \u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9 ** rw [min_eq_left hle, leastGE] ** 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 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n hle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9 \u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9 ** by_cases h : \u2203 j \u2208 Set.Icc 0 (\u03c0 \u03c9), f j \u03c9 \u2208 Set.Ici r ** 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 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n hle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9 h : \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r \u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9 ** refine' hle.trans (Eq.le _) ** 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 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n hle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9 h : \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r \u22a2 leastGE f r n \u03c9 = hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9 ** rw [leastGE, \u2190 hitting_eq_hitting_of_exists (h\u03c0n \u03c9) h] ** 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 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n hle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9 h : \u00ac\u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r \u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9 ** simp only [hitting, if_neg h, le_rfl] ** 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 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n hle : \u00ac\u03c0 \u03c9 \u2264 leastGE f r n \u03c9 \u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9 ** rw [min_eq_right (not_le.1 hle).le, leastGE, leastGE, \u2190\n  hitting_eq_hitting_of_exists (h\u03c0n \u03c9) _] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n hle : \u00ac\u03c0 \u03c9 \u2264 leastGE f r n \u03c9 \u22a2 \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r ** rw [not_le, leastGE, hitting_lt_iff _ (h\u03c0n \u03c9)] at hle ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 \u03c0 : \u03a9 \u2192 \u2115 r : \u211d \u03c9 : \u03a9 n : \u2115 h\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n hle : \u2203 j, j \u2208 Set.Ico 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r \u22a2 \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r ** exact\n  let \u27e8j, hj\u2081, hj\u2082\u27e9 := hle\n  \u27e8j, \u27e8hj\u2081.1, hj\u2081.2.le\u27e9, hj\u2082\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.piecewise_mem_set_pi ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : \u03b1 \u2192 Sort u_4 s : Finset \u03b1 f\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i inst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s) \u03b4 : \u03b1 \u2192 Type u_5 t : Set \u03b1 t' : (i : \u03b1) \u2192 Set (\u03b4 i) f g : (i : \u03b1) \u2192 \u03b4 i hf : f \u2208 Set.pi t t' hg : g \u2208 Set.pi t t' \u22a2 piecewise s f g \u2208 Set.pi t t' ** classical\n  rw [\u2190 piecewise_coe]\n  exact Set.piecewise_mem_pi (\u2191s) hf hg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : \u03b1 \u2192 Sort u_4 s : Finset \u03b1 f\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i inst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s) \u03b4 : \u03b1 \u2192 Type u_5 t : Set \u03b1 t' : (i : \u03b1) \u2192 Set (\u03b4 i) f g : (i : \u03b1) \u2192 \u03b4 i hf : f \u2208 Set.pi t t' hg : g \u2208 Set.pi t t' \u22a2 piecewise s f g \u2208 Set.pi t t' ** rw [\u2190 piecewise_coe] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : \u03b1 \u2192 Sort u_4 s : Finset \u03b1 f\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i inst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s) \u03b4 : \u03b1 \u2192 Type u_5 t : Set \u03b1 t' : (i : \u03b1) \u2192 Set (\u03b4 i) f g : (i : \u03b1) \u2192 \u03b4 i hf : f \u2208 Set.pi t t' hg : g \u2208 Set.pi t t' \u22a2 Set.piecewise (\u2191s) f g \u2208 Set.pi t t' ** exact Set.piecewise_mem_pi (\u2191s) hf hg ** Qed",
        "informal": ""
    },
    {
        "formal": "volume_regionBetween_eq_lintegral' ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc ** rw [Measure.prod_apply] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s h : (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x) \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191volume (Prod.mk x \u207b\u00b9' regionBetween f g s) \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc ** dsimp only [regionBetween, preimage_setOf_eq] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s h : (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x) \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc ** rw [h, lintegral_indicator] <;> simp only [hs, Pi.sub_apply] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s \u22a2 (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x) ** funext x ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s x : \u03b1 \u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = indicator s (fun x => ofReal (g x - f x)) x ** rw [indicator_apply] ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s x : \u03b1 \u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = if x \u2208 s then ofReal (g x - f x) else 0 ** split_ifs with h ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s x : \u03b1 h : x \u2208 s \u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = ofReal (g x - f x) ** have hx : { a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x) } = Ioo (f x) (g x) := by simp [h, Ioo] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s x : \u03b1 h : x \u2208 s hx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x) \u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = ofReal (g x - f x) ** simp only [hx, Real.volume_Ioo, sub_zero] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s x : \u03b1 h : x \u2208 s \u22a2 {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x) ** simp [h, Ioo] ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s x : \u03b1 h : \u00acx \u2208 s \u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = 0 ** have hx : { a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x) } = \u2205 := by simp [h] ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s x : \u03b1 h : \u00acx \u2208 s hx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205 \u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = 0 ** simp only [hx, measure_empty] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s x : \u03b1 h : \u00acx \u2208 s \u22a2 {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205 ** simp [h] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f hg : Measurable g hs : MeasurableSet s \u22a2 MeasurableSet (regionBetween f g s) ** exact measurableSet_regionBetween hf hg hs ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toMeasure_apply_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b1 p : PMF \u03b1 s t : Set \u03b1 a : \u03b1 h : MeasurableSet {a} \u22a2 \u2191\u2191(toMeasure p) {a} = \u2191p a ** simp [toMeasure_apply_eq_toOuterMeasure_apply _ _ h, toOuterMeasure_apply_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.union_get_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Union \u03b1 a b : Part \u03b1 hab : (a \u222a b).Dom \u22a2 get (a \u222a b) hab = get a (_ : a.Dom) \u222a get b (_ : b.Dom) ** simp [union_def] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Union \u03b1 a b : Part \u03b1 hab : (a \u222a b).Dom \u22a2 get (Part.bind a fun y => map (fun x => y \u222a x) b) (_ : (Part.bind a fun y => map (fun x => y \u222a x) b).Dom) =     get a (_ : a.Dom) \u222a get b (_ : b.Dom) ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.cast_sub_one ** R : Type u_1 inst\u271d : Ring R n : \u2115 k : ZMod n \u22a2 \u2191(k - 1) = (if k = 0 then \u2191n else \u2191k) - 1 ** split_ifs with hk ** case pos R : Type u_1 inst\u271d : Ring R n : \u2115 k : ZMod n hk : k = 0 \u22a2 \u2191(k - 1) = \u2191n - 1 ** rw [hk, zero_sub, ZMod.cast_neg_one] ** case neg R : Type u_1 inst\u271d : Ring R n : \u2115 k : ZMod n hk : \u00ack = 0 \u22a2 \u2191(k - 1) = \u2191k - 1 ** cases n ** case neg.zero R : Type u_1 inst\u271d : Ring R k : ZMod Nat.zero hk : \u00ack = 0 \u22a2 \u2191(k - 1) = \u2191k - 1 ** rw [Int.cast_sub, Int.cast_one] ** case neg.succ R : Type u_1 inst\u271d : Ring R n\u271d : \u2115 k : ZMod (Nat.succ n\u271d) hk : \u00ack = 0 \u22a2 \u2191(k - 1) = \u2191k - 1 ** dsimp [ZMod, ZMod.cast, ZMod.val] ** case neg.succ R : Type u_1 inst\u271d : Ring R n\u271d : \u2115 k : ZMod (Nat.succ n\u271d) hk : \u00ack = 0 \u22a2 \u2191\u2191(k - 1) = \u2191\u2191k - 1 ** rw [Fin.coe_sub_one, if_neg] ** case neg.succ R : Type u_1 inst\u271d : Ring R n\u271d : \u2115 k : ZMod (Nat.succ n\u271d) hk : \u00ack = 0 \u22a2 \u2191(\u2191k - 1) = \u2191\u2191k - 1 ** rw [Nat.cast_sub, Nat.cast_one] ** case neg.succ R : Type u_1 inst\u271d : Ring R n\u271d : \u2115 k : ZMod (Nat.succ n\u271d) hk : \u00ack = 0 \u22a2 1 \u2264 \u2191k ** rwa [Fin.ext_iff, Fin.val_zero, \u2190 Ne, \u2190 Nat.one_le_iff_ne_zero] at hk ** case neg.succ.hnc R : Type u_1 inst\u271d : Ring R n\u271d : \u2115 k : ZMod (Nat.succ n\u271d) hk : \u00ack = 0 \u22a2 \u00ack = 0 ** exact hk ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.of_to_int' ** \u03b1 : Type u_1 \u22a2 ofInt' \u21910 = 0 ** dsimp [ofInt', cast_zero] ** \u03b1 : Type u_1 \u22a2 Num.toZNum (Num.ofNat' 0) = 0 ** erw [Num.ofNat'_zero, Num.toZNum] ** \u03b1 : Type u_1 a : PosNum \u22a2 ofInt' \u2191(pos a) = pos a ** rw [cast_pos, \u2190 PosNum.cast_to_nat, \u2190 Num.ofInt'_toZNum, PosNum.of_to_nat] ** \u03b1 : Type u_1 a : PosNum \u22a2 Num.toZNum (Num.pos a) = pos a ** rfl ** \u03b1 : Type u_1 a : PosNum \u22a2 ofInt' \u2191(neg a) = neg a ** rw [cast_neg, ofInt'_neg, \u2190 PosNum.cast_to_nat, \u2190 Num.ofInt'_toZNum, PosNum.of_to_nat] ** \u03b1 : Type u_1 a : PosNum \u22a2 -Num.toZNum (Num.pos a) = neg a ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Stream.next?_toList ** \u03b1 : Type u_1 s : RBNode.Stream \u03b1 \u22a2 Option.map       (fun x =>         match x with         | (a, b) => (a, toList b))       (next? s) =     List.next? (toList s) ** cases s <;> simp [next?, toStream_toList'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.range_IciExtend ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : LinearOrder \u03b1 a b : \u03b1 h : a \u2264 b x : \u03b1 f : \u2191(Ici a) \u2192 \u03b2 \u22a2 range (IciExtend f) = range f ** simp only [IciExtend, range_comp f, range_projIci, range_id', image_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.vadd_ae_eq_self_of_mem_zmultiples ** G\u271d : Type u M : Type v \u03b1\u271d : Type w s\u271d : Set \u03b1\u271d m\u271d : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : Group G\u271d inst\u271d\u2078 : MulAction G\u271d \u03b1\u271d inst\u271d\u2077 : MeasurableSpace G\u271d inst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d c : G\u271d \u03bc\u271d : Measure \u03b1\u271d inst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d G : Type u \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u2074 : AddGroup G inst\u271d\u00b3 : AddAction G \u03b1 inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableVAdd G \u03b1 \u03bc : Measure \u03b1 inst\u271d : VAddInvariantMeasure G \u03b1 \u03bc x y : G hs : x +\u1d65 s =\u1da0[ae \u03bc] s hy : y \u2208 AddSubgroup.zmultiples x \u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s ** letI : MeasurableSpace (Multiplicative G) := (inferInstanceAs (MeasurableSpace G)) ** G\u271d : Type u M : Type v \u03b1\u271d : Type w s\u271d : Set \u03b1\u271d m\u271d : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : Group G\u271d inst\u271d\u2078 : MulAction G\u271d \u03b1\u271d inst\u271d\u2077 : MeasurableSpace G\u271d inst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d c : G\u271d \u03bc\u271d : Measure \u03b1\u271d inst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d G : Type u \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u2074 : AddGroup G inst\u271d\u00b3 : AddAction G \u03b1 inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableVAdd G \u03b1 \u03bc : Measure \u03b1 inst\u271d : VAddInvariantMeasure G \u03b1 \u03bc x y : G hs : x +\u1d65 s =\u1da0[ae \u03bc] s hy : y \u2208 AddSubgroup.zmultiples x this : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G) \u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s ** letI : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=\n  \u27e8fun g => VAddInvariantMeasure.measure_preimage_vadd (Multiplicative.toAdd g)\u27e9 ** G\u271d : Type u M : Type v \u03b1\u271d : Type w s\u271d : Set \u03b1\u271d m\u271d : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : Group G\u271d inst\u271d\u2078 : MulAction G\u271d \u03b1\u271d inst\u271d\u2077 : MeasurableSpace G\u271d inst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d c : G\u271d \u03bc\u271d : Measure \u03b1\u271d inst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d G : Type u \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u2074 : AddGroup G inst\u271d\u00b3 : AddAction G \u03b1 inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableVAdd G \u03b1 \u03bc : Measure \u03b1 inst\u271d : VAddInvariantMeasure G \u03b1 \u03bc x y : G hs : x +\u1d65 s =\u1da0[ae \u03bc] s hy : y \u2208 AddSubgroup.zmultiples x this\u271d : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G) this : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=   { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (\u2191Multiplicative.toAdd g) } \u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s ** letI : MeasurableSMul (Multiplicative G) \u03b1 :=\n  { measurable_const_smul := fun g => measurable_const_vadd (Multiplicative.toAdd g)\n    measurable_smul_const := fun a =>\n      @measurable_vadd_const (Multiplicative G) \u03b1 (inferInstanceAs (VAdd G \u03b1)) _ _\n        (inferInstanceAs (MeasurableVAdd G \u03b1)) a } ** G\u271d : Type u M : Type v \u03b1\u271d : Type w s\u271d : Set \u03b1\u271d m\u271d : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : Group G\u271d inst\u271d\u2078 : MulAction G\u271d \u03b1\u271d inst\u271d\u2077 : MeasurableSpace G\u271d inst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d c : G\u271d \u03bc\u271d : Measure \u03b1\u271d inst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d G : Type u \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u2074 : AddGroup G inst\u271d\u00b3 : AddAction G \u03b1 inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableVAdd G \u03b1 \u03bc : Measure \u03b1 inst\u271d : VAddInvariantMeasure G \u03b1 \u03bc x y : G hs : x +\u1d65 s =\u1da0[ae \u03bc] s hy : y \u2208 AddSubgroup.zmultiples x this\u271d\u00b9 : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G) this\u271d : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=   { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (\u2191Multiplicative.toAdd g) } this : MeasurableSMul (Multiplicative G) \u03b1 :=   { measurable_const_smul := fun g => measurable_const_vadd (\u2191Multiplicative.toAdd g),     measurable_smul_const := fun a => measurable_vadd_const a } \u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s ** exact @smul_ae_eq_self_of_mem_zpowers (Multiplicative G) \u03b1 _ _ _ _ _ _ _ _ _ _ hs hy ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.strongDownwardInductionOn_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 p : Finset \u03b1 \u2192 Sort u_3 s : Finset \u03b1 H : (t\u2081 : Finset \u03b1) \u2192 ({t\u2082 : Finset \u03b1} \u2192 card t\u2082 \u2264 n \u2192 t\u2081 \u2282 t\u2082 \u2192 p t\u2082) \u2192 card t\u2081 \u2264 n \u2192 p t\u2081 \u22a2 (fun a => strongDownwardInductionOn s H a) = H s fun {t} ht x => strongDownwardInductionOn t H ht ** dsimp only [strongDownwardInductionOn] ** \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 p : Finset \u03b1 \u2192 Sort u_3 s : Finset \u03b1 H : (t\u2081 : Finset \u03b1) \u2192 ({t\u2082 : Finset \u03b1} \u2192 card t\u2082 \u2264 n \u2192 t\u2081 \u2282 t\u2082 \u2192 p t\u2082) \u2192 card t\u2081 \u2264 n \u2192 p t\u2081 \u22a2 (fun a => strongDownwardInduction H s a) = H s fun {t} ht x => strongDownwardInduction H t ht ** rw [strongDownwardInduction] ** Qed",
        "informal": ""
    },
    {
        "formal": "WithTop.image_coe_Ioc ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ioc a b = Ioc \u2191a \u2191b ** rw [\u2190 preimage_coe_Ioc, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left\n    (Subset.trans Ioc_subset_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.evaln_prim ** x\u271d : Unit p : \u2115 \u22a2 Nat.Partrec.Code.G       (x\u271d,           List.map             (fun n =>               let a := ofNat (\u2115 \u00d7 Code) n;               List.map (evaln a.1 a.2) (List.range a.1))             (List.range p)).2 =     some       (let a := ofNat (\u2115 \u00d7 Code) p;       List.map (evaln a.1 a.2) (List.range a.1)) ** simp only [G, prod_ofNat_val, ofNat_nat, List.length_map, List.length_range,\n  Nat.pair_unpair, Option.some_inj] ** x\u271d : Unit p : \u2115 \u22a2 List.map       (fun n =>         Nat.rec Option.none           (fun n_1 n_ih =>             rec (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                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range p))                       ((unpair p).1, cf) n                 let y \u2190                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range p))                       ((unpair p).1, cg) n                 some (Nat.pair x y))               (fun cf cg x x => do                 let x \u2190                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range p))                       ((unpair p).1, cg) n                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cf) x)               (fun cf cg x x =>                 Nat.rec                   (Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cf) (unpair n).1)                   (fun n_2 n_ih => do                     let i \u2190                       Nat.Partrec.Code.lup                           (List.map                             (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                             (List.range p))                           (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)                     Nat.Partrec.Code.lup                         (List.map                           (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                           (List.range p))                         ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))                   (unpair n).2)               (fun cf x => do                 let x \u2190                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range p))                       ((unpair p).1, cf) n                 Nat.rec (some (unpair n).2)                     (fun n_2 n_ih =>                       Nat.Partrec.Code.lup                         (List.map                           (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                           (List.range p))                         (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                     x)               (ofNat Code (unpair p).2))           (unpair p).1)       (List.range (unpair p).1) =     List.map (evaln (unpair p).1 (ofNat Code (unpair p).2)) (List.range (unpair p).1) ** refine List.map_congr fun n => ?_ ** x\u271d : Unit p n : \u2115 \u22a2 n \u2208 List.range (unpair p).1 \u2192     Nat.rec Option.none         (fun n_1 n_ih =>           rec (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                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cf) n               let y \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cg) n               some (Nat.pair x y))             (fun cf cg x x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cg) n               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range p))                   ((unpair p).1, cf) x)             (fun cf cg x x =>               Nat.rec                 (Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range p))                   ((unpair p).1, cf) (unpair n).1)                 (fun n_2 n_ih => do                   let i \u2190                     Nat.Partrec.Code.lup                         (List.map                           (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                           (List.range p))                         (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range p))                       ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))                 (unpair n).2)             (fun cf x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cf) n               Nat.rec (some (unpair n).2)                   (fun n_2 n_ih =>                     Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range p))                       (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                   x)             (ofNat Code (unpair p).2))         (unpair p).1 =       evaln (unpair p).1 (ofNat Code (unpair p).2) n ** have : List.range p = List.range (Nat.pair p.unpair.1 (encode (ofNat Code p.unpair.2))) := by\n  simp ** x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) \u22a2 n \u2208 List.range (unpair p).1 \u2192     Nat.rec Option.none         (fun n_1 n_ih =>           rec (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                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cf) n               let y \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cg) n               some (Nat.pair x y))             (fun cf cg x x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cg) n               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range p))                   ((unpair p).1, cf) x)             (fun cf cg x x =>               Nat.rec                 (Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range p))                   ((unpair p).1, cf) (unpair n).1)                 (fun n_2 n_ih => do                   let i \u2190                     Nat.Partrec.Code.lup                         (List.map                           (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                           (List.range p))                         (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range p))                       ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))                 (unpair n).2)             (fun cf x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range p))                     ((unpair p).1, cf) n               Nat.rec (some (unpair n).2)                   (fun n_2 n_ih =>                     Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range p))                       (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                   x)             (ofNat Code (unpair p).2))         (unpair p).1 =       evaln (unpair p).1 (ofNat Code (unpair p).2) n ** rw [this] ** x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) \u22a2 n \u2208 List.range (unpair p).1 \u2192     Nat.rec Option.none         (fun n_1 n_ih =>           rec (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                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))                     ((unpair p).1, cf) n               let y \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))                     ((unpair p).1, cg) n               some (Nat.pair x y))             (fun cf cg x x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))                     ((unpair p).1, cg) n               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))                   ((unpair p).1, cf) x)             (fun cf cg x x =>               Nat.rec                 (Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))                   ((unpair p).1, cf) (unpair n).1)                 (fun n_2 n_ih => do                   let i \u2190                     Nat.Partrec.Code.lup                         (List.map                           (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                           (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))                         (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))                       ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))                 (unpair n).2)             (fun cf x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))                     ((unpair p).1, cf) n               Nat.rec (some (unpair n).2)                   (fun n_2 n_ih =>                     Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))))                       (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                   x)             (ofNat Code (unpair p).2))         (unpair p).1 =       evaln (unpair p).1 (ofNat Code (unpair p).2) n ** generalize p.unpair.1 = k ** x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k : \u2115 \u22a2 n \u2208 List.range k \u2192     Nat.rec Option.none         (fun n_1 n_ih =>           rec (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                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))                     (k, cf) n               let y \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))                     (k, cg) n               some (Nat.pair x y))             (fun cf cg x x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))                     (k, cg) n               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))                   (k, cf) x)             (fun cf cg x x =>               Nat.rec                 (Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))                   (k, cf) (unpair n).1)                 (fun n_2 n_ih => do                   let i \u2190                     Nat.Partrec.Code.lup                         (List.map                           (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                           (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))                         (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2)                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))                       (k, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))                 (unpair n).2)             (fun cf x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))                     (k, cf) n               Nat.rec (some (unpair n).2)                   (fun n_2 n_ih =>                     Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair k (encode (ofNat Code (unpair p).2)))))                       (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                   x)             (ofNat Code (unpair p).2))         k =       evaln k (ofNat Code (unpair p).2) n ** generalize ofNat Code p.unpair.2 = c ** x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k : \u2115 c : Code \u22a2 n \u2208 List.range k \u2192     Nat.rec Option.none         (fun n_1 n_ih =>           rec (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                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode c))))                     (k, cf) n               let y \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode c))))                     (k, cg) n               some (Nat.pair x y))             (fun cf cg x x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode c))))                     (k, cg) n               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair k (encode c))))                   (k, cf) x)             (fun cf cg x x =>               Nat.rec                 (Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair k (encode c))))                   (k, cf) (unpair n).1)                 (fun n_2 n_ih => do                   let i \u2190                     Nat.Partrec.Code.lup                         (List.map                           (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                           (List.range (Nat.pair k (encode c))))                         (n_1, c) (Nat.pair (unpair n).1 n_2)                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair k (encode c))))                       (k, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))                 (unpair n).2)             (fun cf x => do               let x \u2190                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode c))))                     (k, cf) n               Nat.rec (some (unpair n).2)                   (fun n_2 n_ih =>                     Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair k (encode c))))                       (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                   x)             c)         k =       evaln k c n ** intro nk ** x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k : \u2115 c : Code nk : n \u2208 List.range k \u22a2 Nat.rec Option.none       (fun n_1 n_ih =>         rec (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                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair k (encode c))))                   (k, cf) n             let y \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair k (encode c))))                   (k, cg) n             some (Nat.pair x y))           (fun cf cg x x => do             let x \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair k (encode c))))                   (k, cg) n             Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair k (encode c))))                 (k, cf) x)           (fun cf cg x x =>             Nat.rec               (Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair k (encode c))))                 (k, cf) (unpair n).1)               (fun n_2 n_ih => do                 let i \u2190                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair k (encode c))))                       (n_1, c) (Nat.pair (unpair n).1 n_2)                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode c))))                     (k, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))               (unpair n).2)           (fun cf x => do             let x \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair k (encode c))))                   (k, cf) n             Nat.rec (some (unpair n).2)                 (fun n_2 n_ih =>                   Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair k (encode c))))                     (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                 x)           c)       k =     evaln k c n ** cases' k with k' ** case succ x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) c : Code k' : \u2115 nk : n \u2208 List.range (Nat.succ k') \u22a2 Nat.rec Option.none       (fun n_1 n_ih =>         rec (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                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (Nat.succ k') (encode c))))                   (Nat.succ k', cf) n             let y \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (Nat.succ k') (encode c))))                   (Nat.succ k', cg) n             some (Nat.pair x y))           (fun cf cg x x => do             let x \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (Nat.succ k') (encode c))))                   (Nat.succ k', cg) n             Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (Nat.succ k') (encode c))))                 (Nat.succ k', cf) x)           (fun cf cg x x =>             Nat.rec               (Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (Nat.succ k') (encode c))))                 (Nat.succ k', cf) (unpair n).1)               (fun n_2 n_ih => do                 let i \u2190                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair (Nat.succ k') (encode c))))                       (n_1, c) (Nat.pair (unpair n).1 n_2)                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair (Nat.succ k') (encode c))))                     (Nat.succ k', cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))               (unpair n).2)           (fun cf x => do             let x \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (Nat.succ k') (encode c))))                   (Nat.succ k', cf) n             Nat.rec (some (unpair n).2)                 (fun n_2 n_ih =>                   Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair (Nat.succ k') (encode c))))                     (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                 x)           c)       (Nat.succ k') =     evaln (Nat.succ k') c n ** let k := k' + 1 ** case succ x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) c : Code k' : \u2115 nk : n \u2208 List.range (Nat.succ k') k : \u2115 := k' + 1 \u22a2 Nat.rec Option.none       (fun n_1 n_ih =>         rec (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                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (Nat.succ k') (encode c))))                   (Nat.succ k', cf) n             let y \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (Nat.succ k') (encode c))))                   (Nat.succ k', cg) n             some (Nat.pair x y))           (fun cf cg x x => do             let x \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (Nat.succ k') (encode c))))                   (Nat.succ k', cg) n             Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (Nat.succ k') (encode c))))                 (Nat.succ k', cf) x)           (fun cf cg x x =>             Nat.rec               (Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (Nat.succ k') (encode c))))                 (Nat.succ k', cf) (unpair n).1)               (fun n_2 n_ih => do                 let i \u2190                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair (Nat.succ k') (encode c))))                       (n_1, c) (Nat.pair (unpair n).1 n_2)                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair (Nat.succ k') (encode c))))                     (Nat.succ k', cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))               (unpair n).2)           (fun cf x => do             let x \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (Nat.succ k') (encode c))))                   (Nat.succ k', cf) n             Nat.rec (some (unpair n).2)                 (fun n_2 n_ih =>                   Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair (Nat.succ k') (encode c))))                     (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                 x)           c)       (Nat.succ k') =     evaln (Nat.succ k') c n ** simp only [show k'.succ = k from rfl] ** case succ x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) c : Code k' : \u2115 nk : n \u2208 List.range (Nat.succ k') k : \u2115 := k' + 1 \u22a2 rec (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               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cf) n         let y \u2190           Nat.Partrec.Code.lup               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cg) n         some (Nat.pair x y))       (fun cf cg x x => do         let x \u2190           Nat.Partrec.Code.lup               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cg) n         Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode c))))             (k' + 1, cf) x)       (fun cf cg x x =>         Nat.rec           (Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode c))))             (k' + 1, cf) (unpair n).1)           (fun n_1 n_ih => do             let i \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (k' + 1) (encode c))))                   (k', c) (Nat.pair (unpair n).1 n_1)             Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (k' + 1) (encode c))))                 (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))           (unpair n).2)       (fun cf x => do         let x \u2190           Nat.Partrec.Code.lup               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cf) n         Nat.rec (some (unpair n).2)             (fun n_1 n_ih =>               Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (k' + 1) (encode c))))                 (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))             x)       c =     evaln (k' + 1) c n ** simp [Nat.lt_succ_iff] at nk ** case succ x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) c : Code k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' \u22a2 rec (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               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cf) n         let y \u2190           Nat.Partrec.Code.lup               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cg) n         some (Nat.pair x y))       (fun cf cg x x => do         let x \u2190           Nat.Partrec.Code.lup               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cg) n         Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode c))))             (k' + 1, cf) x)       (fun cf cg x x =>         Nat.rec           (Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode c))))             (k' + 1, cf) (unpair n).1)           (fun n_1 n_ih => do             let i \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (k' + 1) (encode c))))                   (k', c) (Nat.pair (unpair n).1 n_1)             Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (k' + 1) (encode c))))                 (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))           (unpair n).2)       (fun cf x => do         let x \u2190           Nat.Partrec.Code.lup               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cf) n         Nat.rec (some (unpair n).2)             (fun n_1 n_ih =>               Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (k' + 1) (encode c))))                 (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))             x)       c =     evaln (k' + 1) c n ** have hg :\n  \u2200 {k' c' n},\n    Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192\n      lup ((List.range (Nat.pair k (encode c))).map fun n =>\n        (List.range n.unpair.1).map (evaln n.unpair.1 (ofNat Code n.unpair.2))) (k', c') n =\n      evaln k' c' n := by\n  intro k\u2081 c\u2081 n\u2081 hl\n  simp [lup, List.get?_range hl, evaln_map, Bind.bind] ** case succ x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) c : Code k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode c))))           (k', c') n =         evaln k' c' n \u22a2 rec (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               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cf) n         let y \u2190           Nat.Partrec.Code.lup               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cg) n         some (Nat.pair x y))       (fun cf cg x x => do         let x \u2190           Nat.Partrec.Code.lup               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cg) n         Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode c))))             (k' + 1, cf) x)       (fun cf cg x x =>         Nat.rec           (Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode c))))             (k' + 1, cf) (unpair n).1)           (fun n_1 n_ih => do             let i \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair (k' + 1) (encode c))))                   (k', c) (Nat.pair (unpair n).1 n_1)             Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (k' + 1) (encode c))))                 (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))           (unpair n).2)       (fun cf x => do         let x \u2190           Nat.Partrec.Code.lup               (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                 (List.range (Nat.pair (k' + 1) (encode c))))               (k' + 1, cf) n         Nat.rec (some (unpair n).2)             (fun n_1 n_ih =>               Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair (k' + 1) (encode c))))                 (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))             x)       c =     evaln (k' + 1) c n ** cases' c with cf cg cf cg cf cg cf <;>\n  simp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure] ** x\u271d : Unit p n : \u2115 \u22a2 List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) ** simp ** case zero x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) c : Code nk : n \u2208 List.range Nat.zero \u22a2 Nat.rec Option.none       (fun n_1 n_ih =>         rec (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                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair Nat.zero (encode c))))                   (Nat.zero, cf) n             let y \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair Nat.zero (encode c))))                   (Nat.zero, cg) n             some (Nat.pair x y))           (fun cf cg x x => do             let x \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair Nat.zero (encode c))))                   (Nat.zero, cg) n             Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair Nat.zero (encode c))))                 (Nat.zero, cf) x)           (fun cf cg x x =>             Nat.rec               (Nat.Partrec.Code.lup                 (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                   (List.range (Nat.pair Nat.zero (encode c))))                 (Nat.zero, cf) (unpair n).1)               (fun n_2 n_ih => do                 let i \u2190                   Nat.Partrec.Code.lup                       (List.map                         (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                         (List.range (Nat.pair Nat.zero (encode c))))                       (n_1, c) (Nat.pair (unpair n).1 n_2)                 Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair Nat.zero (encode c))))                     (Nat.zero, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i)))               (unpair n).2)           (fun cf x => do             let x \u2190               Nat.Partrec.Code.lup                   (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                     (List.range (Nat.pair Nat.zero (encode c))))                   (Nat.zero, cf) n             Nat.rec (some (unpair n).2)                 (fun n_2 n_ih =>                   Nat.Partrec.Code.lup                     (List.map                       (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))                       (List.range (Nat.pair Nat.zero (encode c))))                     (n_1, c) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))                 x)           c)       Nat.zero =     evaln Nat.zero c n ** simp [evaln] ** x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) c : Code k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' \u22a2 \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode c))))           (k', c') n =         evaln k' c' n ** intro k\u2081 c\u2081 n\u2081 hl ** x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) c : Code k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' k\u2081 : \u2115 c\u2081 : Code n\u2081 : \u2115 hl : Nat.pair k\u2081 (encode c\u2081) < Nat.pair k (encode c) \u22a2 Nat.Partrec.Code.lup       (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))         (List.range (Nat.pair k (encode c))))       (k\u2081, c\u2081) n\u2081 =     evaln k\u2081 c\u2081 n\u2081 ** simp [lup, List.get?_range hl, evaln_map, Bind.bind] ** case succ.pair x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (pair cf cg)))))           (k', c') n =         evaln k' c' n \u22a2 (Option.bind       (Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))         (k' + 1, cf) n)       fun x =>       Option.bind         (Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))           (k' + 1, cg) n)         fun y => some (Nat.pair x y)) =     Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n) ** cases' encode_lt_pair cf cg with lf lg ** case succ.pair.intro x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (pair cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (pair cf cg) lg : encode cg < encode (pair cf cg) \u22a2 (Option.bind       (Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))         (k' + 1, cf) n)       fun x =>       Option.bind         (Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))           (k' + 1, cg) n)         fun y => some (Nat.pair x y)) =     Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n) ** rw [hg (Nat.pair_lt_pair_right _ lf), hg (Nat.pair_lt_pair_right _ lg)] ** case succ.pair.intro x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (pair cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (pair cf cg) lg : encode cg < encode (pair cf cg) \u22a2 (Option.bind (evaln k cf n) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =     Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n) ** cases evaln k cf n ** case succ.pair.intro.some x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (pair cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (pair cf cg) lg : encode cg < encode (pair cf cg) val\u271d : \u2115 \u22a2 (Option.bind (some val\u271d) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =     Option.bind (Option.map Nat.pair (some val\u271d)) fun y => Option.map y (evaln (k' + 1) cg n) ** cases evaln k cg n <;> rfl ** case succ.pair.intro.none x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (pair cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (pair cf cg) lg : encode cg < encode (pair cf cg) \u22a2 (Option.bind Option.none fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =     Option.bind (Option.map Nat.pair Option.none) fun y => Option.map y (evaln (k' + 1) cg n) ** rfl ** case succ.comp x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (comp cf cg)))))           (k', c') n =         evaln k' c' n \u22a2 (Option.bind       (Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))         (k' + 1, cg) n)       fun x =>       Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))         (k' + 1, cf) x) =     Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x ** cases' encode_lt_comp cf cg with lf lg ** case succ.comp.intro x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (comp cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (comp cf cg) lg : encode cg < encode (comp cf cg) \u22a2 (Option.bind       (Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))         (k' + 1, cg) n)       fun x =>       Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))         (k' + 1, cf) x) =     Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x ** rw [hg (Nat.pair_lt_pair_right _ lg)] ** case succ.comp.intro x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (comp cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (comp cf cg) lg : encode cg < encode (comp cf cg) \u22a2 (Option.bind (evaln k cg n) fun x =>       Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))         (k' + 1, cf) x) =     Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x ** cases evaln k cg n ** case succ.comp.intro.some x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (comp cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (comp cf cg) lg : encode cg < encode (comp cf cg) val\u271d : \u2115 \u22a2 (Option.bind (some val\u271d) fun x =>       Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))         (k' + 1, cf) x) =     Option.bind (some val\u271d) fun x => evaln (k' + 1) cf x ** simp [hg (Nat.pair_lt_pair_right _ lf)] ** case succ.comp.intro.none x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (comp cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (comp cf cg) lg : encode cg < encode (comp cf cg) \u22a2 (Option.bind Option.none fun x =>       Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))         (k' + 1, cf) x) =     Option.bind Option.none fun x => evaln (k' + 1) cf x ** rfl ** case succ.prec x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (prec cf cg)))))           (k', c') n =         evaln k' c' n \u22a2 Nat.rec       (Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))         (k' + 1, cf) (unpair n).1)       (fun n_1 n_ih =>         Option.bind           (Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k', prec cf cg) (Nat.pair (unpair n).1 n_1))           fun i =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       (unpair n).2 =     Nat.rec (evaln (k' + 1) cf (unpair n).1)       (fun n_1 n_ih =>         Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>           evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       (unpair n).2 ** cases' encode_lt_prec cf cg with lf lg ** case succ.prec.intro x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (prec cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (prec cf cg) lg : encode cg < encode (prec cf cg) \u22a2 Nat.rec       (Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))         (k' + 1, cf) (unpair n).1)       (fun n_1 n_ih =>         Option.bind           (Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k', prec cf cg) (Nat.pair (unpair n).1 n_1))           fun i =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       (unpair n).2 =     Nat.rec (evaln (k' + 1) cf (unpair n).1)       (fun n_1 n_ih =>         Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>           evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       (unpair n).2 ** rw [hg (Nat.pair_lt_pair_right _ lf)] ** case succ.prec.intro x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (prec cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (prec cf cg) lg : encode cg < encode (prec cf cg) \u22a2 Nat.rec (evaln k cf (unpair n).1)       (fun n_1 n_ih =>         Option.bind           (Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k', prec cf cg) (Nat.pair (unpair n).1 n_1))           fun i =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       (unpair n).2 =     Nat.rec (evaln (k' + 1) cf (unpair n).1)       (fun n_1 n_ih =>         Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>           evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       (unpair n).2 ** cases n.unpair.2 ** case succ.prec.intro.succ x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (prec cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (prec cf cg) lg : encode cg < encode (prec cf cg) n\u271d : \u2115 \u22a2 Nat.rec (evaln k cf (unpair n).1)       (fun n_1 n_ih =>         Option.bind           (Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k', prec cf cg) (Nat.pair (unpair n).1 n_1))           fun i =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       (Nat.succ n\u271d) =     Nat.rec (evaln (k' + 1) cf (unpair n).1)       (fun n_1 n_ih =>         Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>           evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       (Nat.succ n\u271d) ** simp only [decode_eq_ofNat, Option.some.injEq] ** case succ.prec.intro.succ x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (prec cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (prec cf cg) lg : encode cg < encode (prec cf cg) n\u271d : \u2115 \u22a2 (Option.bind       (Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))         (k', prec cf cg) (Nat.pair (unpair n).1 n\u271d))       fun i =>       Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))         (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =     Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n\u271d)) fun i =>       evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i)) ** rw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)] ** case succ.prec.intro.succ x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (prec cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (prec cf cg) lg : encode cg < encode (prec cf cg) n\u271d : \u2115 \u22a2 (Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n\u271d)) fun i =>       Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))         (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =     Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n\u271d)) fun i =>       evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i)) ** cases evaln k' _ _ ** case succ.prec.intro.succ.some x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (prec cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (prec cf cg) lg : encode cg < encode (prec cf cg) n\u271d val\u271d : \u2115 \u22a2 (Option.bind (some val\u271d) fun i =>       Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))         (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =     Option.bind (some val\u271d) fun i => evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i)) ** simp [hg (Nat.pair_lt_pair_right _ lg)] ** case succ.prec.intro.zero x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (prec cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (prec cf cg) lg : encode cg < encode (prec cf cg) \u22a2 Nat.rec (evaln k cf (unpair n).1)       (fun n_1 n_ih =>         Option.bind           (Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k', prec cf cg) (Nat.pair (unpair n).1 n_1))           fun i =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))             (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       Nat.zero =     Nat.rec (evaln (k' + 1) cf (unpair n).1)       (fun n_1 n_ih =>         Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>           evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))       Nat.zero ** rfl ** case succ.prec.intro.succ.none x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf cg : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (prec cf cg)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (prec cf cg) lg : encode cg < encode (prec cf cg) n\u271d : \u2115 \u22a2 (Option.bind Option.none fun i =>       Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))         (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n\u271d i))) =     Option.bind Option.none fun i => evaln (k' + 1) cg (Nat.pair (unpair n).1 (Nat.pair n\u271d i)) ** rfl ** case succ.rfind' x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (rfind' cf)))))           (k', c') n =         evaln k' c' n \u22a2 (Option.bind       (Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))         (k' + 1, cf) n)       fun x =>       Nat.rec (some (unpair n).2)         (fun n_1 n_ih =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))             (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))         x) =     Option.bind (evaln (k' + 1) cf n) fun x =>       if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) ** have lf := encode_lt_rfind' cf ** case succ.rfind' x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (rfind' cf)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (rfind' cf) \u22a2 (Option.bind       (Nat.Partrec.Code.lup         (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))           (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))         (k' + 1, cf) n)       fun x =>       Nat.rec (some (unpair n).2)         (fun n_1 n_ih =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))             (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))         x) =     Option.bind (evaln (k' + 1) cf n) fun x =>       if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) ** rw [hg (Nat.pair_lt_pair_right _ lf)] ** case succ.rfind' x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (rfind' cf)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (rfind' cf) \u22a2 (Option.bind (evaln k cf n) fun x =>       Nat.rec (some (unpair n).2)         (fun n_1 n_ih =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))             (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))         x) =     Option.bind (evaln (k' + 1) cf n) fun x =>       if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) ** cases' evaln k cf n with x ** case succ.rfind'.some x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (rfind' cf)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (rfind' cf) x : \u2115 \u22a2 (Option.bind (some x) fun x =>       Nat.rec (some (unpair n).2)         (fun n_1 n_ih =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))             (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))         x) =     Option.bind (some x) fun x =>       if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) ** simp only [decode_eq_ofNat, Option.some.injEq, Option.some_bind] ** case succ.rfind'.some x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (rfind' cf)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (rfind' cf) x : \u2115 \u22a2 Nat.rec (some (unpair n).2)       (fun n_1 n_ih =>         Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))           (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))       x =     if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) ** cases x <;> simp [Nat.succ_ne_zero] ** case succ.rfind'.some.succ x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (rfind' cf)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (rfind' cf) n\u271d : \u2115 \u22a2 Nat.Partrec.Code.lup       (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))         (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))       (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) =     evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) ** rw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)] ** case succ.rfind'.none x\u271d : Unit p n : \u2115 this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2))) k' : \u2115 k : \u2115 := k' + 1 nk : n \u2264 k' cf : Code hg :   \u2200 {k' : \u2115} {c' : Code} {n : \u2115},     Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192       Nat.Partrec.Code.lup           (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))             (List.range (Nat.pair k (encode (rfind' cf)))))           (k', c') n =         evaln k' c' n lf : encode cf < encode (rfind' cf) \u22a2 (Option.bind Option.none fun x =>       Nat.rec (some (unpair n).2)         (fun n_1 n_ih =>           Nat.Partrec.Code.lup             (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))               (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))             (k', rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)))         x) =     Option.bind Option.none fun x =>       if x = 0 then some (unpair n).2 else evaln k' (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1)) ** rfl ** this :   Primrec\u2082 fun x n =>     let a := ofNat (\u2115 \u00d7 Code) n;     List.map (evaln a.1 a.2) (List.range a.1) x\u271d : (\u2115 \u00d7 Code) \u00d7 \u2115 k : \u2115 c : Code n : \u2115 \u22a2 (Option.bind       (List.get?         (let a := ofNat (\u2115 \u00d7 Code) (encode ((k, c), n).1);         List.map (evaln a.1 a.2) (List.range a.1))         ((k, c), n).2)       fun b => (((k, c), n), b).2) =     evaln ((k, c), n).1.1 ((k, c), n).1.2 ((k, c), n).2 ** simp [evaln_map] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pairwise_disjoint_Ioo_add_int_cast ** \u03b1 : Type u_1 inst\u271d : OrderedRing \u03b1 a : \u03b1 \u22a2 Pairwise (Disjoint on fun n => Ioo (a + \u2191n) (a + \u2191n + 1)) ** simpa only [zsmul_one, Int.cast_add, Int.cast_one, \u2190 add_assoc] using\n  pairwise_disjoint_Ioo_add_zsmul a (1 : \u03b1) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_normalize_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 \u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) ** rw [ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds,\n  tendsto_iff_forall_testAgainstNN_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 \u22a2 \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)) ** exact fun f => tendsto_normalize_testAgainstNN_of_tendsto \u03bcs_lim nonzero f ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.ofReal_norm_sub_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 g : { x // x \u2208 Lp \u03b2 1 } \u22a2 ENNReal.ofReal \u2016f - g\u2016 = \u222b\u207b (x : \u03b1), \u2191\u2016\u2191\u2191f x - \u2191\u2191g x\u2016\u208a \u2202\u03bc ** simp_rw [ofReal_norm_eq_lintegral, \u2190 edist_eq_coe_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 g : { x // x \u2208 Lp \u03b2 1 } \u22a2 \u222b\u207b (x : \u03b1), edist (\u2191\u2191(f - g) x) 0 \u2202\u03bc = \u222b\u207b (x : \u03b1), edist (\u2191\u2191f x - \u2191\u2191g x) 0 \u2202\u03bc ** apply lintegral_congr_ae ** 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 g : { x // x \u2208 Lp \u03b2 1 } \u22a2 (fun a => edist (\u2191\u2191(f - g) a) 0) =\u1d50[\u03bc] fun a => edist (\u2191\u2191f a - \u2191\u2191g a) 0 ** filter_upwards [Lp.coeFn_sub f g] with _ ha ** 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 g : { x // x \u2208 Lp \u03b2 1 } a\u271d : \u03b1 ha : \u2191\u2191(f - g) a\u271d = (\u2191\u2191f - \u2191\u2191g) a\u271d \u22a2 edist (\u2191\u2191(f - g) a\u271d) 0 = edist (\u2191\u2191f a\u271d - \u2191\u2191g a\u271d) 0 ** simp only [ha, Pi.sub_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.toDual_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 hs : Finset.Nonempty s \u22a2 \u2191toDual (max' s hs) = min' (image (\u2191toDual) s) (_ : Finset.Nonempty (image (\u2191toDual) 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 hs : Finset.Nonempty s \u22a2 \u2191(\u2191toDual (max' s hs)) = \u2191(min' (image (\u2191toDual) s) (_ : Finset.Nonempty (image (\u2191toDual) s))) ** simp only [max'_eq_sup', id_eq, toDual_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 hs : Finset.Nonempty s \u22a2 inf s (WithTop.some \u2218 fun x => \u2191toDual x) = inf s ((WithTop.some \u2218 fun x => x) \u2218 \u2191toDual) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.cgf_zero_measure ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d \u22a2 cgf X 0 t = 0 ** simp only [cgf, log_zero, mgf_zero_measure] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.singularPart_add_withDensity_rnDeriv_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 singularPart s \u03bc + withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = s ** conv_rhs =>\n  rw [\u2190 toSignedMeasure_toJordanDecomposition s, JordanDecomposition.toSignedMeasure] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 singularPart s \u03bc + withDensity\u1d65 \u03bc (rnDeriv s \u03bc) =     toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart ** rw [singularPart, rnDeriv,\n  withDensity\u1d65_sub' (integrable_toReal_of_lintegral_ne_top _ _)\n    (integrable_toReal_of_lintegral_ne_top _ _),\n  withDensity\u1d65_toReal, withDensity\u1d65_toReal, sub_eq_add_neg, sub_eq_add_neg,\n  add_comm (s.toJordanDecomposition.posPart.singularPart \u03bc).toSignedMeasure, \u2190 add_assoc,\n  add_assoc (-(s.toJordanDecomposition.negPart.singularPart \u03bc).toSignedMeasure),\n  \u2190 toSignedMeasure_add, add_comm, \u2190 add_assoc, \u2190 neg_add, \u2190 toSignedMeasure_add, add_comm,\n  \u2190 sub_eq_add_neg] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 toSignedMeasure         (Measure.singularPart (toJordanDecomposition s).posPart \u03bc +           withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x) -       toSignedMeasure         ((withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x) +           Measure.singularPart (toJordanDecomposition s).negPart \u03bc) =     toSignedMeasure (toJordanDecomposition s).posPart - toSignedMeasure (toJordanDecomposition s).negPart  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hfm \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hfm \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x  \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x  \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4 ** convert rfl ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hfm \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  case hfm \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x  case hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x  \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x \u2202\u03bc \u2260 \u22a4  \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x  \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4 ** all_goals\n  first\n  | exact (lintegral_rnDeriv_lt_top _ _).ne\n  | measurability ** case h.e'_3.h.e'_5.h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 (toJordanDecomposition s).posPart =     Measure.singularPart (toJordanDecomposition s).posPart \u03bc +       withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).posPart \u03bc x ** exact s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add \u03bc ** case h.e'_3.h.e'_6.h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 (toJordanDecomposition s).negPart =     (withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x) +       Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** rw [add_comm] ** case h.e'_3.h.e'_6.h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 (toJordanDecomposition s).negPart =     Measure.singularPart (toJordanDecomposition s).negPart \u03bc +       withDensity \u03bc fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x ** exact s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add \u03bc ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4 ** first\n| exact (lintegral_rnDeriv_lt_top _ _).ne\n| measurability ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \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 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition s \u03bc \u22a2 AEMeasurable fun x => Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x ** measurability ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.map_add_left_Ico ** \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) (Ico a b) = Ico (c + a) (c + b) ** rw [\u2190 coe_inj, coe_map, coe_Ico, coe_Ico] ** \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.Ico a b = Set.Ico (c + a) (c + b) ** exact Set.image_const_add_Ico _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_lintegral_thickenedIndicator_of_isClosed ** \u03a9\u271d : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03a9 inst\u271d\u00b2 : PseudoEMetricSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc F : Set \u03a9 F_closed : IsClosed F \u03b4s : \u2115 \u2192 \u211d \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u03bc F)) ** apply measure_of_cont_bdd_of_tendsto_indicator \u03bc F_closed.measurableSet\n  (fun n => thickenedIndicator (\u03b4s_pos n) F) fun n \u03c9 => thickenedIndicator_le_one (\u03b4s_pos n) F \u03c9 ** \u03a9\u271d : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03a9 inst\u271d\u00b2 : PseudoEMetricSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc F : Set \u03a9 F_closed : IsClosed F \u03b4s : \u2115 \u2192 \u211d \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => \u2191(thickenedIndicator (_ : 0 < \u03b4s n) F)) atTop (\ud835\udcdd (indicator F fun x => 1)) ** have key := thickenedIndicator_tendsto_indicator_closure \u03b4s_pos \u03b4s_lim F ** \u03a9\u271d : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03a9 inst\u271d\u00b2 : PseudoEMetricSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc F : Set \u03a9 F_closed : IsClosed F \u03b4s : \u2115 \u2192 \u211d \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key : Tendsto (fun n => \u2191(thickenedIndicator (_ : 0 < \u03b4s n) F)) atTop (\ud835\udcdd (indicator (closure F) fun x => 1)) \u22a2 Tendsto (fun n => \u2191(thickenedIndicator (_ : 0 < \u03b4s n) F)) atTop (\ud835\udcdd (indicator F fun x => 1)) ** rwa [F_closed.closure_eq] at key ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_comp_smul_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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** rw [integrableOn_Ici_iff_integrableOn_Ioi] at 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ioi a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** have t2 := intervalIntegral_tendsto_integral_Ioi _ hg2 tendsto_id ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ioi a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u t2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x)) \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** have : Ioi (f a) \u2286 f '' Ici a :=\n  Ioi_subset_Ici_self.trans <|\n    IsPreconnected.intermediate_value_Ici isPreconnected_Ici left_mem_Ici\n      (le_principal_iff.mpr <| Ici_mem_atTop _) hf hft ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ioi a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u t2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x)) this : Ioi (f a) \u2286 f '' Ici a \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** have t1 := (intervalIntegral_tendsto_integral_Ioi _ (hg1.mono_set this) tendsto_id).comp hft ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ioi a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u t2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x)) this : Ioi (f a) \u2286 f '' Ici a t1 : Tendsto ((fun i => \u222b (x : \u211d) in f a..id i, g x) \u2218 f) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi (f a), g x)) \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** exact tendsto_nhds_unique (Tendsto.congr' (eventuallyEq_of_mem (Ioi_mem_atTop a) eq) t2) t1 ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** have i1 : Ioo (min a b) (max a b) \u2286 Ioi a := by\n  rw [min_eq_left hb.le]\n  exact Ioo_subset_Ioi_self ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** have i2 : [[a, b]] \u2286 Ici a := by rw [uIcc_of_le hb.le]; exact Icc_subset_Ici_self ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** refine'\n  intervalIntegral.integral_comp_smul_deriv''' (hf.mono i2)\n    (fun x hx => hff' x <| mem_of_mem_of_subset hx i1) (hg_cont.mono <| image_subset _ _)\n    (hg1.mono_set <| image_subset _ _) (hg2.mono_set i2) ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b \u22a2 Ioo (min a b) (max a b) \u2286 Ioi a ** rw [min_eq_left hb.le] ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b \u22a2 Ioo a (max a b) \u2286 Ioi a ** exact Ioo_subset_Ioi_self ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a \u22a2 [[a, b]] \u2286 Ici a ** rw [uIcc_of_le hb.le] ** 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a \u22a2 Icc a b \u2286 Ici a ** exact Icc_subset_Ici_self ** case refine'_1 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 Ioo (min a b) (max a b) \u2286 Ioi a ** rw [min_eq_left hb.le] ** case refine'_1 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 Ioo a (max a b) \u2286 Ioi a ** exact Ioo_subset_Ioi_self ** case refine'_2 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 [[a, b]] \u2286 Ici a ** rw [uIcc_of_le hb.le] ** case refine'_2 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' : \u211d \u2192 \u211d g : \u211d \u2192 E 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 => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 Icc a b \u2286 Ici a ** exact Icc_subset_Ici_self ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.PairingHeapImp.Heap.size_combine ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 \u22a2 size (combine le s) = size s ** unfold combine ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 \u22a2 size       (match s with       | h\u2081@h_1:(node a child h\u2082@h:(node a_1 child_1 s)) => merge le (merge le h\u2081 h\u2082) (combine le s)       | h => h) =     size s ** split ** case h_1 \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool x\u271d : Heap \u03b1 a\u271d\u00b9 : \u03b1 child\u271d\u00b9 : Heap \u03b1 a\u271d : \u03b1 child\u271d s\u271d : Heap \u03b1 \u22a2 size (merge le (merge le (node a\u271d\u00b9 child\u271d\u00b9 (node a\u271d child\u271d s\u271d)) (node a\u271d child\u271d s\u271d)) (combine le s\u271d)) =     size (node a\u271d\u00b9 child\u271d\u00b9 (node a\u271d child\u271d s\u271d)) ** rename_i a\u2081 c\u2081 a\u2082 c\u2082 s ** case h_1 \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool x\u271d : Heap \u03b1 a\u2081 : \u03b1 c\u2081 : Heap \u03b1 a\u2082 : \u03b1 c\u2082 s : Heap \u03b1 \u22a2 size (merge le (merge le (node a\u2081 c\u2081 (node a\u2082 c\u2082 s)) (node a\u2082 c\u2082 s)) (combine le s)) =     size (node a\u2081 c\u2081 (node a\u2082 c\u2082 s)) ** rw [size_merge le (noSibling_merge _ _ _) (noSibling_combine _ _),\n  size_merge_node, size_combine le s] ** case h_1 \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool x\u271d : Heap \u03b1 a\u2081 : \u03b1 c\u2081 : Heap \u03b1 a\u2082 : \u03b1 c\u2082 s : Heap \u03b1 \u22a2 size c\u2081 + size c\u2082 + 2 + size s = size (node a\u2081 c\u2081 (node a\u2082 c\u2082 s)) ** simp_arith [size] ** case h_2 \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s x\u271d\u00b9 : Heap \u03b1 x\u271d : \u2200 (a : \u03b1) (child : Heap \u03b1) (a_1 : \u03b1) (child_1 s_1 : Heap \u03b1), s = node a child (node a_1 child_1 s_1) \u2192 False \u22a2 size s = size s ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.AssocList.foldl_eq ** \u03b4 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 f : \u03b4 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b4 init : \u03b4 l : AssocList \u03b1 \u03b2 \u22a2 foldl f init l =     List.foldl       (fun d x =>         match x with         | (a, b) => f d a b)       init (toList l) ** simp [List.foldl_eq_foldlM, foldl, Id.run] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.le_add_of_neg_add_le_left ** a b c : Int h : -b + a \u2264 c \u22a2 a \u2264 b + c ** rw [Int.add_comm] at h ** a b c : Int h : a + -b \u2264 c \u22a2 a \u2264 b + c ** exact Int.le_add_of_sub_left_le h ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ite_inter_of_inter_eq ** \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 h : s\u2081 \u2229 s = s\u2082 \u2229 s \u22a2 Set.ite t s\u2081 s\u2082 \u2229 s = s\u2081 \u2229 s ** rw [\u2190 ite_inter, \u2190 h, ite_same] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.evaln_complete ** c : Code n x : \u2115 h : x \u2208 eval c n \u22a2 \u2203 k, x \u2208 evaln k c n ** rsuffices \u27e8k, h\u27e9 : \u2203 k, x \u2208 evaln (k + 1) c n ** c : Code n x : \u2115 h : x \u2208 eval c n \u22a2 \u2203 k, x \u2208 evaln (k + 1) c n ** induction c generalizing n x <;> simp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2 ** case zero n x : \u2115 h : x = 0 \u22a2 (\u2203 x, n \u2264 x) \u2227 0 = x  case succ n x : \u2115 h : x = Nat.succ n \u22a2 (\u2203 x, n \u2264 x) \u2227 Nat.succ n = x  case left n x : \u2115 h : x = (unpair n).1 \u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).1 = x  case right n x : \u2115 h : x = (unpair n).2 \u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).2 = x  case pair a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x  case comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x  case prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h :   x \u2208     Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))       (unpair n).2 \u22a2 \u2203 k,     n \u2264 k \u2227       Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)           (fun n_1 n_ih =>             Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>               evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))           (unpair n).2 =         some x  case rfind' a\u271d : Code a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h :   \u2203 a,     (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227         \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227       a + (unpair n).2 = x \u22a2 \u2203 k,     n \u2264 k \u2227       \u2203 a,         evaln (k + 1) a\u271d n = some a \u2227           (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x ** iterate 4 exact \u27e8\u27e8_, le_rfl\u27e9, h.symm\u27e9 ** case pair a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x  case comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x  case prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h :   x \u2208     Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))       (unpair n).2 \u22a2 \u2203 k,     n \u2264 k \u2227       Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)           (fun n_1 n_ih =>             Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>               evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))           (unpair n).2 =         some x  case rfind' a\u271d : Code a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h :   \u2203 a,     (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227         \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227       a + (unpair n).2 = x \u22a2 \u2203 k,     n \u2264 k \u2227       \u2203 a,         evaln (k + 1) a\u271d n = some a \u2227           (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x ** case pair cf cg hf hg =>\n  rcases h with \u27e8x, hx, y, hy, rfl\u27e9\n  rcases hf hx with \u27e8k\u2081, hk\u2081\u27e9; rcases hg hy with \u27e8k\u2082, hk\u2082\u27e9\n  refine' \u27e8max k\u2081 k\u2082, _\u27e9\n  refine'\n    \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _,\n      evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081, _,\n      evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082, rfl\u27e9 ** case comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x  case prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h :   x \u2208     Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))       (unpair n).2 \u22a2 \u2203 k,     n \u2264 k \u2227       Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)           (fun n_1 n_ih =>             Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>               evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))           (unpair n).2 =         some x  case rfind' a\u271d : Code a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h :   \u2203 a,     (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227         \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227       a + (unpair n).2 = x \u22a2 \u2203 k,     n \u2264 k \u2227       \u2203 a,         evaln (k + 1) a\u271d n = some a \u2227           (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x ** case comp cf cg hf hg =>\n  rcases h with \u27e8y, hy, hx\u27e9\n  rcases hg hy with \u27e8k\u2081, hk\u2081\u27e9; rcases hf hx with \u27e8k\u2082, hk\u2082\u27e9\n  refine' \u27e8max k\u2081 k\u2082, _\u27e9\n  exact\n    \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _,\n      evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n      evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\u27e9 ** case intro c : Code n x : \u2115 h\u271d : x \u2208 eval c n k : \u2115 h : x \u2208 evaln (k + 1) c n \u22a2 \u2203 k, x \u2208 evaln k c n ** exact \u27e8k + 1, h\u27e9 ** case right n x : \u2115 h : x = (unpair n).2 \u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).2 = x  case pair a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x  case comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x  case prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h :   x \u2208     Nat.rec (eval a\u271d\u00b9 (unpair n).1) (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).1 (Nat.pair y i)))       (unpair n).2 \u22a2 \u2203 k,     n \u2264 k \u2227       Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).1)           (fun n_1 n_ih =>             Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).1 n_1)) fun i =>               evaln (k + 1) a\u271d (Nat.pair (unpair n).1 (Nat.pair n_1 i)))           (unpair n).2 =         some x  case rfind' a\u271d : Code a_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n n x : \u2115 h :   \u2203 a,     (0 \u2208 eval a\u271d (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227         \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227       a + (unpair n).2 = x \u22a2 \u2203 k,     n \u2264 k \u2227       \u2203 a,         evaln (k + 1) a\u271d n = some a \u2227           (if a = 0 then some (unpair n).2 else evaln k (rfind' a\u271d) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x ** exact \u27e8\u27e8_, le_rfl\u27e9, h.symm\u27e9 ** cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x : \u2115 h : \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x ** rcases h with \u27e8x, hx, y, hy, rfl\u27e9 ** case intro.intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x : \u2115 hx : x \u2208 eval cf n y : \u2115 hy : y \u2208 eval cg n \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y ** rcases hf hx with \u27e8k\u2081, hk\u2081\u27e9 ** case intro.intro.intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x : \u2115 hx : x \u2208 eval cf n y : \u2115 hy : y \u2208 eval cg n k\u2081 : \u2115 hk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y ** rcases hg hy with \u27e8k\u2082, hk\u2082\u27e9 ** case intro.intro.intro.intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x : \u2115 hx : x \u2208 eval cf n y : \u2115 hy : y \u2208 eval cg n k\u2081 : \u2115 hk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n k\u2082 : \u2115 hk\u2082 : y \u2208 evaln (k\u2082 + 1) cg n \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y ** refine' \u27e8max k\u2081 k\u2082, _\u27e9 ** case intro.intro.intro.intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x : \u2115 hx : x \u2208 eval cf n y : \u2115 hy : y \u2208 eval cg n k\u2081 : \u2115 hk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n k\u2082 : \u2115 hk\u2082 : y \u2208 evaln (k\u2082 + 1) cg n \u22a2 n \u2264 max k\u2081 k\u2082 \u2227     \u2203 a,       evaln (max k\u2081 k\u2082 + 1) cf n = some a \u2227 \u2203 a_1, evaln (max k\u2081 k\u2082 + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y ** refine'\n  \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _,\n    evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081, _,\n    evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082, rfl\u27e9 ** cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x : \u2115 h : \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x ** rcases h with \u27e8y, hy, hx\u27e9 ** case intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x y : \u2115 hy : y \u2208 eval cg n hx : x \u2208 eval cf y \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x ** rcases hg hy with \u27e8k\u2081, hk\u2081\u27e9 ** case intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x y : \u2115 hy : y \u2208 eval cg n hx : x \u2208 eval cf y k\u2081 : \u2115 hk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x ** rcases hf hx with \u27e8k\u2082, hk\u2082\u27e9 ** case intro.intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x y : \u2115 hy : y \u2208 eval cg n hx : x \u2208 eval cf y k\u2081 : \u2115 hk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n k\u2082 : \u2115 hk\u2082 : x \u2208 evaln (k\u2082 + 1) cf y \u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x ** refine' \u27e8max k\u2081 k\u2082, _\u27e9 ** case intro.intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x y : \u2115 hy : y \u2208 eval cg n hx : x \u2208 eval cf y k\u2081 : \u2115 hk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n k\u2082 : \u2115 hk\u2082 : x \u2208 evaln (k\u2082 + 1) cf y \u22a2 n \u2264 max k\u2081 k\u2082 \u2227 \u2203 a, evaln (max k\u2081 k\u2082 + 1) cg n = some a \u2227 evaln (max k\u2081 k\u2082 + 1) cf a = some x ** exact\n  \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _,\n    evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n    evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\u27e9 ** cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x : \u2115 h :   x \u2208     Nat.rec (eval cf (unpair n).1) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).1 (Nat.pair y i)))       (unpair n).2 \u22a2 \u2203 k,     n \u2264 k \u2227       Nat.rec (evaln (k + 1) cf (unpair n).1)           (fun n_1 n_ih =>             Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>               evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))           (unpair n).2 =         some x ** revert h ** cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x : \u2115 \u22a2 x \u2208       Nat.rec (eval cf (unpair n).1) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).1 (Nat.pair y i)))         (unpair n).2 \u2192     \u2203 k,       n \u2264 k \u2227         Nat.rec (evaln (k + 1) cf (unpair n).1)             (fun n_1 n_ih =>               Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).1 n_1)) fun i =>                 evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))             (unpair n).2 =           some x ** generalize n.unpair.1 = n\u2081 ** cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x n\u2081 : \u2115 \u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) (unpair n).2 \u2192     \u2203 k,       n \u2264 k \u2227         Nat.rec (evaln (k + 1) cf n\u2081)             (fun n n_ih =>               Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))             (unpair n).2 =           some x ** generalize n.unpair.2 = n\u2082 ** cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n x n\u2081 n\u2082 : \u2115 \u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) n\u2082 \u2192     \u2203 k,       n \u2264 k \u2227         Nat.rec (evaln (k + 1) cf n\u2081)             (fun n n_ih =>               Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))             n\u2082 =           some x ** induction' n\u2082 with m IH generalizing x n <;> simp ** case zero cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n\u271d x\u271d n\u2081 n x : \u2115 \u22a2 x \u2208 eval cf n\u2081 \u2192 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x ** intro h ** case zero cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n\u271d x\u271d n\u2081 n x : \u2115 h : x \u2208 eval cf n\u2081 \u22a2 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x ** rcases hf h with \u27e8k, hk\u27e9 ** case zero.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n\u271d x\u271d n\u2081 n x : \u2115 h : x \u2208 eval cf n\u2081 k : \u2115 hk : x \u2208 evaln (k + 1) cf n\u2081 \u22a2 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x ** exact \u27e8_, le_max_left _ _, evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u27e9 ** case succ cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n\u271d x\u271d n\u2081 m : \u2115 IH :   \u2200 {n x : \u2115},     x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192       \u2203 k,         n \u2264 k \u2227           Nat.rec (evaln (k + 1) cf n\u2081)               (fun n n_ih =>                 Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>                   evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))               m =             some x n x : \u2115 \u22a2 \u2200 (x_1 : \u2115),     x_1 \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192       x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m x_1)) \u2192         \u2203 k,           n \u2264 k \u2227             \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x ** intro y hy hx ** case succ cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n\u271d x\u271d n\u2081 m : \u2115 IH :   \u2200 {n x : \u2115},     x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192       \u2203 k,         n \u2264 k \u2227           Nat.rec (evaln (k + 1) cf n\u2081)               (fun n n_ih =>                 Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>                   evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))               m =             some x n x y : \u2115 hy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m hx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y)) \u22a2 \u2203 k,     n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x ** rcases IH hy with \u27e8k\u2081, nk\u2081, hk\u2081\u27e9 ** case succ.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n\u271d x\u271d n\u2081 m : \u2115 IH :   \u2200 {n x : \u2115},     x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192       \u2203 k,         n \u2264 k \u2227           Nat.rec (evaln (k + 1) cf n\u2081)               (fun n n_ih =>                 Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>                   evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))               m =             some x n x y : \u2115 hy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m hx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y)) k\u2081 : \u2115 nk\u2081 : ?m.1146745 \u2264 k\u2081 hk\u2081 :   Nat.rec (evaln (k\u2081 + 1) cf n\u2081)       (fun n n_ih =>         Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))       m =     some y \u22a2 \u2203 k,     n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x ** rcases hg hx with \u27e8k\u2082, hk\u2082\u27e9 ** case succ.intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n\u271d x\u271d n\u2081 m : \u2115 IH :   \u2200 {n x : \u2115},     x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192       \u2203 k,         n \u2264 k \u2227           Nat.rec (evaln (k + 1) cf n\u2081)               (fun n n_ih =>                 Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>                   evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))               m =             some x n x y : \u2115 hy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m hx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y)) k\u2081 : \u2115 nk\u2081 : ?m.1146745 \u2264 k\u2081 hk\u2081 :   Nat.rec (evaln (k\u2081 + 1) cf n\u2081)       (fun n n_ih =>         Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))       m =     some y k\u2082 : \u2115 hk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y)) \u22a2 \u2203 k,     n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x ** refine'\n  \u27e8(max k\u2081 k\u2082).succ,\n    Nat.le_succ_of_le <| le_max_of_le_left <|\n      le_trans (le_max_left _ (Nat.pair n\u2081 m)) nk\u2081, y,\n    evaln_mono (Nat.succ_le_succ <| le_max_left _ _) _,\n    evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_right _ _) hk\u2082\u27e9 ** case succ.intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n\u271d x\u271d n\u2081 m : \u2115 IH :   \u2200 {n x : \u2115},     x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192       \u2203 k,         n \u2264 k \u2227           Nat.rec (evaln (k + 1) cf n\u2081)               (fun n n_ih =>                 Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>                   evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))               m =             some x n x y : \u2115 hy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m hx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y)) k\u2081 : \u2115 nk\u2081 : max n (Nat.pair n\u2081 m) \u2264 k\u2081 hk\u2081 :   Nat.rec (evaln (k\u2081 + 1) cf n\u2081)       (fun n n_ih =>         Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))       m =     some y k\u2082 : \u2115 hk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y)) \u22a2 y \u2208 evaln (Nat.succ k\u2081) (prec cf cg) (Nat.pair n\u2081 m) ** simp only [evaln._eq_8, bind, unpaired, unpair_pair, Option.mem_def, Option.bind_eq_some,\n  Option.guard_eq_some', exists_and_left, exists_const] ** case succ.intro.intro.intro cf cg : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n hg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n n\u271d x\u271d n\u2081 m : \u2115 IH :   \u2200 {n x : \u2115},     x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192       \u2203 k,         n \u2264 k \u2227           Nat.rec (evaln (k + 1) cf n\u2081)               (fun n n_ih =>                 Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>                   evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))               m =             some x n x y : \u2115 hy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m hx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y)) k\u2081 : \u2115 nk\u2081 : max n (Nat.pair n\u2081 m) \u2264 k\u2081 hk\u2081 :   Nat.rec (evaln (k\u2081 + 1) cf n\u2081)       (fun n n_ih =>         Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))       m =     some y k\u2082 : \u2115 hk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y)) \u22a2 Nat.pair n\u2081 m \u2264 k\u2081 \u2227     Nat.rec (evaln (k\u2081 + 1) cf n\u2081)         (fun n n_ih =>           Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))         m =       some y ** exact \u27e8le_trans (le_max_right _ _) nk\u2081, hk\u2081\u27e9 ** cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n x : \u2115 h :   \u2203 a,     (0 \u2208 eval cf (Nat.pair (unpair n).1 (a + (unpair n).2)) \u2227         \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2227       a + (unpair n).2 = x \u22a2 \u2203 k,     n \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf n = some a \u2227           (if a = 0 then some (unpair n).2 else evaln k (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x ** rcases h with \u27e8y, \u27e8hy\u2081, hy\u2082\u27e9, rfl\u27e9 ** case intro.intro.intro cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2)) hy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0 \u22a2 \u2203 k,     n \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf n = some a \u2227           (if a = 0 then some (unpair n).2 else evaln k (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) =             some (y + (unpair n).2) ** suffices \u2203 k, y + n.unpair.2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair n.unpair.1 n.unpair.2) by\n  simpa [evaln, Bind.bind] ** case intro.intro.intro cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2)) hy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0 \u22a2 \u2203 k, y + (unpair n).2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 (unpair n).2) ** revert hy\u2081 hy\u2082 ** case intro.intro.intro cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y : \u2115 \u22a2 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2)) \u2192     (\u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0) \u2192       \u2203 k, y + (unpair n).2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 (unpair n).2) ** generalize n.unpair.2 = m ** case intro.intro.intro cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y m : \u2115 \u22a2 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192     (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192       \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) ** intro hy\u2081 hy\u2082 ** case intro.intro.intro cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 \u22a2 \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) ** induction' y with y IH generalizing m <;> simp [evaln, Bind.bind] ** cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + (unpair n).2)) hy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)) \u2227 \u00aca = 0 this : \u2203 k, y + (unpair n).2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 (unpair n).2) \u22a2 \u2203 k,     n \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf n = some a \u2227           (if a = 0 then some (unpair n).2 else evaln k (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) =             some (y + (unpair n).2) ** simpa [evaln, Bind.bind] ** case intro.intro.intro.zero cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.zero + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 \u22a2 \u2203 k,     Nat.pair (unpair n).1 m \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227           (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m ** simp at hy\u2081 ** case intro.intro.intro.zero cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 m : \u2115 hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m) \u22a2 \u2203 k,     Nat.pair (unpair n).1 m \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227           (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m ** rcases hf hy\u2081 with \u27e8k, hk\u27e9 ** case intro.intro.intro.zero.intro cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 m : \u2115 hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m) k : \u2115 hk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).1 m) \u22a2 \u2203 k,     Nat.pair (unpair n).1 m \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227           (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m ** exact \u27e8_, Nat.le_of_lt_succ <| evaln_bound hk, _, hk, by simp; rfl\u27e9 ** cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 m : \u2115 hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m) k : \u2115 hk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).1 m) \u22a2 (if 0 = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some m ** simp ** cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 m : \u2115 hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 m) k : \u2115 hk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).1 m) \u22a2 pure m = some m ** rfl ** case intro.intro.intro.succ cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 \u22a2 \u2203 k,     Nat.pair (unpair n).1 m \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227           (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m) ** rcases hy\u2082 (Nat.succ_pos _) with \u27e8a, ha, a0\u27e9 ** case intro.intro.intro.succ.intro.intro cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 \u22a2 \u2203 k,     Nat.pair (unpair n).1 m \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227           (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m) ** rcases hf ha with \u27e8k\u2081, hk\u2081\u27e9 ** case intro.intro.intro.succ.intro.intro.intro cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 k\u2081 : \u2115 hk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m)) \u22a2 \u2203 k,     Nat.pair (unpair n).1 m \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227           (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m) ** rcases IH m.succ (by simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2081)\n    fun {i} hi => by\n    simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using\n      hy\u2082 (Nat.succ_lt_succ hi) with\n  \u27e8k\u2082, hk\u2082\u27e9 ** case intro.intro.intro.succ.intro.intro.intro.intro cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 k\u2081 : \u2115 hk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m)) k\u2082 : \u2115 hk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m)) \u22a2 \u2203 k,     Nat.pair (unpair n).1 m \u2264 k \u2227       \u2203 a,         evaln (k + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227           (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) = some (Nat.succ y + m) ** use (max k\u2081 k\u2082).succ ** case h cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 k\u2081 : \u2115 hk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m)) k\u2082 : \u2115 hk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m)) \u22a2 Nat.pair (unpair n).1 m \u2264 Nat.succ (max k\u2081 k\u2082) \u2227     \u2203 a,       evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227         (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =           some (Nat.succ y + m) ** rw [zero_add] at hk\u2081 ** case h cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 k\u2081 : \u2115 hk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m) k\u2082 : \u2115 hk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m)) \u22a2 Nat.pair (unpair n).1 m \u2264 Nat.succ (max k\u2081 k\u2082) \u2227     \u2203 a,       evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227         (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =           some (Nat.succ y + m) ** use Nat.le_succ_of_le <| le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081 ** case right cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 k\u2081 : \u2115 hk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m) k\u2082 : \u2115 hk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m)) \u22a2 \u2203 a,     evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227       (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =         some (Nat.succ y + m) ** use a ** case h cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 k\u2081 : \u2115 hk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m) k\u2082 : \u2115 hk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m)) \u22a2 evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).1 m) = some a \u2227     (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =       some (Nat.succ y + m) ** use evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_left _ _) hk\u2081 ** case right cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 k\u2081 : \u2115 hk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 m) k\u2082 : \u2115 hk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).1 (Nat.succ m)) \u22a2 (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).1 (m + 1))) =     some (Nat.succ y + m) ** simpa [Nat.succ_eq_add_one, a0, -max_eq_left, -max_eq_right, add_comm, add_left_comm] using\n  evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082 ** cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 k\u2081 : \u2115 hk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m)) \u22a2 0 \u2208 eval cf (Nat.pair (unpair n).1 (y + Nat.succ m)) ** simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2081 ** cf : Code hf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n n y\u271d m\u271d : \u2115 hy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).1 (y\u271d + m\u271d)) hy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m + m\u271d)) \u2227 \u00aca = 0 y : \u2115 IH :   \u2200 (m : \u2115),     0 \u2208 eval cf (Nat.pair (unpair n).1 (y + m)) \u2192       (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0) \u2192         \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).1 m) m : \u2115 hy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).1 (Nat.succ y + m)) hy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (m_1 + m)) \u2227 \u00aca = 0 a : \u2115 ha : a \u2208 eval cf (Nat.pair (unpair n).1 (0 + m)) a0 : \u00aca = 0 k\u2081 : \u2115 hk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).1 (0 + m)) i : \u2115 hi : i < y \u22a2 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).1 (i + Nat.succ m)) \u2227 \u00aca = 0 ** simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using\n  hy\u2082 (Nat.succ_lt_succ hi) ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.sub_le_integral_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 : \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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** obtain rfl | a_lt_b := hab.eq_or_lt ** 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 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** set s := {t | g b - g t \u2264 \u222b u in t..b, \u03c6 u} \u2229 Icc a b ** 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 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** have s_closed : IsClosed s := by\n  have : ContinuousOn (fun t => (g b - g t, \u222b u in t..b, \u03c6 u)) (Icc a b) := by\n    rw [\u2190 uIcc_of_le hab] at hcont \u03c6int \u22a2\n    exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left \u03c6int)\n  simp only [inter_comm]\n  exact this.preimage_closed_of_closed isClosed_Icc isClosed_le_prod ** 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 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** have A : closure (Ioc a b) \u2286 s := by\n  apply s_closed.closure_subset_iff.2\n  intro t ht\n  refine' \u27e8_, \u27e8ht.1.le, ht.2\u27e9\u27e9\n  exact\n    sub_le_integral_of_hasDeriv_right_of_le_Ico ht.2 (hcont.mono (Icc_subset_Icc ht.1.le le_rfl))\n      (fun x hx => hderiv x \u27e8ht.1.trans_le hx.1, hx.2\u27e9)\n      (\u03c6int.mono_set (Icc_subset_Icc ht.1.le le_rfl)) fun x hx => h\u03c6g x \u27e8ht.1.trans_le hx.1, hx.2\u27e9 ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s A : closure (Ioc a b) \u2286 s \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** rw [closure_Ioc a_lt_b.ne] at A ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s A : Icc a b \u2286 s \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** exact (A (left_mem_Icc.2 hab)).1 ** case inl \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 : \u211d hab : a \u2264 a hcont : ContinuousOn g (Icc a a) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a a \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a a) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a a \u2192 g' x \u2264 \u03c6 x \u22a2 g a - g a \u2264 \u222b (y : \u211d) in a..a, \u03c6 y ** simp ** \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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b \u22a2 IsClosed s ** have : ContinuousOn (fun t => (g b - g t, \u222b u in t..b, \u03c6 u)) (Icc a b) := by\n  rw [\u2190 uIcc_of_le hab] at hcont \u03c6int \u22a2\n  exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left \u03c6int) ** \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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b this : ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b this : ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) (Icc a b) \u22a2 IsClosed (Icc a b \u2229 {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u}) ** exact this.preimage_closed_of_closed isClosed_Icc isClosed_le_prod ** \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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b \u22a2 ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) (Icc a b) ** rw [\u2190 uIcc_of_le hab] at hcont \u03c6int \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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 [[a, b]] h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b \u22a2 ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) [[a, b]] ** exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left \u03c6int) ** \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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s \u22a2 closure (Ioc a b) \u2286 s ** apply s_closed.closure_subset_iff.2 ** \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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s \u22a2 Ioc a b \u2286 s ** intro t 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 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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 Ioc a b \u22a2 t \u2208 s ** refine' \u27e8_, \u27e8ht.1.le, ht.2\u27e9\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 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 Ioc a b \u22a2 t \u2208 {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} ** exact\n  sub_le_integral_of_hasDeriv_right_of_le_Ico ht.2 (hcont.mono (Icc_subset_Icc ht.1.le le_rfl))\n    (fun x hx => hderiv x \u27e8ht.1.trans_le hx.1, hx.2\u27e9)\n    (\u03c6int.mono_set (Icc_subset_Icc ht.1.le le_rfl)) fun x hx => h\u03c6g x \u27e8ht.1.trans_le hx.1, hx.2\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.degrees_rename_of_injective ** 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 p : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 h : Injective f \u22a2 degrees (\u2191(rename f) p) = Multiset.map f (degrees p) ** classical\nsimp only [degrees, Multiset.map_finset_sup p.support Finsupp.toMultiset f h,\n  support_rename_of_injective h, Finset.sup_image]\nrefine' Finset.sup_congr rfl fun x _ => _\nexact (Finsupp.toMultiset_map _ _).symm ** 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 p : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 h : Injective f \u22a2 degrees (\u2191(rename f) p) = Multiset.map f (degrees p) ** simp only [degrees, Multiset.map_finset_sup p.support Finsupp.toMultiset f h,\n  support_rename_of_injective h, Finset.sup_image] ** 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 p : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 h : Injective f \u22a2 Finset.sup (support p) ((fun s => \u2191toMultiset s) \u2218 Finsupp.mapDomain f) =     Finset.sup (support p) (Multiset.map f \u2218 \u2191toMultiset) ** refine' Finset.sup_congr rfl fun 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\u271d q p : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 h : Injective f x : \u03c3 \u2192\u2080 \u2115 x\u271d : x \u2208 support p \u22a2 ((fun s => \u2191toMultiset s) \u2218 Finsupp.mapDomain f) x = (Multiset.map f \u2218 \u2191toMultiset) x ** exact (Finsupp.toMultiset_map _ _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.foldr_induction ** \u03b1 : Type u_1 \u03b2 : Type u_2 as : Array \u03b1 motive : Nat \u2192 \u03b2 \u2192 Prop init : \u03b2 h0 : motive (size as) init f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 hf : \u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i] b) \u22a2 motive 0 (foldr f init as (size as)) ** have := SatisfiesM_foldrM (m := Id) (as := as) (f := f) motive h0 ** \u03b1 : Type u_1 \u03b2 : Type u_2 as : Array \u03b1 motive : Nat \u2192 \u03b2 \u2192 Prop init : \u03b2 h0 : motive (size as) init f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 hf : \u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i] b) this :   (\u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 SatisfiesM (motive i.val) (f as[i] b)) \u2192     SatisfiesM (motive 0) (foldrM f init as (size as)) \u22a2 motive 0 (foldr f init as (size as)) ** simp [SatisfiesM_Id_eq] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 as : Array \u03b1 motive : Nat \u2192 \u03b2 \u2192 Prop init : \u03b2 h0 : motive (size as) init f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 hf : \u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i] b) this :   (\u2200 (i : Fin (size as)) (b : \u03b2), motive (i.val + 1) b \u2192 motive i.val (f as[i.val] b)) \u2192     motive 0 (foldrM f init as (size as)) \u22a2 motive 0 (foldr f init as (size as)) ** exact this hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrableOn_Ioi_deriv_of_nonneg ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) \u22a2 IntegrableOn g' (Ioi a) ** refine integrableOn_Ioi_of_intervalIntegral_norm_tendsto (l - g a) a (fun x => ?_) tendsto_id ?_ ** case refine_2 E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) \u22a2 Tendsto (fun i => \u222b (x : \u211d) in a..id i, \u2016g' x\u2016) atTop (\ud835\udcdd (l - g a)) ** apply Tendsto.congr' _ (hg.sub_const _) ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) \u22a2 (fun k => g k - g a) =\u1da0[atTop] fun i => \u222b (x : \u211d) in a..id i, \u2016g' x\u2016 ** filter_upwards [Ioi_mem_atTop a] with x hx ** case h E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a \u22a2 g x - g a = \u222b (x : \u211d) in a..id x, \u2016g' x\u2016 ** have h'x : a \u2264 id x := le_of_lt hx ** case h E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x \u22a2 g x - g a = \u222b (x : \u211d) in a..id x, \u2016g' x\u2016 ** calc\n  g x - g a = \u222b y in a..id x, g' y := by\n    symm\n    apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x\n      (hcont.mono Icc_subset_Ici_self) fun y hy => hderiv y hy.1\n    rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x]\n    exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)\n      (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1\n  _ = \u222b y in a..id x, \u2016g' y\u2016 := by\n    simp_rw [intervalIntegral.integral_of_le h'x]\n    refine' set_integral_congr measurableSet_Ioc fun y hy => _\n    dsimp\n    rw [abs_of_nonneg]\n    exact g'pos _ hy.1 ** case refine_1 E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d \u22a2 IntegrableOn g' (Ioc a (id x)) ** exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)\n  (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1 ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x \u22a2 g x - g a = \u222b (y : \u211d) in a..id x, g' y ** symm ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x \u22a2 \u222b (y : \u211d) in a..id x, g' y = g x - g a ** apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x\n  (hcont.mono Icc_subset_Ici_self) fun y hy => hderiv y hy.1 ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x \u22a2 IntervalIntegrable (fun y => g' y) volume a (id x) ** rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x] ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x \u22a2 IntegrableOn (fun y => g' y) (Ioc a (id x)) ** exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self)\n  (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1 ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x \u22a2 \u222b (y : \u211d) in a..id x, g' y = \u222b (y : \u211d) in a..id x, \u2016g' y\u2016 ** simp_rw [intervalIntegral.integral_of_le h'x] ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x \u22a2 \u222b (y : \u211d) in Ioc a (id x), g' y = \u222b (y : \u211d) in Ioc a (id x), \u2016g' y\u2016 ** refine' set_integral_congr measurableSet_Ioc fun y hy => _ ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x y : \u211d hy : y \u2208 Ioc a (id x) \u22a2 g' y = \u2016g' y\u2016 ** dsimp ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x y : \u211d hy : y \u2208 Ioc a (id x) \u22a2 g' y = |g' y| ** rw [abs_of_nonneg] ** E : Type u_1 f f' : \u211d \u2192 E g g' : \u211d \u2192 \u211d a b l : \u211d m : E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hcont : ContinuousOn g (Ici a) hderiv : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 0 \u2264 g' x hg : Tendsto g atTop (\ud835\udcdd l) x : \u211d hx : x \u2208 Ioi a h'x : a \u2264 id x y : \u211d hy : y \u2208 Ioc a (id x) \u22a2 0 \u2264 g' y ** exact g'pos _ hy.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.ext_of_Iic ** \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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a) \u22a2 \u03bc = \u03bd ** refine' ext_of_Ioc_finite \u03bc \u03bd _ fun a b hlt => _ ** case refine'_2 \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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a) a b : \u03b1 hlt : a < b \u22a2 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b) ** rw [\u2190 Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic,\n  measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic, h a, h b] ** case refine'_1 \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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a) \u22a2 \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ ** rcases exists_countable_dense_bot_top \u03b1 with \u27e8s, hsc, hsd, -, hst\u27e9 ** case refine'_1.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 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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a) s : Set \u03b1 hsc : Set.Countable s hsd : Dense s hst : \u2200 (x : \u03b1), IsTop x \u2192 x \u2208 s this : DirectedOn (fun x x_1 => x \u2264 x_1) s \u22a2 \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ ** simp only [\u2190 biSup_measure_Iic hsc (hsd.exists_ge' hst) this, h] ** case refine'_2 \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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a) a b : \u03b1 hlt : a < b \u22a2 \u2191\u2191\u03bd (Iic a) \u2260 \u22a4 ** rw [\u2190 h a] ** case refine'_2 \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 : \u2200 (a : \u03b1), \u2191\u2191\u03bc (Iic a) = \u2191\u2191\u03bd (Iic a) a b : \u03b1 hlt : a < b \u22a2 \u2191\u2191\u03bc (Iic a) \u2260 \u22a4 ** exact (measure_lt_top \u03bc _).ne ** Qed",
        "informal": ""
    },
    {
        "formal": "ComputablePred.computable_iff_re_compl_re ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03c3 p : \u03b1 \u2192 Prop inst\u271d : DecidablePred p x\u271d : RePred p \u2227 RePred fun a => \u00acp a h\u2081 : RePred p h\u2082 : RePred fun a => \u00acp a \u22a2 Computable fun a => decide (p a) ** obtain \u27e8k, pk, hk\u27e9 :=\n  Partrec.merge (h\u2081.map (Computable.const true).to\u2082) (h\u2082.map (Computable.const false).to\u2082)\n  (by\n    intro a x hx y hy\n    simp at hx hy\n    cases hy.1 hx.1) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03c3 p : \u03b1 \u2192 Prop inst\u271d : DecidablePred p x\u271d : RePred p \u2227 RePred fun a => \u00acp a h\u2081 : RePred p h\u2082 : RePred fun a => \u00acp a \u22a2 \u2200 (a : \u03b1) (x : Bool),     x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2192       \u2200 (y : Bool), y \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ()) \u2192 x = y ** intro a x hx y hy ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03c3 p : \u03b1 \u2192 Prop inst\u271d : DecidablePred p x\u271d : RePred p \u2227 RePred fun a => \u00acp a h\u2081 : RePred p h\u2082 : RePred fun a => \u00acp a a : \u03b1 x : Bool hx : x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) y : Bool hy : y \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ()) \u22a2 x = y ** simp at hx hy ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03c3 p : \u03b1 \u2192 Prop inst\u271d : DecidablePred p x\u271d : RePred p \u2227 RePred fun a => \u00acp a h\u2081 : RePred p h\u2082 : RePred fun a => \u00acp a a : \u03b1 x y : Bool hx : p a \u2227 true = x hy : \u00acp a \u2227 false = y \u22a2 x = y ** cases hy.1 hx.1 ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03c3 p : \u03b1 \u2192 Prop inst\u271d : DecidablePred p x\u271d : RePred p \u2227 RePred fun a => \u00acp a h\u2081 : RePred p h\u2082 : RePred fun a => \u00acp a k : \u03b1 \u2192. Bool pk : Partrec k hk :   \u2200 (a : \u03b1) (x : Bool),     x \u2208 k a \u2194       x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228         x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ()) \u22a2 Computable fun a => decide (p a) ** refine' Partrec.of_eq pk fun n => Part.eq_some_iff.2 _ ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03c3 p : \u03b1 \u2192 Prop inst\u271d : DecidablePred p x\u271d : RePred p \u2227 RePred fun a => \u00acp a h\u2081 : RePred p h\u2082 : RePred fun a => \u00acp a k : \u03b1 \u2192. Bool pk : Partrec k hk :   \u2200 (a : \u03b1) (x : Bool),     x \u2208 k a \u2194       x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228         x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ()) n : \u03b1 \u22a2 (fun a => decide (p a)) n \u2208 k n ** rw [hk] ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03c3 p : \u03b1 \u2192 Prop inst\u271d : DecidablePred p x\u271d : RePred p \u2227 RePred fun a => \u00acp a h\u2081 : RePred p h\u2082 : RePred fun a => \u00acp a k : \u03b1 \u2192. Bool pk : Partrec k hk :   \u2200 (a : \u03b1) (x : Bool),     x \u2208 k a \u2194       x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228         x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ()) n : \u03b1 \u22a2 (fun a => decide (p a)) n \u2208 Part.map (fun b => true) (Part.assert (p n) fun x => Part.some ()) \u2228     (fun a => decide (p a)) n \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) n) fun x => Part.some ()) ** simp only [Part.mem_map_iff, Part.mem_assert_iff, Part.mem_some_iff, exists_prop, and_true,\n  Bool.true_eq_decide_iff, and_self, exists_const, Bool.false_eq_decide_iff] ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03c3 p : \u03b1 \u2192 Prop inst\u271d : DecidablePred p x\u271d : RePred p \u2227 RePred fun a => \u00acp a h\u2081 : RePred p h\u2082 : RePred fun a => \u00acp a k : \u03b1 \u2192. Bool pk : Partrec k hk :   \u2200 (a : \u03b1) (x : Bool),     x \u2208 k a \u2194       x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228         x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ()) n : \u03b1 \u22a2 p n \u2228 \u00acp n ** apply Decidable.em ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.laverage_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 t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hd : AEDisjoint \u03bc s t ht : NullMeasurableSet t \u22a2 \u2a0d\u207b (x : \u03b1) in s \u222a t, f x \u2202\u03bc =     \u2191\u2191\u03bc s / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) * \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc + \u2191\u2191\u03bc t / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) * \u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc ** rw [restrict_union\u2080 hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.all_eq_not_any_not ** \u03b1 : Type u_1 l : List \u03b1 p : \u03b1 \u2192 Bool \u22a2 all l p = !any l fun c => !p c ** rw [Bool.eq_iff_iff] ** \u03b1 : Type u_1 l : List \u03b1 p : \u03b1 \u2192 Bool \u22a2 all l p = true \u2194 (!any l fun c => !p c) = true ** simp ** \u03b1 : Type u_1 l : List \u03b1 p : \u03b1 \u2192 Bool \u22a2 (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true) \u2194 (any l fun c => !p c) = false ** rw [\u2190 Bool.not_eq_true, List.any_eq_true] ** \u03b1 : Type u_1 l : List \u03b1 p : \u03b1 \u2192 Bool \u22a2 (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true) \u2194 \u00ac\u2203 x, x \u2208 l \u2227 (!p x) = true ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_eq_three ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 ncard s = 3 \u2194 \u2203 x y z, x \u2260 y \u2227 x \u2260 z \u2227 y \u2260 z \u2227 s = {x, y, z} ** rw [\u2190encard_eq_three, ncard_def, \u2190Nat.cast_inj (R := \u2115\u221e), Nat.cast_ofNat] ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 \u2191(\u2191ENat.toNat (encard s)) = 3 \u2194 encard s = 3 ** refine' \u27e8fun h \u21a6 _, fun h \u21a6 _\u27e9 ** case refine'_2 \u03b1 : Type u_1 s t : Set \u03b1 h : encard s = 3 \u22a2 \u2191(\u2191ENat.toNat (encard s)) = 3 ** rw [h] ** case refine'_2 \u03b1 : Type u_1 s t : Set \u03b1 h : encard s = 3 \u22a2 \u2191(\u2191ENat.toNat 3) = 3 ** rfl ** case refine'_1 \u03b1 : Type u_1 s t : Set \u03b1 h : \u2191(\u2191ENat.toNat (encard s)) = 3 \u22a2 encard s = 3 ** rwa [ENat.coe_toNat] at h ** case refine'_1 \u03b1 : Type u_1 s t : Set \u03b1 h : \u2191(\u2191ENat.toNat (encard s)) = 3 \u22a2 encard s \u2260 \u22a4 ** rintro h' ** case refine'_1 \u03b1 : Type u_1 s t : Set \u03b1 h : \u2191(\u2191ENat.toNat (encard s)) = 3 h' : encard s = \u22a4 \u22a2 False ** simp [h'] at h ** Qed",
        "informal": ""
    },
    {
        "formal": "List.modify_get?_length ** \u03b1 : Type u_1 f : \u03b1 \u2192 \u03b1 l : List \u03b1 \u22a2 length (modifyHead f l) = length l ** cases l <;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.add_assoc ** m : Nat b : Int k : Nat \u22a2 \u2191m + b + \u2191k = \u2191m + (b + \u2191k) ** rw [Int.add_comm, \u2190 aux1, Int.add_comm k, aux1, Int.add_comm b] ** a : Int n k : Nat \u22a2 a + \u2191n + \u2191k = a + (\u2191n + \u2191k) ** rw [Int.add_comm, Int.add_comm a, \u2190 aux1, Int.add_comm a, Int.add_comm k] ** m n k : Nat \u22a2 -[m+1] + \u2191n + -[k+1] = -[m+1] + (\u2191n + -[k+1]) ** rw [Int.add_comm, \u2190 aux2, Int.add_comm n, \u2190 aux2, Int.add_comm -[m+1]] ** m n k : Nat \u22a2 \u2191m + -[n+1] + -[k+1] = \u2191m + (-[n+1] + -[k+1]) ** rw [Int.add_comm, Int.add_comm m, Int.add_comm m, \u2190 aux2, Int.add_comm -[k+1]] ** m n k : Nat \u22a2 -[m+1] + -[n+1] + -[k+1] = -[m+1] + (-[n+1] + -[k+1]) ** simp [add_succ, Nat.add_comm, Nat.add_left_comm, neg_ofNat_succ] ** x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int m n k : Nat \u22a2 \u2191m + \u2191n + \u2191k = \u2191m + (\u2191n + \u2191k) ** simp [Nat.add_assoc] ** x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int m n k : Nat \u22a2 \u2191m + \u2191n + -[k+1] = \u2191m + (\u2191n + -[k+1]) ** simp [subNatNat_add] ** x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int m n k : Nat \u22a2 -[m+1] + -[n+1] + \u2191k = -[m+1] + (-[n+1] + \u2191k) ** simp [add_succ] ** x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int m n k : Nat \u22a2 subNatNat k (succ (succ (m + n))) = -[m+1] + subNatNat k (succ n) ** rw [Int.add_comm, subNatNat_add_negSucc] ** x\u271d\u00b2 x\u271d\u00b9 x\u271d : Int m n k : Nat \u22a2 subNatNat k (succ (succ (m + n))) = subNatNat k (succ n + succ m) ** simp [add_succ, succ_add, Nat.add_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.infix_cons_iff ** \u03b1\u271d : Type u_1 l\u2081 : List \u03b1\u271d a : \u03b1\u271d l\u2082 : List \u03b1\u271d \u22a2 l\u2081 <:+: a :: l\u2082 \u2194 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082 ** constructor ** case mp \u03b1\u271d : Type u_1 l\u2081 : List \u03b1\u271d a : \u03b1\u271d l\u2082 : List \u03b1\u271d \u22a2 l\u2081 <:+: a :: l\u2082 \u2192 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082 ** rintro \u27e8\u27e8hd, tl\u27e9, t, hl\u2083\u27e9 ** case mp.intro.nil.intro \u03b1\u271d : Type u_1 l\u2081 : List \u03b1\u271d a : \u03b1\u271d l\u2082 t : List \u03b1\u271d hl\u2083 : [] ++ l\u2081 ++ t = a :: l\u2082 \u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082 ** exact Or.inl \u27e8t, hl\u2083\u27e9 ** case mp.intro.cons.intro \u03b1\u271d : Type u_1 l\u2081 : List \u03b1\u271d a : \u03b1\u271d l\u2082 : List \u03b1\u271d head\u271d : \u03b1\u271d tail\u271d t : List \u03b1\u271d hl\u2083 : head\u271d :: tail\u271d ++ l\u2081 ++ t = a :: l\u2082 \u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082 ** simp only [cons_append] at hl\u2083 ** case mp.intro.cons.intro \u03b1\u271d : Type u_1 l\u2081 : List \u03b1\u271d a : \u03b1\u271d l\u2082 : List \u03b1\u271d head\u271d : \u03b1\u271d tail\u271d t : List \u03b1\u271d hl\u2083 : head\u271d :: (tail\u271d ++ l\u2081 ++ t) = a :: l\u2082 \u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082 ** injection hl\u2083 with _ hl\u2084 ** case mp.intro.cons.intro \u03b1\u271d : Type u_1 l\u2081 : List \u03b1\u271d a : \u03b1\u271d l\u2082 : List \u03b1\u271d head\u271d : \u03b1\u271d tail\u271d t : List \u03b1\u271d head_eq\u271d : head\u271d = a hl\u2084 : tail\u271d ++ l\u2081 ++ t = l\u2082 \u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082 ** exact Or.inr \u27e8_, t, hl\u2084\u27e9 ** case mpr \u03b1\u271d : Type u_1 l\u2081 : List \u03b1\u271d a : \u03b1\u271d l\u2082 : List \u03b1\u271d \u22a2 l\u2081 <+: a :: l\u2082 \u2228 l\u2081 <:+: l\u2082 \u2192 l\u2081 <:+: a :: l\u2082 ** rintro (h | hl\u2081) ** case mpr.inl \u03b1\u271d : Type u_1 l\u2081 : List \u03b1\u271d a : \u03b1\u271d l\u2082 : List \u03b1\u271d h : l\u2081 <+: a :: l\u2082 \u22a2 l\u2081 <:+: a :: l\u2082 ** exact h.isInfix ** case mpr.inr \u03b1\u271d : Type u_1 l\u2081 : List \u03b1\u271d a : \u03b1\u271d l\u2082 : List \u03b1\u271d hl\u2081 : l\u2081 <:+: l\u2082 \u22a2 l\u2081 <:+: a :: l\u2082 ** exact infix_cons hl\u2081 ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.eval_polynomial_eval_finSuccEquiv ** R : Type u_1 n : \u2115 x : Fin n \u2192 R inst\u271d : CommSemiring R f : MvPolynomial (Fin (n + 1)) R q : MvPolynomial (Fin n) R \u22a2 \u2191(eval x) (Polynomial.eval q (\u2191(finSuccEquiv R n) f)) = \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f ** simp only [finSuccEquiv_apply, coe_eval\u2082Hom, polynomial_eval_eval\u2082, eval_eval\u2082] ** R : Type u_1 n : \u2115 x : Fin n \u2192 R inst\u271d : CommSemiring R f : MvPolynomial (Fin (n + 1)) R q : MvPolynomial (Fin n) R \u22a2 eval\u2082 (RingHom.comp (eval x) (RingHom.comp (Polynomial.evalRingHom q) (RingHom.comp Polynomial.C C)))       (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =     \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f ** conv in RingHom.comp _ _ =>\n{ refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_\n  simp } ** R : Type u_1 n : \u2115 x : Fin n \u2192 R inst\u271d : CommSemiring R f : MvPolynomial (Fin (n + 1)) R q : MvPolynomial (Fin n) R \u22a2 eval\u2082 (RingHom.id R)       (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =     \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f ** simp only [eval\u2082_id] ** R : Type u_1 n : \u2115 x : Fin n \u2192 R inst\u271d : CommSemiring R f : MvPolynomial (Fin (n + 1)) R q : MvPolynomial (Fin n) R \u22a2 \u2191(eval fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =     \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f ** congr ** case e_a.e_f R : Type u_1 n : \u2115 x : Fin n \u2192 R inst\u271d : CommSemiring R f : MvPolynomial (Fin (n + 1)) R q : MvPolynomial (Fin n) R \u22a2 (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) = fun i =>     Fin.cases (\u2191(eval x) q) x i ** funext i ** case e_a.e_f.h R : Type u_1 n : \u2115 x : Fin n \u2192 R inst\u271d : CommSemiring R f : MvPolynomial (Fin (n + 1)) R q : MvPolynomial (Fin n) R i : Fin (n + 1) \u22a2 \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i)) = Fin.cases (\u2191(eval x) q) x i ** refine Fin.cases (by simp) (by simp) i ** R : Type u_1 n : \u2115 x : Fin n \u2192 R inst\u271d : CommSemiring R f : MvPolynomial (Fin (n + 1)) R q : MvPolynomial (Fin n) R i : Fin (n + 1) \u22a2 \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) 0)) = Fin.cases (\u2191(eval x) q) x 0 ** simp ** R : Type u_1 n : \u2115 x : Fin n \u2192 R inst\u271d : CommSemiring R f : MvPolynomial (Fin (n + 1)) R q : MvPolynomial (Fin n) R i : Fin (n + 1) \u22a2 \u2200 (i : Fin n),     \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) (Fin.succ i))) =       Fin.cases (\u2191(eval x) q) x (Fin.succ i) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.rename_monomial ** \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 d : \u03c3 \u2192\u2080 \u2115 r : R \u22a2 \u2191(rename f) (\u2191(monomial d) r) = \u2191(monomial (Finsupp.mapDomain f d)) r ** rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),\n  Finsupp.prod_mapDomain_index] ** \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 d : \u03c3 \u2192\u2080 \u2115 r : R \u22a2 (\u2191(algebraMap R (MvPolynomial \u03c4 R)) r * Finsupp.prod d fun i k => (X \u2218 f) i ^ k) =     \u2191C r * Finsupp.prod d fun a m => X (f a) ^ m ** rfl ** case h_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 d : \u03c3 \u2192\u2080 \u2115 r : R \u22a2 \u2200 (b : \u03c4), X b ^ 0 = 1 ** exact fun n => pow_zero _ ** case h_add \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 d : \u03c3 \u2192\u2080 \u2115 r : R \u22a2 \u2200 (b : \u03c4) (m\u2081 m\u2082 : \u2115), X b ^ (m\u2081 + m\u2082) = X b ^ m\u2081 * X b ^ m\u2082 ** exact fun n i\u2081 i\u2082 => pow_add _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.HeapNode.WF.realSize_eq ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool a a\u271d\u00b2 : \u03b1 a\u271d\u00b9 a\u271d : HeapNode \u03b1 w\u271d : Nat left\u271d : \u2200 [inst : TotalBLE le], le a a\u271d\u00b2 = true c : WF le a\u271d\u00b2 a\u271d\u00b9 w\u271d s : WF le a a\u271d w\u271d \u22a2 realSize (node a\u271d\u00b2 a\u271d\u00b9 a\u271d) + 1 = 2 ^ (w\u271d + 1) ** rw [realSize, realSize_eq c, Nat.pow_succ, Nat.mul_succ] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool a a\u271d\u00b2 : \u03b1 a\u271d\u00b9 a\u271d : HeapNode \u03b1 w\u271d : Nat left\u271d : \u2200 [inst : TotalBLE le], le a a\u271d\u00b2 = true c : WF le a\u271d\u00b2 a\u271d\u00b9 w\u271d s : WF le a a\u271d w\u271d \u22a2 2 ^ w\u271d + realSize a\u271d + 1 = 2 ^ w\u271d * 1 + 2 ^ w\u271d ** simp [Nat.add_assoc, realSize_eq s] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEStronglyMeasurable'.const_inner ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \ud835\udd5c\u271d : Type u_3 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2\u271d f\u271d g : \u03b1 \u2192 \u03b2\u271d \ud835\udd5c : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : InnerProductSpace \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 hfm : AEStronglyMeasurable' m f \u03bc c : \u03b2 \u22a2 AEStronglyMeasurable' m (fun x => inner c (f x)) \u03bc ** rcases hfm with \u27e8f', hf'_meas, hf_ae\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \ud835\udd5c\u271d : Type u_3 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2\u271d f\u271d g : \u03b1 \u2192 \u03b2\u271d \ud835\udd5c : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : InnerProductSpace \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 c : \u03b2 f' : \u03b1 \u2192 \u03b2 hf'_meas : StronglyMeasurable f' hf_ae : f =\u1d50[\u03bc] f' \u22a2 AEStronglyMeasurable' m (fun x => inner c (f x)) \u03bc ** refine'\n  \u27e8fun x => (inner c (f' x) : \ud835\udd5c), (@stronglyMeasurable_const _ _ m _ c).inner hf'_meas,\n    hf_ae.mono fun x hx => _\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \ud835\udd5c\u271d : Type u_3 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2\u271d f\u271d g : \u03b1 \u2192 \u03b2\u271d \ud835\udd5c : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : InnerProductSpace \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 c : \u03b2 f' : \u03b1 \u2192 \u03b2 hf'_meas : StronglyMeasurable f' hf_ae : f =\u1d50[\u03bc] f' x : \u03b1 hx : f x = f' x \u22a2 (fun x => inner c (f x)) x = (fun x => inner c (f' x)) x ** dsimp only ** case intro.intro \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \ud835\udd5c\u271d : Type u_3 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2\u271d f\u271d g : \u03b1 \u2192 \u03b2\u271d \ud835\udd5c : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : InnerProductSpace \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 c : \u03b2 f' : \u03b1 \u2192 \u03b2 hf'_meas : StronglyMeasurable f' hf_ae : f =\u1d50[\u03bc] f' x : \u03b1 hx : f x = f' x \u22a2 inner c (f x) = inner c (f' x) ** rw [hx] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.mem_map_succ_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 n : \u2115 a a' : \u03b1 b : \u03b2 v : Vector \u03b1 (n + 1) f : \u03b1 \u2192 \u03b2 \u22a2 b \u2208 toList (map f v) \u2194 f (head v) = b \u2228 \u2203 a, a \u2208 toList (tail v) \u2227 f a = b ** rw [mem_succ_iff, head_map, tail_map, mem_map_iff, @eq_comm _ b] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.MutuallySingular.add_left_iff ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 \u22a2 \u03bc\u2081 + \u03bc\u2082 \u27c2\u2098 \u03bd \u2194 \u03bc\u2081 \u27c2\u2098 \u03bd \u2227 \u03bc\u2082 \u27c2\u2098 \u03bd ** rw [\u2190 sum_cond, sum_left, Bool.forall_bool, cond, cond, and_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_exchange' ** \u03b1 : Type u_1 s t : Set \u03b1 a b : \u03b1 ha : \u00aca \u2208 s hb : b \u2208 s \u22a2 ncard (insert a s \\ {b}) = ncard s ** rw [\u2190 ncard_exchange ha hb, \u2190 singleton_union, \u2190 singleton_union, union_diff_distrib,\n  @diff_singleton_eq_self _ b {a} fun h \u21a6 ha (by rwa [\u2190 mem_singleton_iff.mp h])] ** \u03b1 : Type u_1 s t : Set \u03b1 a b : \u03b1 ha : \u00aca \u2208 s hb : b \u2208 s h : b \u2208 {a} \u22a2 a \u2208 s ** rwa [\u2190 mem_singleton_iff.mp h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.totalDegree_multiset_prod ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s : Multiset (MvPolynomial \u03c3 R) \u22a2 totalDegree (Multiset.prod s) \u2264 Multiset.sum (Multiset.map totalDegree s) ** refine' Quotient.inductionOn s fun l => _ ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s : Multiset (MvPolynomial \u03c3 R) l : List (MvPolynomial \u03c3 R) \u22a2 totalDegree (Multiset.prod (Quotient.mk (List.isSetoid (MvPolynomial \u03c3 R)) l)) \u2264     Multiset.sum (Multiset.map totalDegree (Quotient.mk (List.isSetoid (MvPolynomial \u03c3 R)) l)) ** rw [Multiset.quot_mk_to_coe, Multiset.coe_prod, Multiset.coe_map, Multiset.coe_sum] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s : Multiset (MvPolynomial \u03c3 R) l : List (MvPolynomial \u03c3 R) \u22a2 totalDegree (List.prod l) \u2264 List.sum (List.map totalDegree l) ** exact totalDegree_list_prod l ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.PairingHeapImp.Heap.size_tail ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 h : NoSibling s \u22a2 size (tail le s) = size s - 1 ** simp only [Heap.tail] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 h : NoSibling s \u22a2 size (Option.getD (tail? le s) nil) = size s - 1 ** match eq : s.tail? le with\n| none => cases s with cases eq | nil => rfl\n| some tl => simp [Heap.size_tail? h eq]; rfl ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 h : NoSibling s eq : tail? le s = none \u22a2 size (Option.getD none nil) = size s - 1 ** cases s with cases eq | nil => rfl ** case nil.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool h : NoSibling nil \u22a2 size (Option.getD none nil) = size nil - 1 ** rfl ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 h : NoSibling s tl : Heap \u03b1 eq : tail? le s = some tl \u22a2 size (Option.getD (some tl) nil) = size s - 1 ** simp [Heap.size_tail? h eq] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 h : NoSibling s tl : Heap \u03b1 eq : tail? le s = some tl \u22a2 size (Option.getD (some tl) nil) = size tl + 1 - 1 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "DFA.evalFrom_append_singleton ** \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 s : \u03c3 x : List \u03b1 a : \u03b1 \u22a2 evalFrom M s (x ++ [a]) = step M (evalFrom M s x) a ** simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1to1.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 L R : ListBlank \u0393 \u22a2 trTape enc0 (Tape.mk' L R) = trTape' enc0 L R ** simp only [trTape, Tape.mk'_left, Tape.mk'_right\u2080] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.setToL1S_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\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 f g : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 setToL1S T (f + g) = setToL1S T f + setToL1S T g ** 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 f g : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (f + g)) =     SimpleFunc.setToSimpleFunc T (toSimpleFunc f) + SimpleFunc.setToSimpleFunc T (toSimpleFunc g) ** rw [\u2190 SimpleFunc.setToSimpleFunc_add T h_add (SimpleFunc.integrable f)\n    (SimpleFunc.integrable g)] ** \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 f g : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (f + g)) = SimpleFunc.setToSimpleFunc T (toSimpleFunc f + toSimpleFunc g) ** exact\n  SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _)\n    (add_toSimpleFunc f g) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.edist_approxOn_mono ** \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 m n : \u2115 h : m \u2264 n \u22a2 edist (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) (f x) \u2264 edist (\u2191(approxOn f hf s y\u2080 h\u2080 m) x) (f x) ** dsimp only [approxOn, coe_comp, Function.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 m n : \u2115 h : m \u2264 n \u22a2 edist (\u2191(nearestPt (fun k => Nat.rec y\u2080 (fun n n_ih => \u2191(denseSeq (\u2191s) n)) k) n) (f x)) (f x) \u2264     edist (\u2191(nearestPt (fun k => Nat.rec y\u2080 (fun n n_ih => \u2191(denseSeq (\u2191s) n)) k) m) (f x)) (f x) ** exact edist_nearestPt_le _ _ ((nearestPtInd_le _ _ _).trans h) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_eq_zero_iff_eq_empty_of_smulInvariant ** G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u2078 : Group G inst\u271d\u2077 : MulAction G \u03b1 inst\u271d\u2076 : MeasurableSpace G inst\u271d\u2075 : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 inst\u271d\u2074 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : ContinuousConstSMul G \u03b1 inst\u271d\u00b9 : MulAction.IsMinimal G \u03b1 K U : Set \u03b1 inst\u271d : Regular \u03bc h\u03bc : \u03bc \u2260 0 hU : IsOpen U \u22a2 \u2191\u2191\u03bc U = 0 \u2194 U = \u2205 ** rw [\u2190 not_iff_not, \u2190 Ne.def, \u2190 pos_iff_ne_zero,\n  measure_pos_iff_nonempty_of_smulInvariant G h\u03bc hU, nonempty_iff_ne_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.hasCondCdf_ae ** \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, HasCondCdf \u03c1 a ** filter_upwards [monotone_preCdf \u03c1, preCdf_le_one \u03c1, tendsto_preCdf_atTop_one \u03c1,\n  tendsto_preCdf_atBot_zero \u03c1, inf_gt_preCdf \u03c1] with a h1 h2 h3 h4 h5 ** 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 h1 : Monotone fun r => preCdf \u03c1 r a h2 : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h3 : Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) h4 : Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0) h5 : \u2200 (t : \u211a), \u2a05 r, preCdf \u03c1 (\u2191r) a = preCdf \u03c1 t a \u22a2 HasCondCdf \u03c1 a ** exact \u27e8h1, h2, h3, h4, h5\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.IsSFiniteKernel.finset_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 } \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } I : Finset \u03b9 h : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i) \u22a2 IsSFiniteKernel (\u2211 i in I, \u03bas i) ** induction' I using Finset.induction with i I hi_nmem_I h_ind h ** case empty \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 } \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } I : Finset \u03b9 h\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i) h : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsSFiniteKernel (\u03bas i) \u22a2 IsSFiniteKernel (\u2211 i in \u2205, \u03bas i) ** rw [Finset.sum_empty] ** case empty \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 } \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } I : Finset \u03b9 h\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i) h : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsSFiniteKernel (\u03bas i) \u22a2 IsSFiniteKernel 0 ** infer_instance ** case insert \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 } \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } I\u271d : Finset \u03b9 h\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i) i : \u03b9 I : Finset \u03b9 hi_nmem_I : \u00aci \u2208 I h_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i) h : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1) \u22a2 IsSFiniteKernel (\u2211 i in insert i I, \u03bas i) ** rw [Finset.sum_insert hi_nmem_I] ** case insert \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 } \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } I\u271d : Finset \u03b9 h\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i) i : \u03b9 I : Finset \u03b9 hi_nmem_I : \u00aci \u2208 I h_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i) h : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1) \u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x) ** haveI : IsSFiniteKernel (\u03bas i) := h i (Finset.mem_insert_self _ _) ** case insert \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 } \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } I\u271d : Finset \u03b9 h\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i) i : \u03b9 I : Finset \u03b9 hi_nmem_I : \u00aci \u2208 I h_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i) h : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1) this : IsSFiniteKernel (\u03bas i) \u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x) ** have : IsSFiniteKernel (\u2211 x : \u03b9 in I, \u03bas x) :=\n  h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI) ** case insert \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 } \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } I\u271d : Finset \u03b9 h\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i) i : \u03b9 I : Finset \u03b9 hi_nmem_I : \u00aci \u2208 I h_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i) h : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1) this\u271d : IsSFiniteKernel (\u03bas i) this : IsSFiniteKernel (\u2211 x in I, \u03bas x) \u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x) ** exact IsSFiniteKernel.add _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.mem_of_mem_tail ** \u03b1 : Type u_1 \u03b2 : Type u_2 n : \u2115 a a' : \u03b1 v : Vector \u03b1 n ha : a \u2208 toList (tail v) \u22a2 a \u2208 toList v ** induction' n with n _ ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 n : \u2115 a a' : \u03b1 v\u271d : Vector \u03b1 n ha\u271d : a \u2208 toList (tail v\u271d) v : Vector \u03b1 Nat.zero ha : a \u2208 toList (tail v) \u22a2 a \u2208 toList v ** exact False.elim (Vector.not_mem_zero a v.tail ha) ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 n\u271d : \u2115 a a' : \u03b1 v\u271d : Vector \u03b1 n\u271d ha\u271d : a \u2208 toList (tail v\u271d) n : \u2115 n_ih\u271d : \u2200 (v : Vector \u03b1 n), a \u2208 toList (tail v) \u2192 a \u2208 toList v v : Vector \u03b1 (Nat.succ n) ha : a \u2208 toList (tail v) \u22a2 a \u2208 toList v ** exact (mem_succ_iff a v).2 (Or.inr ha) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.extend_iUnion ** \u03b1 : Type u_1 P : Set \u03b1 \u2192 Prop m : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e P0 : P \u2205 m0 : m \u2205 P0 = 0 PU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i) mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),     Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i)) msU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i)) m_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082 \u03b2 : Type u_2 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hd : Pairwise (Disjoint on f) hm : \u2200 (i : \u03b2), P (f i) \u22a2 extend m (\u22c3 i, f i) = \u2211' (i : \u03b2), extend m (f i) ** cases nonempty_encodable \u03b2 ** case intro \u03b1 : Type u_1 P : Set \u03b1 \u2192 Prop m : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e P0 : P \u2205 m0 : m \u2205 P0 = 0 PU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i) mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),     Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i)) msU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i)) m_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082 \u03b2 : Type u_2 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hd : Pairwise (Disjoint on f) hm : \u2200 (i : \u03b2), P (f i) val\u271d : Encodable \u03b2 \u22a2 extend m (\u22c3 i, f i) = \u2211' (i : \u03b2), extend m (f i) ** rw [\u2190 Encodable.iUnion_decode\u2082, \u2190 tsum_iUnion_decode\u2082] ** case intro \u03b1 : Type u_1 P : Set \u03b1 \u2192 Prop m : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e P0 : P \u2205 m0 : m \u2205 P0 = 0 PU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i) mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),     Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i)) msU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i)) m_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082 \u03b2 : Type u_2 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hd : Pairwise (Disjoint on f) hm : \u2200 (i : \u03b2), P (f i) val\u271d : Encodable \u03b2 \u22a2 extend m (\u22c3 i, \u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) = \u2211' (i : \u2115), extend m (\u22c3 b \u2208 Encodable.decode\u2082 \u03b2 i, f b) ** exact\n  extend_iUnion_nat PU (fun n => Encodable.iUnion_decode\u2082_cases P0 hm)\n    (mU _ (Encodable.iUnion_decode\u2082_disjoint_on hd)) ** case intro.m0 \u03b1 : Type u_1 P : Set \u03b1 \u2192 Prop m : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e P0 : P \u2205 m0 : m \u2205 P0 = 0 PU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 i, f i) mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),     Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i)) msU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)), m (\u22c3 i, f i) (_ : P (\u22c3 i, f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i)) m_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082 \u03b2 : Type u_2 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 Set \u03b1 hd : Pairwise (Disjoint on f) hm : \u2200 (i : \u03b2), P (f i) val\u271d : Encodable \u03b2 \u22a2 extend m \u2205 = 0 ** exact extend_empty P0 m0 ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.fmod_add_fdiv ** a\u271d : Nat \u22a2 0 + 0 = 0 ** simp ** m n : Nat \u22a2 fmod \u2191(succ m) -[n+1] + -[n+1] * fdiv \u2191(succ m) -[n+1] = \u2191(succ m) ** show subNatNat (m % succ n) n + (\u2191(succ n * (m / succ n)) + n + 1) = (m + 1) ** m n : Nat \u22a2 subNatNat (m % succ n) n + (\u2191(succ n * (m / succ n)) + \u2191n + 1) = \u2191m + 1 ** rw [Int.add_comm _ n, \u2190 Int.add_assoc, \u2190 Int.add_assoc,\n  Int.subNatNat_eq_coe, Int.sub_add_cancel] ** a\u271d : Nat \u22a2 fmod -[a\u271d+1] 0 + 0 * fdiv -[a\u271d+1] 0 = -[a\u271d+1] ** rw [fmod_zero] ** a\u271d : Nat \u22a2 -[a\u271d+1] + 0 * fdiv -[a\u271d+1] 0 = -[a\u271d+1] ** rfl ** m n : Nat \u22a2 fmod -[m+1] \u2191(succ n) + \u2191(succ n) * fdiv -[m+1] \u2191(succ n) = -[m+1] ** show subNatNat .. - (\u2191(succ n * (m / succ n)) + \u2191(succ n)) = -\u2191(succ m) ** m n : Nat \u22a2 subNatNat (succ n) (succ (m % succ n)) - (\u2191(succ n * (m / succ n)) + \u2191(succ n)) = -\u2191(succ m) ** rw [Int.subNatNat_eq_coe, \u2190 Int.sub_sub, \u2190 Int.neg_sub, Int.sub_sub, Int.sub_sub_self] ** m n : Nat \u22a2 fmod -[m+1] -[n+1] + -[n+1] * fdiv -[m+1] -[n+1] = -[m+1] ** show -(\u2191(succ m % succ n) : Int) + -\u2191(succ n * (succ m / succ n)) = -\u2191(succ m) ** m n : Nat \u22a2 -\u2191(succ m % succ n) + -\u2191(succ n * (succ m / succ n)) = -\u2191(succ m) ** rw [\u2190 Int.neg_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "Semiquot.liftOn_ofMem ** \u03b1 : Type u_1 \u03b2 : Type u_2 q : Semiquot \u03b1 f : \u03b1 \u2192 \u03b2 h : \u2200 (a : \u03b1), a \u2208 q \u2192 \u2200 (b : \u03b1), b \u2208 q \u2192 f a = f b a : \u03b1 aq : a \u2208 q \u22a2 liftOn q f h = f a ** revert h ** \u03b1 : Type u_1 \u03b2 : Type u_2 q : Semiquot \u03b1 f : \u03b1 \u2192 \u03b2 a : \u03b1 aq : a \u2208 q \u22a2 \u2200 (h : \u2200 (a : \u03b1), a \u2208 q \u2192 \u2200 (b : \u03b1), b \u2208 q \u2192 f a = f b), liftOn q f h = f a ** rw [eq_mk_of_mem aq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 q : Semiquot \u03b1 f : \u03b1 \u2192 \u03b2 a : \u03b1 aq : a \u2208 q \u22a2 \u2200 (h : \u2200 (a_1 : \u03b1), a_1 \u2208 mk aq \u2192 \u2200 (b : \u03b1), b \u2208 mk aq \u2192 f a_1 = f b), liftOn (mk aq) f h = f a ** intro ** \u03b1 : Type u_1 \u03b2 : Type u_2 q : Semiquot \u03b1 f : \u03b1 \u2192 \u03b2 a : \u03b1 aq : a \u2208 q h\u271d : \u2200 (a_1 : \u03b1), a_1 \u2208 mk aq \u2192 \u2200 (b : \u03b1), b \u2208 mk aq \u2192 f a_1 = f b \u22a2 liftOn (mk aq) f h\u271d = f a ** rfl ** 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F \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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F 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\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b9 : NormedAddCommGroup F' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F' inst\u271d\u2079 : NormedSpace \u211d F' inst\u271d\u2078 : CompleteSpace F' inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedAddCommGroup G' inst\u271d\u2075 : NormedSpace \u211d G' inst\u271d\u2074 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b2 : SigmaFinite (Measure.trim \u03bc hm) inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : SMulCommClass \u211d \ud835\udd5c F c : \ud835\udd5c x : F 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": "Int.mod_one ** a : Int \u22a2 mod a 1 = 0 ** simp [mod_def, Int.div_one, Int.one_mul, Int.sub_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_iSup_directed ** \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f \u22a2 \u222b\u207b (a : \u03b1), \u2a06 b, f b a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc ** simp_rw [\u2190 iSup_apply] ** \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f \u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc ** let p : \u03b1 \u2192 (\u03b2 \u2192 ENNReal) \u2192 Prop := fun x f' => Directed LE.le f' ** \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' \u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc ** have hp : \u2200\u1d50 x \u2202\u03bc, p x fun i => f i x := by\n  filter_upwards [] with x i j\n  obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed i j\n  exact \u27e8z, hz\u2081 x, hz\u2082 x\u27e9 ** \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_directed : Directed LE.le (aeSeq hf p) \u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc ** convert lintegral_iSup_directed_of_measurable (aeSeq.measurable hf p) h_ae_seq_directed using 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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x ** filter_upwards [] with x i j ** 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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' x : \u03b1 i j : \u03b2 \u22a2 \u2203 z, (fun i => f i x) i \u2264 (fun i => f i x) z \u2227 (fun i => f i x) j \u2264 (fun i => f i x) z ** obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed i j ** case h.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 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' x : \u03b1 i j z : \u03b2 hz\u2081 : f i \u2264 f z hz\u2082 : f j \u2264 f z \u22a2 \u2203 z, (fun i => f i x) i \u2264 (fun i => f i x) z \u2227 (fun i => f i x) j \u2264 (fun i => f i x) z ** exact \u27e8z, hz\u2081 x, hz\u2082 x\u27e9 ** \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x \u22a2 Directed LE.le (aeSeq hf p) ** intro b\u2081 b\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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x b\u2081 b\u2082 : \u03b2 \u22a2 \u2203 z, aeSeq hf p b\u2081 \u2264 aeSeq hf p z \u2227 aeSeq hf p b\u2082 \u2264 aeSeq hf p z ** obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed b\u2081 b\u2082 ** case intro.intro.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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x b\u2081 b\u2082 z : \u03b2 hz\u2081 : f b\u2081 \u2264 f z hz\u2082 : f b\u2082 \u2264 f z \u22a2 aeSeq hf p b\u2082 \u2264 aeSeq hf p z ** intro x ** case intro.intro.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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x b\u2081 b\u2082 z : \u03b2 hz\u2081 : f b\u2081 \u2264 f z hz\u2082 : f b\u2082 \u2264 f z x : \u03b1 \u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x ** by_cases hx : x \u2208 aeSeqSet hf p ** 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 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x b\u2081 b\u2082 z : \u03b2 hz\u2081 : f b\u2081 \u2264 f z hz\u2082 : f b\u2082 \u2264 f z x : \u03b1 hx : x \u2208 aeSeqSet hf p \u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x ** repeat' rw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx] ** 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 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x b\u2081 b\u2082 z : \u03b2 hz\u2081 : f b\u2081 \u2264 f z hz\u2082 : f b\u2082 \u2264 f z x : \u03b1 hx : x \u2208 aeSeqSet hf p \u22a2 f b\u2082 x \u2264 f z x ** apply_rules [hz\u2081, hz\u2082] ** 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 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x b\u2081 b\u2082 z : \u03b2 hz\u2081 : f b\u2081 \u2264 f z hz\u2082 : f b\u2082 \u2264 f z x : \u03b1 hx : x \u2208 aeSeqSet hf p \u22a2 f b\u2082 x \u2264 aeSeq hf p z x ** rw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf 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 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x b\u2081 b\u2082 z : \u03b2 hz\u2081 : f b\u2081 \u2264 f z hz\u2082 : f b\u2082 \u2264 f z x : \u03b1 hx : \u00acx \u2208 aeSeqSet hf p \u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x ** simp only [aeSeq, hx, if_false] ** 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 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x b\u2081 b\u2082 z : \u03b2 hz\u2081 : f b\u2081 \u2264 f z hz\u2082 : f b\u2082 \u2264 f z x : \u03b1 hx : \u00acx \u2208 aeSeqSet hf p \u22a2 Nonempty.some (_ : Nonempty \u211d\u22650\u221e) \u2264 Nonempty.some (_ : Nonempty \u211d\u22650\u221e) ** exact le_rfl ** case h.e'_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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_directed : Directed LE.le (aeSeq hf p) \u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 b, aeSeq hf p b a \u2202?m.1158777 ** simp_rw [\u2190 iSup_apply] ** case h.e'_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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_directed : Directed LE.le (aeSeq hf p) \u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc =     \u222b\u207b (a : \u03b1), iSup (fun i => aeSeq hf (fun x f' => Directed LE.le f') i) a \u2202?m.1158777 ** rw [lintegral_congr_ae (aeSeq.iSup hf hp).symm] ** 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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_directed : Directed LE.le (aeSeq hf p) \u22a2 \u2a06 b, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc = \u2a06 b, \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc ** congr 1 ** case h.e'_3.e_s \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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_directed : Directed LE.le (aeSeq hf p) \u22a2 (fun b => \u222b\u207b (a : \u03b1), f b a \u2202\u03bc) = fun b => \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc ** ext1 b ** case h.e'_3.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 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_directed : Directed LE.le (aeSeq hf p) b : \u03b2 \u22a2 \u222b\u207b (a : \u03b1), f b a \u2202\u03bc = \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc ** rw [lintegral_congr_ae] ** case h.e'_3.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 inst\u271d : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_directed : Directed LE.le (aeSeq hf p) b : \u03b2 \u22a2 (fun a => f b a) =\u1d50[\u03bc] fun a => aeSeq hf p b a ** apply EventuallyEq.symm ** case h.e'_3.e_s.h.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 : Countable \u03b2 f : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (b : \u03b2), AEMeasurable (f b) h_directed : Directed (fun x x_1 => x \u2264 x_1) f p : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_directed : Directed LE.le (aeSeq hf p) b : \u03b2 \u22a2 (fun a => aeSeq hf p b a) =\u1d50[\u03bc] fun a => f b a ** refine' aeSeq.aeSeq_n_eq_fun_n_ae hf hp _ ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.variance_smul' ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 A : Type u_2 inst\u271d\u00b9 : CommSemiring A inst\u271d : Algebra A \u211d c : A X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 variance (c \u2022 X) \u03bc = c ^ 2 \u2022 variance X \u03bc ** convert variance_smul (algebraMap A \u211d c) X \u03bc using 1 ** case h.e'_2 \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 A : Type u_2 inst\u271d\u00b9 : CommSemiring A inst\u271d : Algebra A \u211d c : A X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 variance (c \u2022 X) \u03bc = variance (\u2191(algebraMap A \u211d) c \u2022 X) \u03bc ** congr ** case h.e'_2.e_X \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 A : Type u_2 inst\u271d\u00b9 : CommSemiring A inst\u271d : Algebra A \u211d c : A X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 c \u2022 X = \u2191(algebraMap A \u211d) c \u2022 X ** simp only [algebraMap_smul] ** case h.e'_3 \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 A : Type u_2 inst\u271d\u00b9 : CommSemiring A inst\u271d : Algebra A \u211d c : A X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 c ^ 2 \u2022 variance X \u03bc = \u2191(algebraMap A \u211d) c ^ 2 * variance X \u03bc ** simp only [Algebra.smul_def, map_pow] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.range_restrictPreimage ** \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 range (restrictPreimage t f) = Subtype.val \u207b\u00b9' range f ** delta Set.restrictPreimage ** \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 range (MapsTo.restrict f (f \u207b\u00b9' t) t (_ : MapsTo f (f \u207b\u00b9' t) t)) = Subtype.val \u207b\u00b9' range f ** rw [MapsTo.range_restrict, Set.image_preimage_eq_inter_range, Set.preimage_inter,\n  Subtype.coe_preimage_self, Set.univ_inter] ** Qed",
        "informal": ""
    },
    {
        "formal": "WType.elim_injective ** \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 \u03b3 : Type u_3 f\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3 f\u03b3_injective : Function.Injective f\u03b3 a\u2081 : \u03b1 f\u2081 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a a\u2082 : \u03b1 f\u2082 : \u03b2 a\u2082 \u2192 WType fun a => \u03b2 a h : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2082 f\u2082) \u22a2 mk a\u2081 f\u2081 = mk a\u2082 f\u2082 ** obtain \u27e8rfl, h\u27e9 := Sigma.mk.inj_iff.mp (f\u03b3_injective h) ** case intro \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 \u03b3 : Type u_3 f\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3 f\u03b3_injective : Function.Injective f\u03b3 a\u2081 : \u03b1 f\u2081 f\u2082 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a h\u271d : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2081 f\u2082) h : HEq (fun b => elim \u03b3 f\u03b3 (f\u2081 b)) fun b => elim \u03b3 f\u03b3 (f\u2082 b) \u22a2 mk a\u2081 f\u2081 = mk a\u2081 f\u2082 ** congr with x ** case intro.e_f.h \u03b1 : Type u_1 \u03b2 : \u03b1 \u2192 Type u_2 \u03b3 : Type u_3 f\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3 f\u03b3_injective : Function.Injective f\u03b3 a\u2081 : \u03b1 f\u2081 f\u2082 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a h\u271d : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2081 f\u2082) h : HEq (fun b => elim \u03b3 f\u03b3 (f\u2081 b)) fun b => elim \u03b3 f\u03b3 (f\u2082 b) x : \u03b2 a\u2081 \u22a2 f\u2081 x = f\u2082 x ** exact elim_injective \u03b3 f\u03b3 f\u03b3_injective (congr_fun (eq_of_heq h) x : _) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.fundamentalInterior_smul ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2074 : Group G inst\u271d\u00b3 : MulAction G \u03b1 s : Set \u03b1 x : \u03b1 inst\u271d\u00b2 : Group H inst\u271d\u00b9 : MulAction H \u03b1 inst\u271d : SMulCommClass H G \u03b1 g : H \u22a2 fundamentalInterior G (g \u2022 s) = g \u2022 fundamentalInterior G s ** simp_rw [fundamentalInterior, smul_set_sdiff, smul_set_iUnion, smul_comm g (_ : G) (_ : Set \u03b1)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.norm_integral_le_integral_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 : 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 E hf : Integrable \u2191f \u22a2 \u2016integral \u03bc f\u2016 \u2264 integral \u03bc (map norm f) ** refine' (norm_setToSimpleFunc_le_integral_norm _ (fun s _ _ => _) hf).trans (one_mul _).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 : 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 E hf : Integrable \u2191f s : Set \u03b1 x\u271d\u00b9 : MeasurableSet s x\u271d : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2016weightedSMul \u03bc s\u2016 \u2264 1 * ENNReal.toReal (\u2191\u2191\u03bc s) ** exact (norm_weightedSMul_le s).trans (one_mul _).symm.le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.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 comp\u2082 g hg f\u2081 f\u2082 = mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) ** rw [comp\u2082_eq_pair, pair_eq_mk, comp_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 mk (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) =     mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.PairingHeapImp.Heap.size_tail?_lt ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 \u22a2 tail? le s = some s' \u2192 size s' < size 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 size s' < size 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 size s' < size s ** match eq\u2082 : s.deleteMin le, eq with\n| some (a, tl), rfl => exact size_deleteMin_lt 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 size ((fun x => x.snd) (a, tl)) < size s ** exact size_deleteMin_lt eq\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.upperCrossingTime_succ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : Preorder \u03b9 inst\u271d\u00b9 : OrderBot \u03b9 inst\u271d : InfSet \u03b9 a b : \u211d f : \u03b9 \u2192 \u03a9 \u2192 \u211d N : \u03b9 n m : \u2115 \u03c9 : \u03a9 \u22a2 upperCrossingTime a b f N (n + 1) \u03c9 =     hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n \u03c9) N \u03c9) N \u03c9 ** rw [upperCrossingTime] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Ico_union_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 d : \u03b1 h\u2081 : min a b \u2264 max c d h\u2082 : min c d \u2264 max a b \u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d) ** rw [\u2190 coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico h\u2081 h\u2082] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_fun_fst ** \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 \u2192 E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.1 \u2202Measure.prod \u03bc \u03bd = ENNReal.toReal (\u2191\u2191\u03bd univ) \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** rw [\u2190 integral_prod_swap] ** \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 \u2192 E \u22a2 \u222b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z).1 \u2202Measure.prod \u03bd \u03bc = ENNReal.toReal (\u2191\u2191\u03bd univ) \u2022 \u222b (x : \u03b1), f x \u2202\u03bc ** apply integral_fun_snd ** 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 \u22a2 \u2191(\u2211 i in I, \u03ba i) a = \u2211 i in I, \u2191(\u03ba i) a ** rw [coe_finset_sum, Finset.sum_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.sign_eq_sign ** \u03b1 : Type u_1 n : \u2124 \u22a2 sign n = \u2191(\u2191SignType.sign n) ** obtain (n | _) | _ := n <;> simp [sign, Int.sign_neg, negSucc_lt_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.snorm'_sum_norm_sub_le_tsum_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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc p : \u211d hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N n : \u2115 \u22a2 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 ** let f_norm_diff i x := \u2016f (i + 1) x - f i x\u2016 ** \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 f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc p : \u211d hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N n : \u2115 f_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016 \u22a2 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 ** have hgf_norm_diff :\n  \u2200 n,\n    (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) =\n      \u2211 i in Finset.range (n + 1), f_norm_diff i :=\n  fun n => funext fun x => by simp ** \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 f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc p : \u211d hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N n : \u2115 f_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016 hgf_norm_diff :   \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i \u22a2 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 ** rw [hgf_norm_diff] ** \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 f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc p : \u211d hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N n : \u2115 f_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016 hgf_norm_diff :   \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i \u22a2 snorm' (\u2211 i in Finset.range (n + 1), f_norm_diff i) p \u03bc \u2264 \u2211' (i : \u2115), B i ** refine' (snorm'_sum_le (fun i _ => ((hf (i + 1)).sub (hf i)).norm) hp1).trans _ ** \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 f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc p : \u211d hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N n : \u2115 f_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016 hgf_norm_diff :   \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i \u22a2 \u2211 i in Finset.range (n + 1), snorm' (fun x => \u2016(f (i + 1) - f i) x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i ** simp_rw [snorm'_norm] ** \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 f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc p : \u211d hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N n : \u2115 f_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016 hgf_norm_diff :   \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i \u22a2 \u2211 x in Finset.range (n + 1), snorm' (fun a => (f (x + 1) - f x) a) p \u03bc \u2264 \u2211' (i : \u2115), B i ** refine' (Finset.sum_le_sum _).trans (sum_le_tsum _ (fun m _ => zero_le _) ENNReal.summable) ** \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 f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc p : \u211d hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N n : \u2115 f_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016 hgf_norm_diff :   \u2200 (n : \u2115), (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) = \u2211 i in Finset.range (n + 1), f_norm_diff i \u22a2 \u2200 (i : \u2115), i \u2208 Finset.range (n + 1) \u2192 snorm' (fun a => (f (i + 1) - f i) a) p \u03bc \u2264 B i ** exact fun m _ => (h_cau m (m + 1) m (Nat.le_succ m) (le_refl m)).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 : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc p : \u211d hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N n\u271d : \u2115 f_norm_diff : \u2115 \u2192 \u03b1 \u2192 \u211d := fun i x => \u2016f (i + 1) x - f i x\u2016 n : \u2115 x : \u03b1 \u22a2 \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016 = Finset.sum (Finset.range (n + 1)) (fun i => f_norm_diff i) x ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.chaar_empty ** 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 \u22a5 = 0 ** let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5 ** 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 \u22a5 \u22a2 chaar K\u2080 \u22a5 = 0 ** have : Continuous eval := continuous_apply \u22a5 ** 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 \u22a5 this : Continuous eval \u22a2 chaar K\u2080 \u22a5 = 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 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5 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 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5 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 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5 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 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5 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, _, rfl\u27e9 ** 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 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5 this : Continuous eval U : Set G left\u271d : U \u2208 {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0} ** apply prehaar_empty ** 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 \u22a5 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 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5 this : Continuous eval \u22a2 IsClosed {0} ** exact isClosed_singleton ** 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 c : \u211d f : { x // x \u2208 Lp E 1 } \u22a2 \u2191(setToL1 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) f = c \u2022 \u2191(setToL1 hT) f ** suffices setToL1 (hT.smul c) = 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 c : \u211d f : { x // x \u2208 Lp E 1 } \u22a2 setToL1 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) = c \u2022 setToL1 hT ** refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc (hT.smul c)) _ _ _ _ _ ** \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 c : \u211d f : { x // x \u2208 Lp E 1 } \u22a2 ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d) =     setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * 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 c : \u211d 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 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) f ** suffices c \u2022 setToL1 hT f = setToL1SCLM \u03b1 E \u03bc (hT.smul c) 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 c : \u211d 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 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) f ** rw [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_left c 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 c : \u211d f : { x // x \u2208 Lp E 1 } this : setToL1 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C)) = c \u2022 setToL1 hT \u22a2 \u2191(setToL1 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) 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 c : \u211d 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 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) f \u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f =     \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * 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 c : \u211d 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 (_ : DominatedFinMeasAdditive \u03bc (fun s => c \u2022 T s) (\u2016c\u2016 * C))) f \u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f = c \u2022 \u2191(setToL1 hT) \u2191f ** congr ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.toMeasure_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 inst\u271d \u2264 OuterMeasure.caratheodory \u22a4 ** rw [OuterMeasure.top_caratheodory] ** \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 inst\u271d \u2264 \u22a4 ** exact le_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\u271d s' t : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191\u22a4 s \u2264 \u2191\u2191(OuterMeasure.toMeasure \u22a4 (_ : inst\u271d \u2264 OuterMeasure.caratheodory \u22a4)) s ** cases' s.eq_empty_or_nonempty with h h <;>\n  simp [h, toMeasure_apply \u22a4 _ hs, OuterMeasure.top_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_iUnion\u2080 ** \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 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 Set \u03b1 hd : Pairwise (AEDisjoint \u03bc on f) h : \u2200 (i : \u03b9), NullMeasurableSet (f i) \u22a2 \u2191\u2191\u03bc (\u22c3 i, f i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (f i) ** rcases exists_subordinate_pairwise_disjoint h hd with \u27e8t, _ht_sub, ht_eq, htm, htd\u27e9 ** case intro.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 t\u271d : Set \u03b1 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 Set \u03b1 hd : Pairwise (AEDisjoint \u03bc on f) h : \u2200 (i : \u03b9), NullMeasurableSet (f i) t : \u03b9 \u2192 Set \u03b1 _ht_sub : \u2200 (i : \u03b9), t i \u2286 f i ht_eq : \u2200 (i : \u03b9), f i =\u1d50[\u03bc] t i htm : \u2200 (i : \u03b9), MeasurableSet (t i) htd : Pairwise (Disjoint on t) \u22a2 \u2191\u2191\u03bc (\u22c3 i, f i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (f i) ** calc\n  \u03bc (\u22c3 i, f i) = \u03bc (\u22c3 i, t i) := measure_congr (EventuallyEq.countable_iUnion ht_eq)\n  _ = \u2211' i, \u03bc (t i) := (measure_iUnion htd htm)\n  _ = \u2211' i, \u03bc (f i) := tsum_congr fun i => measure_congr (ht_eq _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.expand_one ** \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 \u22a2 expand 1 = AlgHom.id R (MvPolynomial \u03c3 R) ** ext1 f ** case hf \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 : \u03c3 \u22a2 \u2191(expand 1) (X f) = \u2191(AlgHom.id R (MvPolynomial \u03c3 R)) (X f) ** rw [expand_one_apply, AlgHom.id_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBSet.find?_insert ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp v v' : \u03b1 \u22a2 find? (insert t v) v' = if cmp v' v = Ordering.eq then some v else find? t v' ** split <;> [exact find?_insert_of_eq t \u2039_\u203a; exact find?_insert_of_ne t \u2039_\u203a] ** Qed",
        "informal": ""
    },
    {
        "formal": "aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet ** \u03b9 : Sort u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b1 p : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop hf : \u2200 (i : \u03b9), AEMeasurable (f i) x : \u03b1 hx : x \u2208 aeSeqSet hf p i : \u03b9 \u22a2 aeSeq hf p i x = f i x ** simp only [aeSeq_eq_mk_of_mem_aeSeqSet hf hx i, mk_eq_fun_of_mem_aeSeqSet hf hx i] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ProbabilityMeasure.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 : ProbabilityMeasure \u03a9 \u03bcs : \u03b9 \u2192 ProbabilityMeasure \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 ** apply FiniteMeasure.limsup_measure_closed_le_of_tendsto\n  ((ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds L).mp \u03bcs_lim) F_closed ** Qed",
        "informal": ""
    },
    {
        "formal": "List.sections_eq_nil_of_isEmpty ** \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) h : any (l :: L) isEmpty = true \u22a2 sections (l :: L) = [] ** simp only [any, foldr, Bool.or_eq_true] at h ** \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) h : isEmpty l = true \u2228 foldr (fun a r => isEmpty a || r) false L = true \u22a2 sections (l :: L) = [] ** match l, h with\n| [], .inl rfl => simp; induction sections L <;> simp [*]\n| l, .inr h => simp [sections, sections_eq_nil_of_isEmpty h] ** \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) h : isEmpty l = true \u2228 foldr (fun a r => isEmpty a || r) false L = true \u22a2 sections ([] :: L) = [] ** simp ** \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) h : isEmpty l = true \u2228 foldr (fun a r => isEmpty a || r) false L = true \u22a2 (List.bind (sections L) fun s => []) = [] ** induction sections L <;> simp [*] ** \u03b1 : Type u_1 l\u271d : List \u03b1 L : List (List \u03b1) h\u271d : isEmpty l\u271d = true \u2228 foldr (fun a r => isEmpty a || r) false L = true l : List \u03b1 h : foldr (fun a r => isEmpty a || r) false L = true \u22a2 sections (l :: L) = [] ** simp [sections, sections_eq_nil_of_isEmpty h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.fin_succ ** \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 \u22a2 Primrec fun a => \u2191(Fin.succ a) ** simp [succ.comp fin_val] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.subNatNat_add_add ** m n k : Nat \u22a2 subNatNat (m + k) (n + k) = subNatNat m n ** apply subNatNat_elim m n (fun m n i => subNatNat (m + k) (n + k) = i) ** case hp m n k : Nat \u22a2 \u2200 (i n : Nat), subNatNat (n + i + k) (n + k) = \u2191i ** intro i j ** case hp m n k i j : Nat \u22a2 subNatNat (j + i + k) (j + k) = \u2191i ** rw [Nat.add_assoc, Nat.add_comm i k, \u2190 Nat.add_assoc] ** case hp m n k i j : Nat \u22a2 subNatNat (j + k + i) (j + k) = \u2191i ** exact subNatNat_add_left ** case hn m n k : Nat \u22a2 \u2200 (i m : Nat), subNatNat (m + k) (m + i + 1 + k) = -[i+1] ** intro i j ** case hn m n k i j : Nat \u22a2 subNatNat (j + k) (j + i + 1 + k) = -[i+1] ** rw [Nat.add_assoc j i 1, Nat.add_comm j (i+1), Nat.add_assoc, Nat.add_comm (i+1) (j+k)] ** case hn m n k i j : Nat \u22a2 subNatNat (j + k) (j + k + (i + 1)) = -[i+1] ** exact subNatNat_add_right ** Qed",
        "informal": ""
    },
    {
        "formal": "Language.kstar_eq_iSup_pow ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x : List \u03b1 l : Language \u03b1 \u22a2 l\u2217 = \u2a06 i, l ^ i ** ext x ** case h \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 x \u2208 l\u2217 \u2194 x \u2208 \u2a06 i, l ^ i ** simp only [mem_kstar, mem_iSup, mem_pow] ** case h \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 L, x = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l) \u2194 \u2203 i S, x = join S \u2227 length S = i \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** constructor ** case h.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 L, x = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l) \u2192 \u2203 i S, x = join S \u2227 length S = i \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** rintro \u27e8S, rfl, hS\u27e9 ** case h.mp.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x : List \u03b1 l : Language \u03b1 S : List (List \u03b1) hS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l \u22a2 \u2203 i S_1, join S = join S_1 \u2227 length S_1 = i \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l ** exact \u27e8_, S, rfl, rfl, hS\u27e9 ** case h.mpr \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 i S, x = join S \u2227 length S = i \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2192 \u2203 L, x = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l ** rintro \u27e8_, S, rfl, rfl, hS\u27e9 ** case h.mpr.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x : List \u03b1 l : Language \u03b1 S : List (List \u03b1) hS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l \u22a2 \u2203 L, join S = join L \u2227 \u2200 (y : List \u03b1), y \u2208 L \u2192 y \u2208 l ** exact \u27e8S, rfl, hS\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Ico_eq_image_ssubsets ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t \u22a2 Ico s t = image ((fun x x_1 => x \u222a x_1) s) (ssubsets (t \\ s)) ** ext u ** case a \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t u : Finset \u03b1 \u22a2 u \u2208 Ico s t \u2194 u \u2208 image ((fun x x_1 => x \u222a x_1) s) (ssubsets (t \\ s)) ** simp_rw [mem_Ico, mem_image, mem_ssubsets] ** case a \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t u : Finset \u03b1 \u22a2 s \u2264 u \u2227 u < t \u2194 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u ** constructor ** case a.mp \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t u : Finset \u03b1 \u22a2 s \u2264 u \u2227 u < t \u2192 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u ** rintro \u27e8hs, ht\u27e9 ** case a.mp.intro \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t u : Finset \u03b1 hs : s \u2264 u ht : u < t \u22a2 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u ** exact \u27e8u \\ s, sdiff_lt_sdiff_right ht hs, sup_sdiff_cancel_right hs\u27e9 ** case a.mpr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t u : Finset \u03b1 \u22a2 (\u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u) \u2192 s \u2264 u \u2227 u < t ** rintro \u27e8v, hv, rfl\u27e9 ** case a.mpr.intro.intro \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : s \u2286 t v : Finset \u03b1 hv : v \u2282 t \\ s \u22a2 s \u2264 s \u222a v \u2227 s \u222a v < t ** exact \u27e8le_sup_left, sup_lt_of_lt_sdiff_left hv h\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "parallelepiped_eq_sum_segment ** \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 = \u2211 i : \u03b9, segment \u211d 0 (v i) ** ext ** 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 x\u271d : E \u22a2 x\u271d \u2208 parallelepiped v \u2194 x\u271d \u2208 \u2211 i : \u03b9, segment \u211d 0 (v i) ** simp only [mem_parallelepiped_iff, Set.mem_finset_sum, Finset.mem_univ, forall_true_left,\n  segment_eq_image, smul_zero, zero_add, \u2190 Set.pi_univ_Icc, Set.mem_univ_pi] ** 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 x\u271d : E \u22a2 (\u2203 t h, x\u271d = \u2211 i : \u03b9, t i \u2022 v i) \u2194 \u2203 g h, \u2211 i : \u03b9, g i = x\u271d ** constructor ** case h.mpr \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 x\u271d : E \u22a2 (\u2203 g h, \u2211 i : \u03b9, g i = x\u271d) \u2192 \u2203 t h, x\u271d = \u2211 i : \u03b9, t i \u2022 v i ** rintro \u27e8g, hg, rfl\u27e9 ** case h.mpr.intro.intro \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 g : \u03b9 \u2192 E hg : \u2200 {i : \u03b9}, g i \u2208 (fun a => a \u2022 v i) '' Icc 0 1 \u22a2 \u2203 t h, \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t i \u2022 v i ** choose t ht hg using @hg ** case h.mpr.intro.intro \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 g : \u03b9 \u2192 E t : {i : \u03b9} \u2192 \u211d ht : \u2200 {i : \u03b9}, t \u2208 Icc 0 1 hg : \u2200 {i : \u03b9}, (fun a => a \u2022 v i) t = g i \u22a2 \u2203 t h, \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t i \u2022 v i ** refine \u27e8@t, @ht, ?_\u27e9 ** case h.mpr.intro.intro \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 g : \u03b9 \u2192 E t : {i : \u03b9} \u2192 \u211d ht : \u2200 {i : \u03b9}, t \u2208 Icc 0 1 hg : \u2200 {i : \u03b9}, (fun a => a \u2022 v i) t = g i \u22a2 \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t \u2022 v i ** simp_rw [hg] ** case h.mp \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 x\u271d : E \u22a2 (\u2203 t h, x\u271d = \u2211 i : \u03b9, t i \u2022 v i) \u2192 \u2203 g h, \u2211 i : \u03b9, g i = x\u271d ** rintro \u27e8t, ht, rfl\u27e9 ** case h.mp.intro.intro \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 t : \u03b9 \u2192 \u211d ht : \u2200 (i : \u03b9), t i \u2208 Icc (OfNat.ofNat 0 i) (OfNat.ofNat 1 i) \u22a2 \u2203 g h, \u2211 i : \u03b9, g i = \u2211 i : \u03b9, t i \u2022 v i ** exact \u27e8t \u2022 v, fun {i} => \u27e8t i, ht _, by simp\u27e9, rfl\u27e9 ** \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 t : \u03b9 \u2192 \u211d ht : \u2200 (i : \u03b9), t i \u2208 Icc (OfNat.ofNat 0 i) (OfNat.ofNat 1 i) i : \u03b9 \u22a2 (fun a => a \u2022 v i) (t i) = (t \u2022 v) i ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg ** \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 : \u03b1 \u2192 \u211d hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u22a2 0 \u2264\u1d50[\u03bc] f ** rcases hf.1 with \u27e8f', hf'_meas, hf_ae\u27e9 ** case intro.intro \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 : \u03b1 \u2192 \u211d hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc f' : \u03b1 \u2192 \u211d hf'_meas : StronglyMeasurable f' hf_ae : f =\u1d50[\u03bc] f' \u22a2 0 \u2264\u1d50[\u03bc] f ** have hf'_integrable : Integrable f' \u03bc := Integrable.congr hf hf_ae ** case intro.intro \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 : \u03b1 \u2192 \u211d hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc f' : \u03b1 \u2192 \u211d hf'_meas : StronglyMeasurable f' hf_ae : f =\u1d50[\u03bc] f' hf'_integrable : Integrable f' \u22a2 0 \u2264\u1d50[\u03bc] f ** have hf'_zero : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 0 \u2264 \u222b x in s, f' x \u2202\u03bc := by\n  intro s hs h's\n  rw [set_integral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)]\n  exact hf_zero s hs h's ** case intro.intro \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 : \u03b1 \u2192 \u211d hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc f' : \u03b1 \u2192 \u211d hf'_meas : StronglyMeasurable f' hf_ae : f =\u1d50[\u03bc] f' hf'_integrable : Integrable f' hf'_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f' x \u2202\u03bc \u22a2 0 \u2264\u1d50[\u03bc] f ** exact\n  (ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable hf'_meas hf'_integrable\n        hf'_zero).trans\n    hf_ae.symm.le ** \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 : \u03b1 \u2192 \u211d hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc f' : \u03b1 \u2192 \u211d hf'_meas : StronglyMeasurable f' hf_ae : f =\u1d50[\u03bc] f' hf'_integrable : Integrable f' \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f' x \u2202\u03bc ** intro s hs h's ** \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 : \u03b1 \u2192 \u211d hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc f' : \u03b1 \u2192 \u211d hf'_meas : StronglyMeasurable f' hf_ae : f =\u1d50[\u03bc] f' hf'_integrable : Integrable f' s : Set \u03b1 hs : MeasurableSet s h's : \u2191\u2191\u03bc s < \u22a4 \u22a2 0 \u2264 \u222b (x : \u03b1) in s, f' x \u2202\u03bc ** rw [set_integral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)] ** \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 : \u03b1 \u2192 \u211d hf : Integrable f hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc f' : \u03b1 \u2192 \u211d hf'_meas : StronglyMeasurable f' hf_ae : f =\u1d50[\u03bc] f' hf'_integrable : Integrable f' s : Set \u03b1 hs : MeasurableSet s h's : \u2191\u2191\u03bc s < \u22a4 \u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc ** exact hf_zero s hs h's ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.progMeasurable_of_tendsto' ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m \u03b3 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b2 fltr : Filter \u03b3 inst\u271d\u00b9 : NeBot fltr inst\u271d : IsCountablyGenerated fltr U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 h : \u2200 (l : \u03b3), ProgMeasurable f (U l) h_tendsto : Tendsto U fltr (\ud835\udcdd u) \u22a2 ProgMeasurable f u ** intro i ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m \u03b3 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b2 fltr : Filter \u03b3 inst\u271d\u00b9 : NeBot fltr inst\u271d : IsCountablyGenerated fltr U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 h : \u2200 (l : \u03b3), ProgMeasurable f (U l) h_tendsto : Tendsto U fltr (\ud835\udcdd u) i : \u03b9 \u22a2 StronglyMeasurable fun p => u (\u2191p.1) p.2 ** apply @stronglyMeasurable_of_tendsto (Set.Iic i \u00d7 \u03a9) \u03b2 \u03b3\n  (MeasurableSpace.prod _ (f i)) _ _ fltr _ _ _ _ fun l => h l i ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m \u03b3 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b2 fltr : Filter \u03b3 inst\u271d\u00b9 : NeBot fltr inst\u271d : IsCountablyGenerated fltr U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 h : \u2200 (l : \u03b3), ProgMeasurable f (U l) h_tendsto : Tendsto U fltr (\ud835\udcdd u) i : \u03b9 \u22a2 Tendsto (fun l p => U l (\u2191p.1) p.2) fltr (\ud835\udcdd fun p => u (\u2191p.1) p.2) ** rw [tendsto_pi_nhds] at h_tendsto \u22a2 ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m \u03b3 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b2 fltr : Filter \u03b3 inst\u271d\u00b9 : NeBot fltr inst\u271d : IsCountablyGenerated fltr U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 h : \u2200 (l : \u03b3), ProgMeasurable f (U l) h_tendsto : \u2200 (x : \u03b9), Tendsto (fun i => U i x) fltr (\ud835\udcdd (u x)) i : \u03b9 \u22a2 \u2200 (x : \u2191(Set.Iic i) \u00d7 \u03a9), Tendsto (fun i_1 => U i_1 (\u2191x.1) x.2) fltr (\ud835\udcdd (u (\u2191x.1) x.2)) ** intro x ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m \u03b3 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b2 fltr : Filter \u03b3 inst\u271d\u00b9 : NeBot fltr inst\u271d : IsCountablyGenerated fltr U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 h : \u2200 (l : \u03b3), ProgMeasurable f (U l) h_tendsto : \u2200 (x : \u03b9), Tendsto (fun i => U i x) fltr (\ud835\udcdd (u x)) i : \u03b9 x : \u2191(Set.Iic i) \u00d7 \u03a9 \u22a2 Tendsto (fun i_1 => U i_1 (\u2191x.1) x.2) fltr (\ud835\udcdd (u (\u2191x.1) x.2)) ** specialize h_tendsto x.fst ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m \u03b3 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b2 fltr : Filter \u03b3 inst\u271d\u00b9 : NeBot fltr inst\u271d : IsCountablyGenerated fltr U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 h : \u2200 (l : \u03b3), ProgMeasurable f (U l) i : \u03b9 x : \u2191(Set.Iic i) \u00d7 \u03a9 h_tendsto : Tendsto (fun i_1 => U i_1 \u2191x.1) fltr (\ud835\udcdd (u \u2191x.1)) \u22a2 Tendsto (fun i_1 => U i_1 (\u2191x.1) x.2) fltr (\ud835\udcdd (u (\u2191x.1) x.2)) ** rw [tendsto_nhds] at h_tendsto \u22a2 ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m \u03b3 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b9 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b2 fltr : Filter \u03b3 inst\u271d\u00b9 : NeBot fltr inst\u271d : IsCountablyGenerated fltr U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 h : \u2200 (l : \u03b3), ProgMeasurable f (U l) i : \u03b9 x : \u2191(Set.Iic i) \u00d7 \u03a9 h_tendsto : \u2200 (s : Set (\u03a9 \u2192 \u03b2)), IsOpen s \u2192 u \u2191x.1 \u2208 s \u2192 (fun i_1 => U i_1 \u2191x.1) \u207b\u00b9' s \u2208 fltr \u22a2 \u2200 (s : Set \u03b2), IsOpen s \u2192 u (\u2191x.1) x.2 \u2208 s \u2192 (fun i_1 => U i_1 (\u2191x.1) x.2) \u207b\u00b9' s \u2208 fltr ** exact fun s hs h_mem => h_tendsto {g | g x.snd \u2208 s} (hs.preimage (continuous_apply x.snd)) h_mem ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.append_get_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Append \u03b1 a b : Part \u03b1 hab : (a ++ b).Dom \u22a2 get (a ++ b) hab = get a (_ : a.Dom) ++ get b (_ : b.Dom) ** simp [append_def] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Append \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.natAbs_lt_iff_sq_lt ** 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_lt_iff_mul_self_lt ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.pdf.integral_fun_mul_eq_integral ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f \u22a2 \u222b (x : E), f x * ENNReal.toReal (pdf X \u2119 x) \u2202\u03bc = \u222b (x : \u03a9), f (X x) \u2202\u2119 ** by_cases hpdf : Integrable (fun x => f x * (pdf X \u2119 \u03bc x).toReal) \u03bc ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 \u222b (x : E), f x * ENNReal.toReal (pdf X \u2119 x) \u2202\u03bc = \u222b (x : \u03a9), f (X x) \u2202\u2119 ** rw [\u2190 integral_map (HasPDF.measurable X \u2119 \u03bc).aemeasurable hf.aestronglyMeasurable,\n  map_eq_withDensity_pdf X \u2119 \u03bc, integral_eq_lintegral_pos_part_sub_lintegral_neg_part hpdf,\n  integral_eq_lintegral_pos_part_sub_lintegral_neg_part,\n  lintegral_withDensity_eq_lintegral_mul _ (measurable_pdf X \u2119 \u03bc) hf.neg.ennreal_ofReal,\n  lintegral_withDensity_eq_lintegral_mul _ (measurable_pdf X \u2119 \u03bc) hf.ennreal_ofReal] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 ENNReal.toReal (\u222b\u207b (a : E), ENNReal.ofReal (f a * ENNReal.toReal (pdf X \u2119 a)) \u2202\u03bc) -       ENNReal.toReal (\u222b\u207b (a : E), ENNReal.ofReal (-(f a * ENNReal.toReal (pdf X \u2119 a))) \u2202\u03bc) =     ENNReal.toReal (\u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc) -       ENNReal.toReal (\u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc) ** congr 2 ** case pos.e_a.e_a \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 \u222b\u207b (a : E), ENNReal.ofReal (f a * ENNReal.toReal (pdf X \u2119 a)) \u2202\u03bc =     \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc ** have : \u2200 x, ENNReal.ofReal (f x * (pdf X \u2119 \u03bc x).toReal) =\n    ENNReal.ofReal (pdf X \u2119 \u03bc x).toReal * ENNReal.ofReal (f x) := fun x \u21a6 by\n  rw [mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg] ** case pos.e_a.e_a \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) this :   \u2200 (x : E),     ENNReal.ofReal (f x * ENNReal.toReal (pdf X \u2119 x)) =       ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (f x) \u22a2 \u222b\u207b (a : E), ENNReal.ofReal (f a * ENNReal.toReal (pdf X \u2119 a)) \u2202\u03bc =     \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc ** simp_rw [this] ** case pos.e_a.e_a \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) this :   \u2200 (x : E),     ENNReal.ofReal (f x * ENNReal.toReal (pdf X \u2119 x)) =       ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (f x) \u22a2 \u222b\u207b (a : E), ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 a)) * ENNReal.ofReal (f a) \u2202\u03bc =     \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (f x)) a \u2202\u03bc ** exact lintegral_congr_ae (Filter.EventuallyEq.mul ofReal_toReal_ae_eq (ae_eq_refl _)) ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) x : E \u22a2 ENNReal.ofReal (f x * ENNReal.toReal (pdf X \u2119 x)) = ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (f x) ** rw [mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg] ** case pos.e_a.e_a \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 \u222b\u207b (a : E), ENNReal.ofReal (-(f a * ENNReal.toReal (pdf X \u2119 a))) \u2202\u03bc =     \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc ** have :\n  \u2200 x,\n    ENNReal.ofReal (-(f x * (pdf X \u2119 \u03bc x).toReal)) =\n      ENNReal.ofReal (pdf X \u2119 \u03bc x).toReal * ENNReal.ofReal (-f x) := by\n  intro x\n  rw [neg_mul_eq_neg_mul, mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg] ** case pos.e_a.e_a \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) this :   \u2200 (x : E),     ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =       ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x) \u22a2 \u222b\u207b (a : E), ENNReal.ofReal (-(f a * ENNReal.toReal (pdf X \u2119 a))) \u2202\u03bc =     \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc ** simp_rw [this] ** case pos.e_a.e_a \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) this :   \u2200 (x : E),     ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =       ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x) \u22a2 \u222b\u207b (a : E), ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 a)) * ENNReal.ofReal (-f a) \u2202\u03bc =     \u222b\u207b (a : E), (pdf X \u2119 * fun x => ENNReal.ofReal (-f x)) a \u2202\u03bc ** exact lintegral_congr_ae (Filter.EventuallyEq.mul ofReal_toReal_ae_eq (ae_eq_refl _)) ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 \u2200 (x : E),     ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =       ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x) ** intro x ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) x : E \u22a2 ENNReal.ofReal (-(f x * ENNReal.toReal (pdf X \u2119 x))) =     ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x)) * ENNReal.ofReal (-f x) ** rw [neg_mul_eq_neg_mul, mul_comm, ENNReal.ofReal_mul ENNReal.toReal_nonneg] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 Integrable fun y => f y ** refine' \u27e8hf.aestronglyMeasurable, _\u27e9 ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 HasFiniteIntegral fun y => f y ** rw [HasFiniteIntegral,\n  lintegral_withDensity_eq_lintegral_mul _ (measurable_pdf _ _ _)\n    hf.nnnorm.coe_nnreal_ennreal] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 \u222b\u207b (a : E), (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) a \u2202\u03bc < \u22a4 ** have : (fun x => (pdf X \u2119 \u03bc * fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)) x) =\u1d50[\u03bc]\n    fun x => \u2016f x * (pdf X \u2119 \u03bc x).toReal\u2016\u208a := by\n  simp_rw [\u2190 smul_eq_mul, nnnorm_smul, ENNReal.coe_mul]\n  rw [smul_eq_mul, mul_comm]\n  refine' Filter.EventuallyEq.mul (ae_eq_refl _) (ae_eq_trans ofReal_toReal_ae_eq.symm _)\n  simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) this : (fun x => (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x * ENNReal.toReal (pdf X \u2119 x)\u2016\u208a \u22a2 \u222b\u207b (a : E), (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) a \u2202\u03bc < \u22a4 ** rw [lintegral_congr_ae this] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) this : (fun x => (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x * ENNReal.toReal (pdf X \u2119 x)\u2016\u208a \u22a2 \u222b\u207b (a : E), \u2191\u2016f a * ENNReal.toReal (pdf X \u2119 a)\u2016\u208a \u2202\u03bc < \u22a4 ** exact hpdf.2 ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 (fun x => (pdf X \u2119 * fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x * ENNReal.toReal (pdf X \u2119 x)\u2016\u208a ** simp_rw [\u2190 smul_eq_mul, nnnorm_smul, ENNReal.coe_mul] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 (fun x => (pdf X \u2119 \u2022 fun x => \u2191\u2016f x\u2016\u208a) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x\u2016\u208a * \u2191\u2016ENNReal.toReal (pdf X \u2119 x)\u2016\u208a ** rw [smul_eq_mul, mul_comm] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 (fun x => ((fun x => \u2191\u2016f x\u2016\u208a) * pdf X \u2119) x) =\u1da0[ae \u03bc] fun x => \u2191\u2016f x\u2016\u208a * \u2191\u2016ENNReal.toReal (pdf X \u2119 x)\u2016\u208a ** refine' Filter.EventuallyEq.mul (ae_eq_refl _) (ae_eq_trans ofReal_toReal_ae_eq.symm _) ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : Integrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 (fun x => ENNReal.ofReal (ENNReal.toReal (pdf X \u2119 x))) =\u1da0[ae \u03bc] fun x => \u2191\u2016ENNReal.toReal (pdf X \u2119 x)\u2016\u208a ** simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl] ** case neg \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : \u00acIntegrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 \u222b (x : E), f x * ENNReal.toReal (pdf X \u2119 x) \u2202\u03bc = \u222b (x : \u03a9), f (X x) \u2202\u2119 ** rw [integral_undef hpdf, integral_undef] ** case neg \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d\u00b9 : IsFiniteMeasure \u2119 X : \u03a9 \u2192 E inst\u271d : HasPDF X \u2119 f : E \u2192 \u211d hf : Measurable f hpdf : \u00acIntegrable fun x => f x * ENNReal.toReal (pdf X \u2119 x) \u22a2 \u00acIntegrable fun x => f (X x) ** rwa [\u2190 integrable_iff_integrable_mul_pdf hf] at hpdf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.sInf_apply ** \u03b1 : Type u_1 m : Set (OuterMeasure \u03b1) s : Set \u03b1 h : Set.Nonempty m \u22a2 \u2191(sInf m) s = \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 \u03bc \u2208 m, \u2191\u03bc (t n) ** simp_rw [sInf_eq_boundedBy_sInfGen, boundedBy_apply, iSup_sInfGen_nonempty h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.neg_smul ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u2074 : Ring \u03b1 inst\u271d\u00b3 : AddCommGroup \u03b2 inst\u271d\u00b2 : Module \u03b1 \u03b2 inst\u271d\u00b9 : DecidableEq \u03b2 s : Finset \u03b1 t : Finset \u03b2 a : \u03b1 inst\u271d : DecidableEq \u03b1 \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\u2074 : Ring \u03b1 inst\u271d\u00b3 : AddCommGroup \u03b2 inst\u271d\u00b2 : Module \u03b1 \u03b2 inst\u271d\u00b9 : DecidableEq \u03b2 s : Finset \u03b1 t : Finset \u03b2 a : \u03b1 inst\u271d : DecidableEq \u03b1 \u22a2 image (fun x => -x) s \u2022 t = image (fun x => -x) (s \u2022 t) ** exact image\u2082_image_left_comm neg_smul ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.lcm_assoc ** i j k : \u2124 \u22a2 lcm (\u2191(lcm i j)) k = lcm i \u2191(lcm j k) ** rw [Int.lcm, Int.lcm, Int.lcm, Int.lcm, natAbs_ofNat, natAbs_ofNat] ** i j k : \u2124 \u22a2 Nat.lcm (Nat.lcm (natAbs i) (natAbs j)) (natAbs k) = Nat.lcm (natAbs i) (Nat.lcm (natAbs j) (natAbs k)) ** apply Nat.lcm_assoc ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.zneg_bit1 ** \u03b1 : Type u_1 n : ZNum \u22a2 -ZNum.bit1 n = ZNum.bitm1 (-n) ** cases n <;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "measurableSet_pi_of_nonempty ** \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 s : Set \u03b4 t : (i : \u03b4) \u2192 Set (\u03c0 i) hs : Set.Countable s h : Set.Nonempty (Set.pi s t) \u22a2 MeasurableSet (Set.pi s t) \u2194 \u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i) ** classical\n  rcases h with \u27e8f, hf\u27e9\n  refine' \u27e8fun hst i hi => _, MeasurableSet.pi hs\u27e9\n  convert measurable_update f (a := i) hst\n  rw [update_preimage_pi hi]\n  exact fun j hj _ => hf j hj ** \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 s : Set \u03b4 t : (i : \u03b4) \u2192 Set (\u03c0 i) hs : Set.Countable s h : Set.Nonempty (Set.pi s t) \u22a2 MeasurableSet (Set.pi s t) \u2194 \u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i) ** rcases h with \u27e8f, hf\u27e9 ** case 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\u271d u : Set \u03b1 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 s : Set \u03b4 t : (i : \u03b4) \u2192 Set (\u03c0 i) hs : Set.Countable s f : (i : \u03b4) \u2192 \u03c0 i hf : f \u2208 Set.pi s t \u22a2 MeasurableSet (Set.pi s t) \u2194 \u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i) ** refine' \u27e8fun hst i hi => _, MeasurableSet.pi hs\u27e9 ** case 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\u271d u : Set \u03b1 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 s : Set \u03b4 t : (i : \u03b4) \u2192 Set (\u03c0 i) hs : Set.Countable s f : (i : \u03b4) \u2192 \u03c0 i hf : f \u2208 Set.pi s t hst : MeasurableSet (Set.pi s t) i : \u03b4 hi : i \u2208 s \u22a2 MeasurableSet (t i) ** convert measurable_update f (a := i) hst ** case h.e'_3 \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 s : Set \u03b4 t : (i : \u03b4) \u2192 Set (\u03c0 i) hs : Set.Countable s f : (i : \u03b4) \u2192 \u03c0 i hf : f \u2208 Set.pi s t hst : MeasurableSet (Set.pi s t) i : \u03b4 hi : i \u2208 s \u22a2 t i = update f i \u207b\u00b9' Set.pi s t ** rw [update_preimage_pi hi] ** case h.e'_3 \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 s : Set \u03b4 t : (i : \u03b4) \u2192 Set (\u03c0 i) hs : Set.Countable s f : (i : \u03b4) \u2192 \u03c0 i hf : f \u2208 Set.pi s t hst : MeasurableSet (Set.pi s t) i : \u03b4 hi : i \u2208 s \u22a2 \u2200 (j : \u03b4), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j ** exact fun j hj _ => hf j hj ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.bindOnSupport_pure ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p\u271d \u2192 PMF \u03b2 p : PMF \u03b1 \u22a2 (bindOnSupport p fun a x => pure a) = p ** simp only [PMF.bind_pure, PMF.bindOnSupport_eq_bind] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hitting_le_iff_of_lt ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 inst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 inst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1 m i : \u03b9 hi : i < m \u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s ** by_cases h_exists : \u2203 j \u2208 Set.Icc n m, u j \u03c9 \u2208 s ** case pos \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 inst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 inst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1 m i : \u03b9 hi : i < m h_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s \u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s ** rw [hitting_le_iff_of_exists h_exists] ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 inst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 inst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1 m i : \u03b9 hi : i < m h_exists : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s \u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s ** simp_rw [hitting, if_neg h_exists] ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 inst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 inst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1 m i : \u03b9 hi : i < m h_exists : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s \u22a2 m \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s ** push_neg at h_exists ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 inst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 inst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1 m i : \u03b9 hi : i < m h_exists : \u2200 (j : \u03b9), j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s \u22a2 m \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s ** simp only [not_le.mpr hi, Set.mem_Icc, false_iff_iff, not_exists, not_and, and_imp] ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 inst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Set \u03b2 n i\u271d : \u03b9 \u03c9 : \u03a9 inst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1 m i : \u03b9 hi : i < m h_exists : \u2200 (j : \u03b9), j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s \u22a2 \u2200 (x : \u03b9), n \u2264 x \u2192 x \u2264 i \u2192 \u00acu x \u03c9 \u2208 s ** exact fun k hkn hki => h_exists k \u27e8hkn, hki.trans hi.le\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.getElem_ofFn ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat h : i < size (ofFn f) \u22a2 0 \u2264 n ** simp ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat h : i < size (ofFn f) \u22a2 0 = size (mkEmpty n) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_lt_ack_of_nat_primrec ** f : \u2115 \u2192 \u2115 hf : Nat.Primrec f \u22a2 \u2203 m, \u2200 (n : \u2115), f n < ack m n ** induction' hf with f g hf hg IHf IHg f g hf hg IHf IHg f g hf hg IHf IHg ** case pair f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g IHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n IHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n \u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n  case comp f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g IHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n IHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n \u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n  case prec f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g IHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n IHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n \u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n ** all_goals cases' IHf with a ha; cases' IHg with b hb ** case zero f : \u2115 \u2192 \u2115 \u22a2 \u2203 m, \u2200 (n : \u2115), (fun x => 0) n < ack m n ** exact \u27e80, ack_pos 0\u27e9 ** case succ f : \u2115 \u2192 \u2115 \u22a2 \u2203 m, \u2200 (n : \u2115), succ n < ack m n ** refine' \u27e81, fun n => _\u27e9 ** case succ f : \u2115 \u2192 \u2115 n : \u2115 \u22a2 succ n < ack 1 n ** rw [succ_eq_one_add] ** case succ f : \u2115 \u2192 \u2115 n : \u2115 \u22a2 1 + n < ack 1 n ** apply add_lt_ack ** case left f : \u2115 \u2192 \u2115 \u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => (unpair n).1) n < ack m n ** refine' \u27e80, fun n => _\u27e9 ** case left f : \u2115 \u2192 \u2115 n : \u2115 \u22a2 (fun n => (unpair n).1) n < ack 0 n ** rw [ack_zero, lt_succ_iff] ** case left f : \u2115 \u2192 \u2115 n : \u2115 \u22a2 (fun n => (unpair n).1) n \u2264 n ** exact unpair_left_le n ** case right f : \u2115 \u2192 \u2115 \u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => (unpair n).2) n < ack m n ** refine' \u27e80, fun n => _\u27e9 ** case right f : \u2115 \u2192 \u2115 n : \u2115 \u22a2 (fun n => (unpair n).2) n < ack 0 n ** rw [ack_zero, lt_succ_iff] ** case right f : \u2115 \u2192 \u2115 n : \u2115 \u22a2 (fun n => (unpair n).2) n \u2264 n ** exact unpair_right_le n ** case prec f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g IHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n IHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n \u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n ** cases' IHf with a ha ** case prec.intro f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g IHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n \u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n ** cases' IHg with b hb ** case pair.intro.intro f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n \u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n ** refine'\n  \u27e8max a b + 3, fun n =>\n    (pair_lt_max_add_one_sq _ _).trans_le <|\n      (pow_le_pow_of_le_left (add_le_add_right _ _) 2).trans <|\n        ack_add_one_sq_lt_ack_add_three _ _\u27e9 ** case pair.intro.intro f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n n : \u2115 \u22a2 max (f n) (g n) \u2264 ack (max a b) n ** rw [max_ack_left] ** case pair.intro.intro f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n n : \u2115 \u22a2 max (f n) (g n) \u2264 max (ack a n) (ack b n) ** exact max_le_max (ha n).le (hb n).le ** case comp.intro.intro f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n \u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n ** exact\n  \u27e8max a b + 2, fun n =>\n    (ha _).trans <| (ack_strictMono_right a <| hb n).trans <| ack_ack_lt_ack_max_add_two a b n\u27e9 ** case prec.intro.intro f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n this : \u2200 {m n : \u2115}, rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) \u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n ** exact \u27e8max a b + 9, fun n => this.trans_le <| ack_mono_right _ <| unpair_add_le n\u27e9 ** f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n \u22a2 \u2200 {m n : \u2115}, rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) ** intro m n ** f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 \u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) ** induction' n with n IH ** case zero f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m : \u2115 \u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) zero < ack (max a b + 9) (m + zero) ** apply (ha m).trans (ack_strictMono_left m <| (le_max_left a b).trans_lt _) ** f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m : \u2115 \u22a2 max a b < max a b + 9 ** linarith ** case succ f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) \u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) (succ n) < ack (max a b + 9) (m + succ n) ** simp only [ge_iff_le] ** case succ f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) \u22a2 g (pair m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) < ack (max a b + 9) (m + succ n) ** apply (hb _).trans ((ack_pair_lt _ _ _).trans_le _) ** f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) \u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n) ** cases' lt_or_le _ m with h\u2081 h\u2081 ** case inr f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) \u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n) ** rw [max_eq_right h\u2081] ** case inr f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) \u22a2 ack (b + 4) (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n) ** apply (ack_pair_lt _ _ _).le.trans ** case inr f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) \u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n) ** cases' lt_or_le _ n with h\u2082 h\u2082 ** case inr.inr f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) h\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n \u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n) ** rw [max_eq_right h\u2082] ** case inr.inr f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) h\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n \u22a2 ack (b + 4 + 4) (rec (f m) (fun y IH => g (pair m (pair y IH))) n) \u2264 ack (max a b + 9) (m + succ n) ** apply (ack_strictMono_right _ IH).le.trans ** case inr.inr f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) h\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n \u22a2 ack (b + 4 + 4) (ack (max a b + 9) (m + n)) \u2264 ack (max a b + 9) (m + succ n) ** rw [add_succ m, add_succ _ 8, succ_eq_add_one, succ_eq_add_one,\n    ack_succ_succ (_ + 8), add_assoc] ** case inr.inr f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) h\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n \u22a2 ack (b + (4 + 4)) (ack (max a b + 8 + 1) (m + n)) \u2264 ack (max a b + 8) (ack (max a b + 8 + 1) (m + n)) ** exact ack_mono_left _ (Nat.add_le_add (le_max_right a b) le_rfl) ** case inl f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : ?m.127778 < m \u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n) ** rw [max_eq_left h\u2081.le] ** case inl f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m \u22a2 ack (b + 4) m \u2264 ack (max a b + 9) (m + succ n) ** exact ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num)\n                 (self_le_add_right m _) ** f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m \u22a2 4 \u2264 9 ** norm_num ** case inr.inl f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) h\u2082 : ?m.128145 < n \u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n) ** rw [max_eq_left h\u2082.le, add_assoc] ** case inr.inl f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) h\u2082 : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n \u22a2 ack (b + (4 + 4)) n \u2264 ack (max a b + 9) (m + succ n) ** exact\n  ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num)\n    ((le_succ n).trans <| self_le_add_left _ _) ** f\u271d f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g a : \u2115 ha : \u2200 (n : \u2115), f n < ack a n b : \u2115 hb : \u2200 (n : \u2115), g n < ack b n m n : \u2115 IH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n) h\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) h\u2082 : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n \u22a2 4 + 4 \u2264 9 ** norm_num ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_fintype ** \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 : MeasurableSingletonClass \u03b1 inst\u271d : Fintype \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u2211 x : \u03b1, f x * \u2191\u2191\u03bc {x} ** rw [\u2190 lintegral_finset, Finset.coe_univ, Measure.restrict_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.empty_of_count_eq_zero' ** \u03b1 : Type u_1 \u03b2 : Type ?u.16305 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 s_mble : MeasurableSet s hsc : \u2191\u2191count s = 0 \u22a2 s = \u2205 ** have hs : s.Finite := by\n  rw [\u2190 count_apply_lt_top' s_mble, hsc]\n  exact WithTop.zero_lt_top ** \u03b1 : Type u_1 \u03b2 : Type ?u.16305 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 s_mble : MeasurableSet s hsc : \u2191\u2191count s = 0 hs : Set.Finite s \u22a2 s = \u2205 ** simpa [count_apply_finite' hs s_mble] using hsc ** \u03b1 : Type u_1 \u03b2 : Type ?u.16305 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 s_mble : MeasurableSet s hsc : \u2191\u2191count s = 0 \u22a2 Set.Finite s ** rw [\u2190 count_apply_lt_top' s_mble, hsc] ** \u03b1 : Type u_1 \u03b2 : Type ?u.16305 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 s_mble : MeasurableSet s hsc : \u2191\u2191count s = 0 \u22a2 0 < \u22a4 ** exact WithTop.zero_lt_top ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.exists_eq_insert_iff_ncard ** \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 (\u2203 a x, insert a s = t) \u2194 s \u2286 t \u2227 ncard s + 1 = ncard t ** cases' t.finite_or_infinite with ht ht ** case inr \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d ht : Set.Infinite t \u22a2 (\u2203 a x, insert a s = t) \u2194 s \u2286 t \u2227 ncard s + 1 = ncard t ** simp only [ht.ncard, exists_prop, add_eq_zero, and_false, iff_false, not_exists, not_and] ** case inr \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d ht : Set.Infinite t \u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u00acinsert x s = t ** rintro x - rfl ** case inr \u03b1 : Type u_1 s : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d x : \u03b1 ht : Set.Infinite (insert x s) \u22a2 False ** exact ht (hs.insert x) ** case inl \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d ht : Set.Finite t \u22a2 (\u2203 a x, insert a s = t) \u2194 s \u2286 t \u2227 ncard s + 1 = ncard t ** rw [ncard_eq_toFinset_card _ hs, ncard_eq_toFinset_card _ ht,\n  \u2190@Finite.toFinset_subset_toFinset _ _ _ hs ht, \u2190Finset.exists_eq_insert_iff] ** case inl \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d ht : Set.Finite t \u22a2 (\u2203 a x, insert a s = t) \u2194 \u2203 a x, insert a (Finite.toFinset hs) = Finite.toFinset ht ** convert Iff.rfl using 2 ** case h.e'_2.h.e'_2 \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d ht : Set.Finite t \u22a2 (fun a => \u2203 x, insert a (Finite.toFinset hs) = Finite.toFinset ht) = fun a => \u2203 x, insert a s = t ** simp ** case h.e'_2.h.e'_2 \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d ht : Set.Finite t \u22a2 (fun a => \u00aca \u2208 s \u2227 insert a (Finite.toFinset hs) = Finite.toFinset ht) = fun a => \u00aca \u2208 s \u2227 insert a s = t ** ext x ** case h.e'_2.h.e'_2.h.a \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d ht : Set.Finite t x : \u03b1 \u22a2 \u00acx \u2208 s \u2227 insert x (Finite.toFinset hs) = Finite.toFinset ht \u2194 \u00acx \u2208 s \u2227 insert x s = t ** simp [Finset.ext_iff, Set.ext_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.sigmaFinite_trim_bot_iff ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 \u22a2 SigmaFinite (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) \u2194 IsFiniteMeasure \u03bc ** rw [sigmaFinite_bot_iff] ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 \u22a2 IsFiniteMeasure (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) \u2194 IsFiniteMeasure \u03bc ** refine' \u27e8fun h => \u27e8_\u27e9, fun h => \u27e8_\u27e9\u27e9 <;> have h_univ := h.measure_univ_lt_top ** case refine'_1 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h : IsFiniteMeasure (Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) h_univ : \u2191\u2191(Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) Set.univ < \u22a4 \u22a2 \u2191\u2191\u03bc Set.univ < \u22a4 ** rwa [trim_measurableSet_eq bot_le MeasurableSet.univ] at h_univ ** case refine'_2 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h : IsFiniteMeasure \u03bc h_univ : \u2191\u2191\u03bc Set.univ < \u22a4 \u22a2 \u2191\u2191(Measure.trim \u03bc (_ : \u22a5 \u2264 m0)) Set.univ < \u22a4 ** rwa [trim_measurableSet_eq bot_le MeasurableSet.univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.SatelliteConfig.exists_normalized ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 \u22a2 \u2203 c', (\u2200 (n : Fin (Nat.succ N)), \u2016c' n\u2016 \u2264 2) \u2227 \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** let c' : Fin N.succ \u2192 E := fun i => if \u2016a.c i\u2016 \u2264 2 then a.c i else (2 / \u2016a.c i\u2016) \u2022 a.c i ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 \u22a2 \u2203 c', (\u2200 (n : Fin (Nat.succ N)), \u2016c' n\u2016 \u2264 2) \u2227 \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** refine' \u27e8c', fun n => norm_c'_le n, fun i j inej => _\u27e9 ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j \u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** wlog hij : \u2016a.c i\u2016 \u2264 \u2016a.c j\u2016 generalizing i j ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 \u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** rcases le_or_lt \u2016a.c j\u2016 2 with (Hj | Hj) ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i \u22a2 \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 ** intro i ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i i : Fin (Nat.succ N) \u22a2 \u2016c' i\u2016 \u2264 2 ** simp only ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i i : Fin (Nat.succ N) \u22a2 \u2016if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i\u2016 \u2264 2 ** split_ifs with h ** case neg E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i i : Fin (Nat.succ N) h : \u00ac\u2016c a i\u2016 \u2264 2 \u22a2 \u2016(2 / \u2016c a i\u2016) \u2022 c a i\u2016 \u2264 2 ** by_cases hi : \u2016a.c i\u2016 = 0 <;> field_simp [norm_smul, hi] ** case pos E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i i : Fin (Nat.succ N) h : \u00ac\u2016c a i\u2016 \u2264 2 hi : \u2016c a i\u2016 = 0 \u22a2 0 \u2264 2 ** norm_num ** case pos E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i i : Fin (Nat.succ N) h : \u2016c a i\u2016 \u2264 2 \u22a2 \u2016c a i\u2016 \u2264 2 ** exact h ** case inr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j this : \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 \u2016c a i\u2016 \u2264 \u2016c a j\u2016 \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 hij : \u00ac\u2016c a i\u2016 \u2264 \u2016c a j\u2016 \u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** rw [norm_sub_rev] ** case inr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j this : \u2200 (i j : Fin (Nat.succ N)), i \u2260 j \u2192 \u2016c a i\u2016 \u2264 \u2016c a j\u2016 \u2192 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 hij : \u00ac\u2016c a i\u2016 \u2264 \u2016c a j\u2016 \u22a2 1 - \u03b4 \u2264 \u2016c' j - c' i\u2016 ** exact this j i inej.symm (le_of_not_le hij) ** case inl E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : \u2016c a j\u2016 \u2264 2 \u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** simp_rw [Hj, hij.trans Hj, if_true] ** case inl E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : \u2016c a j\u2016 \u2264 2 \u22a2 1 - \u03b4 \u2264 \u2016c a i - c a j\u2016 ** exact exists_normalized_aux1 a lastr h\u03c4 \u03b4 h\u03b41 h\u03b42 i j inej ** case inr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : 2 < \u2016c a j\u2016 \u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** have H'j : \u2016a.c j\u2016 \u2264 2 \u2194 False := by simpa only [not_le, iff_false_iff] using Hj ** case inr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : 2 < \u2016c a j\u2016 H'j : \u2016c a j\u2016 \u2264 2 \u2194 False \u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** rcases le_or_lt \u2016a.c i\u2016 2 with (Hi | Hi) ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : 2 < \u2016c a j\u2016 \u22a2 \u2016c a j\u2016 \u2264 2 \u2194 False ** simpa only [not_le, iff_false_iff] using Hj ** case inr.inl E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : 2 < \u2016c a j\u2016 H'j : \u2016c a j\u2016 \u2264 2 \u2194 False Hi : \u2016c a i\u2016 \u2264 2 \u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** simp_rw [Hi, if_true, H'j, if_false] ** case inr.inl E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : 2 < \u2016c a j\u2016 H'j : \u2016c a j\u2016 \u2264 2 \u2194 False Hi : \u2016c a i\u2016 \u2264 2 \u22a2 1 - \u03b4 \u2264 \u2016c a i - (2 / \u2016c a j\u2016) \u2022 c a j\u2016 ** exact exists_normalized_aux2 a lastc lastr h\u03c4 \u03b4 h\u03b41 h\u03b42 i j inej Hi Hj ** case inr.inr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : 2 < \u2016c a j\u2016 H'j : \u2016c a j\u2016 \u2264 2 \u2194 False Hi : 2 < \u2016c a i\u2016 \u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** have H'i : \u2016a.c i\u2016 \u2264 2 \u2194 False := by simpa only [not_le, iff_false_iff] using Hi ** case inr.inr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : 2 < \u2016c a j\u2016 H'j : \u2016c a j\u2016 \u2264 2 \u2194 False Hi : 2 < \u2016c a i\u2016 H'i : \u2016c a i\u2016 \u2264 2 \u2194 False \u22a2 1 - \u03b4 \u2264 \u2016c' i - c' j\u2016 ** simp_rw [H'i, if_false, H'j, if_false] ** case inr.inr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : 2 < \u2016c a j\u2016 H'j : \u2016c a j\u2016 \u2264 2 \u2194 False Hi : 2 < \u2016c a i\u2016 H'i : \u2016c a i\u2016 \u2264 2 \u2194 False \u22a2 1 - \u03b4 \u2264 \u2016(2 / \u2016c a i\u2016) \u2022 c a i - (2 / \u2016c a j\u2016) \u2022 c a j\u2016 ** exact exists_normalized_aux3 a lastc lastr h\u03c4 \u03b4 h\u03b41 i j inej Hi hij ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E N : \u2115 \u03c4 : \u211d a : SatelliteConfig E N \u03c4 lastc : c a (last N) = 0 lastr : r a (last N) = 1 h\u03c4 : 1 \u2264 \u03c4 \u03b4 : \u211d h\u03b41 : \u03c4 \u2264 1 + \u03b4 / 4 h\u03b42 : \u03b4 \u2264 1 c' : Fin (Nat.succ N) \u2192 E := fun i => if \u2016c a i\u2016 \u2264 2 then c a i else (2 / \u2016c a i\u2016) \u2022 c a i norm_c'_le : \u2200 (i : Fin (Nat.succ N)), \u2016c' i\u2016 \u2264 2 i j : Fin (Nat.succ N) inej : i \u2260 j hij : \u2016c a i\u2016 \u2264 \u2016c a j\u2016 Hj : 2 < \u2016c a j\u2016 H'j : \u2016c a j\u2016 \u2264 2 \u2194 False Hi : 2 < \u2016c a i\u2016 \u22a2 \u2016c a i\u2016 \u2264 2 \u2194 False ** simpa only [not_le, iff_false_iff] using Hi ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.indep_iSup_directed_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) \u22a2 Indep (\u2a06 a, \u2a06 n \u2208 ns a, s n) (limsup s f) ** refine' indep_iSup_of_directed_le _ _ _ _ ** case refine'_1 \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) \u22a2 \u2200 (i : \u03b1), Indep (\u2a06 n \u2208 ns i, s n) (limsup s f) ** exact fun a => indep_biSup_limsup h_le h_indep hf (hnsp a) ** case refine'_2 \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) \u22a2 \u2200 (i : \u03b1), \u2a06 n \u2208 ns i, s n \u2264 m0 ** exact fun a => iSup\u2082_le fun n _ => h_le n ** case refine'_3 \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) \u22a2 limsup s f \u2264 m0 ** exact limsup_le_iSup.trans (iSup_le h_le) ** case refine'_4 \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) \u22a2 Directed (fun x x_1 => x \u2264 x_1) fun a => \u2a06 n \u2208 ns a, s n ** intro a b ** case refine'_4 \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) a b : \u03b1 \u22a2 \u2203 z,     (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) a) ((fun a => \u2a06 n \u2208 ns a, s n) z) \u2227       (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) b) ((fun a => \u2a06 n \u2208 ns a, s n) z) ** obtain \u27e8c, hc\u27e9 := hns a b ** case refine'_4.intro \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) a b c : \u03b1 hc : (fun x x_1 => x \u2264 x_1) (ns a) (ns c) \u2227 (fun x x_1 => x \u2264 x_1) (ns b) (ns c) \u22a2 \u2203 z,     (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) a) ((fun a => \u2a06 n \u2208 ns a, s n) z) \u2227       (fun x x_1 => x \u2264 x_1) ((fun a => \u2a06 n \u2208 ns a, s n) b) ((fun a => \u2a06 n \u2208 ns a, s n) z) ** refine' \u27e8c, _, _\u27e9 <;> refine' iSup_mono fun n => iSup_mono' fun hn => \u27e8_, le_rfl\u27e9 ** case refine'_4.intro.refine'_1 \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) a b c : \u03b1 hc : (fun x x_1 => x \u2264 x_1) (ns a) (ns c) \u2227 (fun x x_1 => x \u2264 x_1) (ns b) (ns c) n : \u03b9 hn : n \u2208 ns a \u22a2 n \u2208 ns c ** exact hc.1 hn ** case refine'_4.intro.refine'_2 \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) a b c : \u03b1 hc : (fun x x_1 => x \u2264 x_1) (ns a) (ns c) \u2227 (fun x x_1 => x \u2264 x_1) (ns b) (ns c) n : \u03b9 hn : n \u2208 ns b \u22a2 n \u2208 ns c ** exact hc.2 hn ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Nontrivial.sdiff_singleton_nonempty ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t u v : Finset \u03b1 a b c : \u03b1 s : Finset \u03b1 hS : Finset.Nontrivial s \u22a2 Finset.Nonempty (s \\ {c}) ** rw [Finset.sdiff_nonempty, Finset.subset_singleton_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t u v : Finset \u03b1 a b c : \u03b1 s : Finset \u03b1 hS : Finset.Nontrivial s \u22a2 \u00ac(s = \u2205 \u2228 s = {c}) ** push_neg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t u v : Finset \u03b1 a b c : \u03b1 s : Finset \u03b1 hS : Finset.Nontrivial s \u22a2 s \u2260 \u2205 \u2227 s \u2260 {c} ** exact \u27e8by rintro rfl; exact Finset.not_nontrivial_empty hS, hS.ne_singleton\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t u v : Finset \u03b1 a b c : \u03b1 s : Finset \u03b1 hS : Finset.Nontrivial s \u22a2 s \u2260 \u2205 ** rintro rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t u v : Finset \u03b1 a b c : \u03b1 hS : Finset.Nontrivial \u2205 \u22a2 False ** exact Finset.not_nontrivial_empty hS ** Qed",
        "informal": ""
    },
    {
        "formal": "integrableOn_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\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s ** obtain \u27e8u, u_open, x\u2080u, hu\u27e9 : \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 x \u2208 u \u2229 s, g x \u2208 ball (g x\u2080) 1 ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1  case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 \u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s ** exact mem_nhdsWithin.1 (hcg (ball_mem_nhds _ zero_lt_one)) ** case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 \u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s ** filter_upwards [tendstoUniformlyOn_iff.1 (hl\u03c6 u u_open x\u2080u) 1 zero_lt_one, 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\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 \u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s ** have A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) \u03bc := by\n  refine' Integrable.smul_of_top_right (hmg.mono (diff_subset _ _) le_rfl) _\n  apply\n    mem\u2112p_top_of_bound\n      ((integrable_of_integral_eq_one h'i).aestronglyMeasurable.mono_set (diff_subset _ _)) 1\n  filter_upwards [self_mem_ae_restrict (hs.diff u_open.measurableSet)] with x hx\n  simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le ** case h \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) B : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u) \u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s ** convert A.union B ** case h.e'_6 \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) B : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u) \u22a2 s = s \\ u \u222a s \u2229 u ** simp only [diff_union_inter] ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 \u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) ** refine' Integrable.smul_of_top_right (hmg.mono (diff_subset _ _) le_rfl) _ ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 \u22a2 Mem\u2112p (fun x => \u03c6 i x) \u22a4 ** apply\n  mem\u2112p_top_of_bound\n    ((integrable_of_integral_eq_one h'i).aestronglyMeasurable.mono_set (diff_subset _ _)) 1 ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \\ u), \u2016\u03c6 i x\u2016 \u2264 1 ** filter_upwards [self_mem_ae_restrict (hs.diff u_open.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\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 x : \u03b1 hx : x \u2208 s \\ u \u22a2 \u2016\u03c6 i x\u2016 \u2264 1 ** simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) \u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u) ** apply Integrable.smul_of_top_left ** case h\u03c6 \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) \u22a2 Integrable fun x => \u03c6 i x ** exact IntegrableOn.mono_set (integrable_of_integral_eq_one h'i) (inter_subset_left _ _) ** case hf \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) \u22a2 Mem\u2112p (fun x => g x) \u22a4 ** apply\n  mem\u2112p_top_of_bound (hmg.mono_set (inter_subset_left _ _)).aestronglyMeasurable (\u2016g x\u2080\u2016 + 1) ** case hf \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \u2229 u), \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1 ** filter_upwards [self_mem_ae_restrict (hs.inter u_open.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\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) x : \u03b1 hx : x \u2208 s \u2229 u \u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1 ** rw [inter_comm] at hx ** case h \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d hs : MeasurableSet s hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u hu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1 i : \u03b9 hi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1 h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1 A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) x : \u03b1 hx : x \u2208 u \u2229 s \u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1 ** exact (norm_lt_of_mem_ball (hu x hx)).le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.upcrossingStrat_le_one ** \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 \u22a2 upcrossingStrat a b f N n \u03c9 \u2264 1 ** rw [upcrossingStrat, \u2190 Set.indicator_finset_biUnion_apply] ** \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 \u22a2 Set.indicator (\u22c3 i \u2208 Finset.range N, Set.Ico (lowerCrossingTime a b f N i \u03c9) (upperCrossingTime a b f N (i + 1) \u03c9)) 1       n \u2264     1 ** exact Set.indicator_le_self' (fun _ _ => zero_le_one) _ ** 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 : \u03a9 \u2131 : Filtration \u2115 m0 \u22a2 \u2200 (i : \u2115),     i \u2208 Finset.range N \u2192       \u2200 (j : \u2115),         j \u2208 Finset.range N \u2192           i \u2260 j \u2192             Disjoint (Set.Ico (lowerCrossingTime a b f N i \u03c9) (upperCrossingTime a b f N (i + 1) \u03c9))               (Set.Ico (lowerCrossingTime a b f N j \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9)) ** intro i _ j _ hij ** 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 : \u03a9 \u2131 : Filtration \u2115 m0 i : \u2115 a\u271d\u00b9 : i \u2208 Finset.range N j : \u2115 a\u271d : j \u2208 Finset.range N hij : i \u2260 j \u22a2 Disjoint (Set.Ico (lowerCrossingTime a b f N i \u03c9) (upperCrossingTime a b f N (i + 1) \u03c9))     (Set.Ico (lowerCrossingTime a b f N j \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9)) ** rw [Set.Ico_disjoint_Ico] ** 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 : \u03a9 \u2131 : Filtration \u2115 m0 i : \u2115 a\u271d\u00b9 : i \u2208 Finset.range N j : \u2115 a\u271d : j \u2208 Finset.range N hij : i \u2260 j \u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264     max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9) ** obtain hij' | hij' := lt_or_gt_of_ne hij ** case h.inl \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 i : \u2115 a\u271d\u00b9 : i \u2208 Finset.range N j : \u2115 a\u271d : j \u2208 Finset.range N hij : i \u2260 j hij' : i < j \u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264     max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9) ** rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :\n  upperCrossingTime a b f N _ \u03c9 \u2264 upperCrossingTime a b f N _ \u03c9),\n  max_eq_right (lowerCrossingTime_mono hij'.le :\n    lowerCrossingTime a b f N _ _ \u2264 lowerCrossingTime _ _ _ _ _ _)] ** case h.inl \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 i : \u2115 a\u271d\u00b9 : i \u2208 Finset.range N j : \u2115 a\u271d : j \u2208 Finset.range N hij : i \u2260 j hij' : i < j \u22a2 upperCrossingTime a b f N (Nat.succ i) \u03c9 \u2264 lowerCrossingTime a b f N j \u03c9 ** refine' le_trans upperCrossingTime_le_lowerCrossingTime\n  (lowerCrossingTime_mono (Nat.succ_le_of_lt hij')) ** case h.inr \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 i : \u2115 a\u271d\u00b9 : i \u2208 Finset.range N j : \u2115 a\u271d : j \u2208 Finset.range N hij : i \u2260 j hij' : i > j \u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264     max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9) ** rw [gt_iff_lt] at hij' ** case h.inr \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 i : \u2115 a\u271d\u00b9 : i \u2208 Finset.range N j : \u2115 a\u271d : j \u2208 Finset.range N hij : i \u2260 j hij' : j < i \u22a2 min (upperCrossingTime a b f N (i + 1) \u03c9) (upperCrossingTime a b f N (j + 1) \u03c9) \u2264     max (lowerCrossingTime a b f N i \u03c9) (lowerCrossingTime a b f N j \u03c9) ** rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) :\n  upperCrossingTime a b f N _ \u03c9 \u2264 upperCrossingTime a b f N _ \u03c9),\n  max_eq_left (lowerCrossingTime_mono hij'.le :\n    lowerCrossingTime a b f N _ _ \u2264 lowerCrossingTime _ _ _ _ _ _)] ** case h.inr \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 i : \u2115 a\u271d\u00b9 : i \u2208 Finset.range N j : \u2115 a\u271d : j \u2208 Finset.range N hij : i \u2260 j hij' : j < i \u22a2 upperCrossingTime a b f N (Nat.succ j) \u03c9 \u2264 lowerCrossingTime a b f N i \u03c9 ** refine' le_trans upperCrossingTime_le_lowerCrossingTime\n  (lowerCrossingTime_mono (Nat.succ_le_of_lt hij')) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.indep_iSup_of_directed_le ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m \u22a2 Indep (\u2a06 i, m i) m' \u03ba ** let p : \u03b9 \u2192 Set (Set \u03a9) := fun n => { t | MeasurableSet[m n] t } ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} \u22a2 Indep (\u2a06 i, m i) m' \u03ba ** have hp : \u2200 n, IsPiSystem (p n) := fun n => @isPiSystem_measurableSet \u03a9 (m n) ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) \u22a2 Indep (\u2a06 i, m i) m' \u03ba ** have h_gen_n : \u2200 n, m n = generateFrom (p n) := fun n =>\n  (@generateFrom_measurableSet \u03a9 (m n)).symm ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) \u22a2 Indep (\u2a06 i, m i) m' \u03ba ** have hp_supr_pi : IsPiSystem (\u22c3 n, p n) := isPiSystem_iUnion_of_directed_le p hp hm ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) hp_supr_pi : IsPiSystem (\u22c3 n, p n) \u22a2 Indep (\u2a06 i, m i) m' \u03ba ** let p' := { t : Set \u03a9 | MeasurableSet[m'] t } ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) hp_supr_pi : IsPiSystem (\u22c3 n, p n) p' : Set (Set \u03a9) := {t | MeasurableSet t} \u22a2 Indep (\u2a06 i, m i) m' \u03ba ** have hp'_pi : IsPiSystem p' := @isPiSystem_measurableSet \u03a9 m' ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) hp_supr_pi : IsPiSystem (\u22c3 n, p n) p' : Set (Set \u03a9) := {t | MeasurableSet t} hp'_pi : IsPiSystem p' \u22a2 Indep (\u2a06 i, m i) m' \u03ba ** have h_gen' : m' = generateFrom p' := (@generateFrom_measurableSet \u03a9 m').symm ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) hp_supr_pi : IsPiSystem (\u22c3 n, p n) p' : Set (Set \u03a9) := {t | MeasurableSet t} hp'_pi : IsPiSystem p' h_gen' : m' = generateFrom p' \u22a2 Indep (\u2a06 i, m i) m' \u03ba ** have h_pi_system_indep : IndepSets (\u22c3 n, p n) p' \u03ba \u03bc := by\n  refine IndepSets.iUnion ?_\n  conv at h_indep =>\n    intro i\n    rw [h_gen_n i, h_gen']\n  exact fun n => (h_indep n).indepSets ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) hp_supr_pi : IsPiSystem (\u22c3 n, p n) p' : Set (Set \u03a9) := {t | MeasurableSet t} hp'_pi : IsPiSystem p' h_gen' : m' = generateFrom p' h_pi_system_indep : IndepSets (\u22c3 n, p n) p' \u03ba \u22a2 Indep (\u2a06 i, m i) m' \u03ba ** refine' IndepSets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi _ h_gen' h_pi_system_indep ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) hp_supr_pi : IsPiSystem (\u22c3 n, p n) p' : Set (Set \u03a9) := {t | MeasurableSet t} hp'_pi : IsPiSystem p' h_gen' : m' = generateFrom p' h_pi_system_indep : IndepSets (\u22c3 n, p n) p' \u03ba \u22a2 \u2a06 i, m i = generateFrom (\u22c3 n, p n) ** exact (generateFrom_iUnion_measurableSet _).symm ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) hp_supr_pi : IsPiSystem (\u22c3 n, p n) p' : Set (Set \u03a9) := {t | MeasurableSet t} hp'_pi : IsPiSystem p' h_gen' : m' = generateFrom p' \u22a2 IndepSets (\u22c3 n, p n) p' \u03ba ** refine IndepSets.iUnion ?_ ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_indep : \u2200 (i : \u03b9), Indep (m i) m' \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) hp_supr_pi : IsPiSystem (\u22c3 n, p n) p' : Set (Set \u03a9) := {t | MeasurableSet t} hp'_pi : IsPiSystem p' h_gen' : m' = generateFrom p' \u22a2 \u2200 (n : \u03b9), IndepSets (p n) p' \u03ba ** conv at h_indep =>\n  intro i\n  rw [h_gen_n i, h_gen'] ** \u03b1 : Type u_1 \u03a9\u271d : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9\u271d \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03a9\u271d } \u03bc\u271d : Measure \u03b1 \u03a9 : Type u_4 m : \u03b9 \u2192 MeasurableSpace \u03a9 m' m0 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba h_le : \u2200 (i : \u03b9), m i \u2264 m0 h_le' : m' \u2264 m0 hm : Directed (fun x x_1 => x \u2264 x_1) m p : \u03b9 \u2192 Set (Set \u03a9) := fun n => {t | MeasurableSet t} hp : \u2200 (n : \u03b9), IsPiSystem (p n) h_gen_n : \u2200 (n : \u03b9), m n = generateFrom (p n) hp_supr_pi : IsPiSystem (\u22c3 n, p n) p' : Set (Set \u03a9) := {t | MeasurableSet t} h_indep : \u2200 (i : \u03b9), Indep (generateFrom (p i)) (generateFrom p') \u03ba hp'_pi : IsPiSystem p' h_gen' : m' = generateFrom p' \u22a2 \u2200 (n : \u03b9), IndepSets (p n) p' \u03ba ** exact fun n => (h_indep n).indepSets ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrable_withDensity_iff_integrable_coe_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 hf : Measurable f g : \u03b1 \u2192 E \u22a2 Integrable g \u2194 Integrable fun x => \u2191(f x) \u2022 g x ** by_cases H : AEStronglyMeasurable (fun x : \u03b1 => (f x : \u211d) \u2022 g x) \u03bc ** 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 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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u22a2 Integrable g \u2194 Integrable fun x => \u2191(f x) \u2022 g x ** simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, HasFiniteIntegral, H,\n  true_and_iff] ** 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 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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a \u2202\u03bc < \u22a4 ** rw [lintegral_withDensity_eq_lintegral_mul\u2080' hf.coe_nnreal_ennreal.aemeasurable] ** 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 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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u22a2 \u222b\u207b (a : \u03b1), ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a \u2202\u03bc < \u22a4 ** rw [iff_iff_eq] ** 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 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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u22a2 (\u222b\u207b (a : \u03b1), ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) a \u2202\u03bc < \u22a4) = (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a \u2202\u03bc < \u22a4) ** congr ** case pos.e_a.e_f \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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u22a2 (fun a => ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) a) = fun a => \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a ** ext1 x ** case pos.e_a.e_f.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\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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc x : \u03b1 \u22a2 ((fun x => \u2191(f x)) * fun a => \u2191\u2016g a\u2016\u208a) x = \u2191\u2016\u2191(f x) \u2022 g x\u2016\u208a ** simp only [nnnorm_smul, NNReal.nnnorm_eq, coe_mul, Pi.mul_apply] ** 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 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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u22a2 AEMeasurable fun a => \u2191\u2016g a\u2016\u208a ** rw [aemeasurable_withDensity_ennreal_iff hf] ** 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 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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u22a2 AEMeasurable fun x => \u2191(f x) * \u2191\u2016g x\u2016\u208a ** convert H.ennnorm using 1 ** case h.e'_5 \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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u22a2 (fun x => \u2191(f x) * \u2191\u2016g x\u2016\u208a) = fun a => \u2191\u2016\u2191(f a) \u2022 g a\u2016\u208a ** ext1 x ** case h.e'_5.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\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 hf : Measurable f g : \u03b1 \u2192 E H : AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc x : \u03b1 \u22a2 \u2191(f x) * \u2191\u2016g x\u2016\u208a = \u2191\u2016\u2191(f x) \u2022 g x\u2016\u208a ** simp only [nnnorm_smul, NNReal.nnnorm_eq, coe_mul] ** 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 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 hf : Measurable f g : \u03b1 \u2192 E H : \u00acAEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u22a2 Integrable g \u2194 Integrable fun x => \u2191(f x) \u2022 g x ** simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, H, false_and_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.pi_pi_aux ** \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) hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** refine' le_antisymm _ _ ** case refine'_1 \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) hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** rw [Measure.pi, toMeasure_apply _ _ (MeasurableSet.pi countable_univ fun i _ => hs i)] ** case refine'_1 \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) hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 \u2191(OuterMeasure.pi fun i => \u2191(\u03bc i)) (Set.pi univ fun i => s i) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** apply OuterMeasure.pi_pi_le ** case refine'_2 \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) hs : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) \u2264 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) ** haveI : Encodable \u03b9 := Fintype.toEncodable \u03b9 ** case refine'_2 \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) hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : Encodable \u03b9 \u22a2 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) \u2264 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) ** simp_rw [\u2190 pi'_pi \u03bc s, Measure.pi,\n  toMeasure_apply _ _ (MeasurableSet.pi countable_univ fun i _ => hs i)] ** case refine'_2 \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) hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : Encodable \u03b9 \u22a2 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u2191(OuterMeasure.pi fun i => \u2191(\u03bc i)) (Set.pi univ fun i => s i) ** suffices (pi' \u03bc).toOuterMeasure \u2264 OuterMeasure.pi fun i => (\u03bc i).toOuterMeasure by exact this _ ** case refine'_2 \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) hs : \u2200 (i : \u03b9), MeasurableSet (s i) this : Encodable \u03b9 \u22a2 \u2191(pi' \u03bc) \u2264 OuterMeasure.pi fun i => \u2191(\u03bc i) ** clear hs s ** case refine'_2 \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) this : Encodable \u03b9 \u22a2 \u2191(pi' \u03bc) \u2264 OuterMeasure.pi fun i => \u2191(\u03bc i) ** rw [OuterMeasure.le_pi] ** case refine'_2 \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) this : Encodable \u03b9 \u22a2 \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), Set.Nonempty (Set.pi univ s) \u2192 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** intro s _ ** case refine'_2 \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) this : Encodable \u03b9 s : (i : \u03b9) \u2192 Set (\u03b1 i) a\u271d : Set.Nonempty (Set.pi univ s) \u22a2 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** exact (pi'_pi \u03bc s).le ** \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) hs : \u2200 (i : \u03b9), MeasurableSet (s i) this\u271d : Encodable \u03b9 this : \u2191(pi' \u03bc) \u2264 OuterMeasure.pi fun i => \u2191(\u03bc i) \u22a2 \u2191\u2191(pi' \u03bc) (Set.pi univ s) \u2264 \u2191(OuterMeasure.pi fun i => \u2191(\u03bc i)) (Set.pi univ fun i => s i) ** exact this _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.enumerate_inj ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 n\u2081 n\u2082 : \u2115 a : \u03b1 s : Set \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s h\u2081 : enumerate sel s n\u2081 = some a h\u2082 : enumerate sel s n\u2082 = some a \u22a2 n\u2081 = n\u2082 ** rcases le_total n\u2081 n\u2082 with (hn|hn) ** case inl \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 n\u2081 n\u2082 : \u2115 a : \u03b1 s : Set \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s h\u2081 : enumerate sel s n\u2081 = some a h\u2082 : enumerate sel s n\u2082 = some a hn : n\u2081 \u2264 n\u2082 \u22a2 n\u2081 = n\u2082  case inr \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 n\u2081 n\u2082 : \u2115 a : \u03b1 s : Set \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s h\u2081 : enumerate sel s n\u2081 = some a h\u2082 : enumerate sel s n\u2082 = some a hn : n\u2082 \u2264 n\u2081 \u22a2 n\u2081 = n\u2082 ** on_goal 2 => swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082 ** case inl \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 n\u2081 n\u2082 : \u2115 a : \u03b1 s : Set \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s h\u2081 : enumerate sel s n\u2081 = some a h\u2082 : enumerate sel s n\u2082 = some a hn : n\u2081 \u2264 n\u2082 \u22a2 n\u2081 = n\u2082  case inr \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 n\u2082 n\u2081 : \u2115 a : \u03b1 s : Set \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s h\u2082 : enumerate sel s n\u2082 = some a h\u2081 : enumerate sel s n\u2081 = some a hn : n\u2081 \u2264 n\u2082 \u22a2 n\u2082 = n\u2081 ** all_goals\n  rcases Nat.le.dest hn with \u27e8m, rfl\u27e9\n  clear hn\n  induction n\u2081 generalizing s\n  case zero =>\n    cases m\n    case zero => rfl\n    case succ m =>\n      have h' : enumerate sel (s \\ {a}) m = some a := by\n        simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\n      have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'\n      simp_all [Set.mem_diff_singleton]\n  case succ k ih =>\n    cases h : sel s\n    \n    case none =>\n      simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n    case some _ =>\n      simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n                     Nat.add_succ, enumerate._eq_2, Nat.succ.injEq]\n      exact ih h\u2081 h\u2082 ** case inr \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 n\u2081 n\u2082 : \u2115 a : \u03b1 s : Set \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s h\u2081 : enumerate sel s n\u2081 = some a h\u2082 : enumerate sel s n\u2082 = some a hn : n\u2082 \u2264 n\u2081 \u22a2 n\u2081 = n\u2082 ** swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082 ** case inr \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 n\u2082 n\u2081 : \u2115 a : \u03b1 s : Set \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s h\u2082 : enumerate sel s n\u2082 = some a h\u2081 : enumerate sel s n\u2081 = some a hn : n\u2081 \u2264 n\u2082 \u22a2 n\u2082 = n\u2081 ** rcases Nat.le.dest hn with \u27e8m, rfl\u27e9 ** case inr.intro \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 n\u2081 : \u2115 a : \u03b1 s : Set \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s h\u2081 : enumerate sel s n\u2081 = some a m : \u2115 h\u2082 : enumerate sel s (n\u2081 + m) = some a hn : n\u2081 \u2264 n\u2081 + m \u22a2 n\u2081 + m = n\u2081 ** clear hn ** case inr.intro \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 n\u2081 : \u2115 a : \u03b1 s : Set \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s h\u2081 : enumerate sel s n\u2081 = some a m : \u2115 h\u2082 : enumerate sel s (n\u2081 + m) = some a \u22a2 n\u2081 + m = n\u2081 ** induction n\u2081 generalizing s ** case inr.intro.zero \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m : \u2115 s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a h\u2082 : enumerate sel s (Nat.zero + m) = some a \u22a2 Nat.zero + m = Nat.zero  case inr.intro.succ \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m n\u271d : \u2115 n_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d + m = n\u271d s : Set \u03b1 h\u2081 : enumerate sel s (Nat.succ n\u271d) = some a h\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a \u22a2 Nat.succ n\u271d + m = Nat.succ n\u271d ** case zero =>\n  cases m\n  case zero => rfl\n  case succ m =>\n    have h' : enumerate sel (s \\ {a}) m = some a := by\n      simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\n    have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'\n    simp_all [Set.mem_diff_singleton] ** case inr.intro.succ \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m n\u271d : \u2115 n_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d + m = n\u271d s : Set \u03b1 h\u2081 : enumerate sel s (Nat.succ n\u271d) = some a h\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a \u22a2 Nat.succ n\u271d + m = Nat.succ n\u271d ** case succ k ih =>\n  cases h : sel s\n  \n  case none =>\n    simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n  case some _ =>\n    simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n                   Nat.add_succ, enumerate._eq_2, Nat.succ.injEq]\n    exact ih h\u2081 h\u2082 ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m : \u2115 s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a h\u2082 : enumerate sel s (Nat.zero + m) = some a \u22a2 Nat.zero + m = Nat.zero ** cases m ** case zero \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a h\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a \u22a2 Nat.zero + Nat.zero = Nat.zero  case succ \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a n\u271d : \u2115 h\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a \u22a2 Nat.zero + Nat.succ n\u271d = Nat.zero ** case zero => rfl ** case succ \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a n\u271d : \u2115 h\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a \u22a2 Nat.zero + Nat.succ n\u271d = Nat.zero ** case succ m =>\n  have h' : enumerate sel (s \\ {a}) m = some a := by\n    simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\n  have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'\n  simp_all [Set.mem_diff_singleton] ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a h\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a \u22a2 Nat.zero + Nat.zero = Nat.zero ** rfl ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a m : \u2115 h\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a \u22a2 Nat.zero + Nat.succ m = Nat.zero ** have h' : enumerate sel (s \\ {a}) m = some a := by\n  simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082 ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a m : \u2115 h\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a h' : enumerate sel (s \\ {a}) m = some a \u22a2 Nat.zero + Nat.succ m = Nat.zero ** have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h' ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a m : \u2115 h\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a h' : enumerate sel (s \\ {a}) m = some a this : a \u2208 s \\ {a} \u22a2 Nat.zero + Nat.succ m = Nat.zero ** simp_all [Set.mem_diff_singleton] ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s s : Set \u03b1 h\u2081 : enumerate sel s Nat.zero = some a m : \u2115 h\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a \u22a2 enumerate sel (s \\ {a}) m = some a ** simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add] ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s s : Set \u03b1 m : \u2115 h\u2081 : sel s = some a h\u2082 :   (do       let a \u2190 some a       enumerate sel (s \\ {a}) m) =     some a \u22a2 enumerate sel (s \\ {a}) m = some a ** exact h\u2082 ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m k : \u2115 ih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k s : Set \u03b1 h\u2081 : enumerate sel s (Nat.succ k) = some a h\u2082 : enumerate sel s (Nat.succ k + m) = some a \u22a2 Nat.succ k + m = Nat.succ k ** cases h : sel s ** case none \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m k : \u2115 ih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k s : Set \u03b1 h\u2081 : enumerate sel s (Nat.succ k) = some a h\u2082 : enumerate sel s (Nat.succ k + m) = some a h : sel s = none \u22a2 Nat.succ k + m = Nat.succ k  case some \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m k : \u2115 ih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k s : Set \u03b1 h\u2081 : enumerate sel s (Nat.succ k) = some a h\u2082 : enumerate sel s (Nat.succ k + m) = some a val\u271d : \u03b1 h : sel s = some val\u271d \u22a2 Nat.succ k + m = Nat.succ k ** case none =>\n  simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h] ** case some \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m k : \u2115 ih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k s : Set \u03b1 h\u2081 : enumerate sel s (Nat.succ k) = some a h\u2082 : enumerate sel s (Nat.succ k + m) = some a val\u271d : \u03b1 h : sel s = some val\u271d \u22a2 Nat.succ k + m = Nat.succ k ** case some _ =>\n  simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n                 Nat.add_succ, enumerate._eq_2, Nat.succ.injEq]\n  exact ih h\u2081 h\u2082 ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m k : \u2115 ih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k s : Set \u03b1 h\u2081 : enumerate sel s (Nat.succ k) = some a h\u2082 : enumerate sel s (Nat.succ k + m) = some a h : sel s = none \u22a2 Nat.succ k + m = Nat.succ k ** simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h] ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m k : \u2115 ih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k s : Set \u03b1 h\u2081 : enumerate sel s (Nat.succ k) = some a h\u2082 : enumerate sel s (Nat.succ k + m) = some a val\u271d : \u03b1 h : sel s = some val\u271d \u22a2 Nat.succ k + m = Nat.succ k ** simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n               Nat.add_succ, enumerate._eq_2, Nat.succ.injEq] ** \u03b1 : Type u_1 sel : Set \u03b1 \u2192 Option \u03b1 a : \u03b1 h_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s m k : \u2115 s : Set \u03b1 val\u271d : \u03b1 ih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (m + k) = some a \u2192 m + k = k h\u2081 :   (do       let a \u2190 some val\u271d       enumerate sel (s \\ {a}) k) =     some a h\u2082 :   (do       let a \u2190 some val\u271d       enumerate sel (s \\ {a}) (m + k)) =     some a h : sel s = some val\u271d \u22a2 m + k = k ** exact ih h\u2081 h\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_comp_add_mul ** \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\u271d : \u211d f : \u211d \u2192 E hc : c \u2260 0 d : \u211d \u22a2 \u222b (x : \u211d) in a..b, f (d + c * x) = c\u207b\u00b9 \u2022 \u222b (x : \u211d) in d + c * a..d + c * b, f x ** rw [\u2190 integral_comp_add_left, \u2190 integral_comp_mul_left _ hc] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.mem_list_prod ** 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\u271d : \u03b1 m n : \u2115 l : List (Set \u03b1) a : \u03b1 \u22a2 a \u2208 List.prod l \u2194 \u2203 l', List.prod (List.map (fun x => \u2191x.snd) l') = a \u2227 List.map Sigma.fst l' = l ** induction' l using List.ofFnRec with n f ** case h 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\u271d : \u03b1 m n\u271d : \u2115 a : \u03b1 n : \u2115 f : Fin n \u2192 Set \u03b1 \u22a2 a \u2208 List.prod (List.ofFn f) \u2194     \u2203 l', List.prod (List.map (fun x => \u2191x.snd) l') = a \u2227 List.map Sigma.fst l' = List.ofFn f ** simp only [mem_prod_list_ofFn, List.exists_iff_exists_tuple, List.map_ofFn, Function.comp,\n  List.ofFn_inj', Sigma.mk.inj_iff, and_left_comm, exists_and_left, exists_eq_left, heq_eq_eq] ** case h 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\u271d : \u03b1 m n\u271d : \u2115 a : \u03b1 n : \u2115 f : Fin n \u2192 Set \u03b1 \u22a2 (\u2203 f_1, List.prod (List.ofFn fun i => \u2191(f_1 i)) = a) \u2194     \u2203 x, List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = a \u2227 (fun x_1 => (x x_1).fst) = f ** constructor ** case h.mp 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\u271d : \u03b1 m n\u271d : \u2115 a : \u03b1 n : \u2115 f : Fin n \u2192 Set \u03b1 \u22a2 (\u2203 f_1, List.prod (List.ofFn fun i => \u2191(f_1 i)) = a) \u2192     \u2203 x, List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = a \u2227 (fun x_1 => (x x_1).fst) = f ** rintro \u27e8fi, rfl\u27e9 ** case h.mp.intro 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 f : Fin n \u2192 Set \u03b1 fi : (i : Fin n) \u2192 \u2191(f i) \u22a2 \u2203 x,     List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = List.prod (List.ofFn fun i => \u2191(fi i)) \u2227       (fun x_1 => (x x_1).fst) = f ** exact \u27e8fun i \u21a6 \u27e8_, fi i\u27e9, rfl, rfl\u27e9 ** case h.mpr 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\u271d : \u03b1 m n\u271d : \u2115 a : \u03b1 n : \u2115 f : Fin n \u2192 Set \u03b1 \u22a2 (\u2203 x, List.prod (List.ofFn fun x_1 => \u2191(x x_1).snd) = a \u2227 (fun x_1 => (x x_1).fst) = f) \u2192     \u2203 f_1, List.prod (List.ofFn fun i => \u2191(f_1 i)) = a ** rintro \u27e8fi, rfl, rfl\u27e9 ** case h.mpr.intro.intro 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 fi : Fin n \u2192 (s : Set \u03b1) \u00d7 \u2191s \u22a2 \u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = List.prod (List.ofFn fun x => \u2191(fi x).snd) ** exact \u27e8fun i \u21a6 _, rfl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "circleIntegral.integral_sub_inv_smul_sub_smul ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c w : \u2102 R : \u211d \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z ** rcases eq_or_ne R 0 with (rfl | hR) ** 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 w : \u2102 R : \u211d hR : R \u2260 0 \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z ** have : (circleMap c R \u207b\u00b9' {w}).Countable := (countable_singleton _).preimage_circleMap c hR ** 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 w : \u2102 R : \u211d hR : R \u2260 0 this : Set.Countable (circleMap c R \u207b\u00b9' {w}) \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z ** refine' intervalIntegral.integral_congr_ae ((this.ae_not_mem _).mono fun \u03b8 h\u03b8 _' => _) ** 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 w : \u2102 R : \u211d hR : R \u2260 0 this : Set.Countable (circleMap c R \u207b\u00b9' {w}) \u03b8 : \u211d h\u03b8 : \u00ac\u03b8 \u2208 circleMap c R \u207b\u00b9' {w} _' : \u03b8 \u2208 \u0399 0 (2 * \u03c0) \u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =     deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8) ** change circleMap c R \u03b8 \u2260 w at h\u03b8 ** 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 w : \u2102 R : \u211d hR : R \u2260 0 this : Set.Countable (circleMap c R \u207b\u00b9' {w}) \u03b8 : \u211d _' : \u03b8 \u2208 \u0399 0 (2 * \u03c0) h\u03b8 : circleMap c R \u03b8 \u2260 w \u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =     deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8) ** simp only [inv_smul_smul\u2080 (sub_ne_zero.2 <| h\u03b8)] ** 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 w : \u2102 \u22a2 (\u222e (z : \u2102) in C(c, 0), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, 0), f z ** simp only [integral_radius_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "essSup_map_measure ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : CompleteLattice \u03b2 \u03b3 : Type u_3 m\u03b3 : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b3 g : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : SecondCountableTopology \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 hg : AEMeasurable g hf : AEMeasurable f \u22a2 essSup g (Measure.map f \u03bc) = essSup (g \u2218 f) \u03bc ** rw [essSup_congr_ae hg.ae_eq_mk, essSup_map_measure_of_measurable hg.measurable_mk hf] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : CompleteLattice \u03b2 \u03b3 : Type u_3 m\u03b3 : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b3 g : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : SecondCountableTopology \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 hg : AEMeasurable g hf : AEMeasurable f \u22a2 essSup (AEMeasurable.mk g hg \u2218 f) \u03bc = essSup (g \u2218 f) \u03bc ** refine' essSup_congr_ae _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : CompleteLattice \u03b2 \u03b3 : Type u_3 m\u03b3 : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b3 g : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : SecondCountableTopology \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 hg : AEMeasurable g hf : AEMeasurable f \u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f ** have h_eq := ae_of_ae_map hf hg.ae_eq_mk ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : CompleteLattice \u03b2 \u03b3 : Type u_3 m\u03b3 : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b3 g : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : SecondCountableTopology \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 hg : AEMeasurable g hf : AEMeasurable f h_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g (f x) = AEMeasurable.mk g hg (f x) \u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f ** rw [\u2190 EventuallyEq] at h_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : CompleteLattice \u03b2 \u03b3 : Type u_3 m\u03b3 : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b3 g : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : SecondCountableTopology \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 hg : AEMeasurable g hf : AEMeasurable f h_eq : (fun x => g (f x)) =\u1d50[\u03bc] fun x => AEMeasurable.mk g hg (f x) \u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f ** exact h_eq.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.ext_of_generateFrom_of_cover_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 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 h_inter : IsPiSystem S h_sub : T \u2286 S hc : Set.Countable T hU : \u22c3\u2080 T = univ htop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 h_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s \u22a2 \u03bc = \u03bd ** refine' ext_of_generateFrom_of_cover h_gen hc h_inter hU htop _ fun t ht => h_eq t (h_sub 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 : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S h_inter : IsPiSystem S h_sub : T \u2286 S hc : Set.Countable T hU : \u22c3\u2080 T = univ htop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 h_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s \u22a2 \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) ** intro t ht s 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\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S h_inter : IsPiSystem S h_sub : T \u2286 S hc : Set.Countable T hU : \u22c3\u2080 T = univ htop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 h_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s t : Set \u03b1 ht : t \u2208 T s : Set \u03b1 hs : s \u2208 S \u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) ** cases' (s \u2229 t).eq_empty_or_nonempty with H H ** case 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\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S h_inter : IsPiSystem S h_sub : T \u2286 S hc : Set.Countable T hU : \u22c3\u2080 T = univ htop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 h_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s t : Set \u03b1 ht : t \u2208 T s : Set \u03b1 hs : s \u2208 S H : s \u2229 t = \u2205 \u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) ** simp only [H, measure_empty] ** case 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\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S h_inter : IsPiSystem S h_sub : T \u2286 S hc : Set.Countable T hU : \u22c3\u2080 T = univ htop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 h_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s t : Set \u03b1 ht : t \u2208 T s : Set \u03b1 hs : s \u2208 S H : Set.Nonempty (s \u2229 t) \u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) ** exact h_eq _ (h_inter _ hs _ (h_sub ht) H) ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.contains ** l m r : List Char c : Char x\u271d : Substring h : ValidFor l m r x\u271d \u22a2 Substring.contains x\u271d c = true \u2194 c \u2208 m ** simp [Substring.contains, h.any, String.contains] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm'_le_snorm'_mul_rpow_measure_univ ** \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 : \u211d hp0_lt : 0 < p hpq : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc \u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** have hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt 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\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q : \u211d hp0_lt : 0 < p hpq : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q \u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** by_cases hpq_eq : p = q ** 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 : \u211d hp0_lt : 0 < p hpq : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q \u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** have hpq : p < q := lt_of_le_of_ne hpq hpq_eq ** 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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q \u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** let g := fun _ : \u03b1 => (1 : \u211d\u22650\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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 \u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** have h_rw : (\u222b\u207b a, (\u2016f a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) = \u222b\u207b a, ((\u2016f a\u2016\u208a : \u211d\u22650\u221e) * g a) ^ p \u2202\u03bc :=\n  lintegral_congr fun a => by simp ** 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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc \u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** repeat' rw [snorm'] ** 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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** rw [h_rw] ** 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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc \u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264     (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** let r := p * q / (q - p) ** 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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc r : \u211d := p * q / (q - p) \u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264     (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** have hpqr : 1 / p = 1 / q + 1 / r := by\n  field_simp [(ne_of_lt hp0_lt).symm, (ne_of_lt hq0_lt).symm]\n  ring ** 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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc r : \u211d := p * q / (q - p) hpqr : 1 / p = 1 / q + 1 / r \u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264     (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** calc\n  (\u222b\u207b a : \u03b1, (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n      (\u222b\u207b a : \u03b1, \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b a : \u03b1, g a ^ r \u2202\u03bc) ^ (1 / r) :=\n    ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr \u03bc hf.ennnorm aemeasurable_const\n  _ = (\u222b\u207b a : \u03b1, \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u03bc Set.univ ^ (1 / p - 1 / q) := by\n    rw [hpqr]; simp ** 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 : \u211d hp0_lt : 0 < p hpq : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : p = q \u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** rw [hpq_eq, sub_self, ENNReal.rpow_zero, mul_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\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 a : \u03b1 \u22a2 \u2191\u2016f a\u2016\u208a ^ p = (\u2191\u2016f a\u2016\u208a * g a) ^ p ** simp ** 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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** rw [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\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc r : \u211d := p * q / (q - p) \u22a2 1 / p = 1 / q + 1 / r ** field_simp [(ne_of_lt hp0_lt).symm, (ne_of_lt hq0_lt).symm] ** \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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc r : \u211d := p * q / (q - p) \u22a2 p * q = q * p ** ring ** \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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc r : \u211d := p * q / (q - p) hpqr : 1 / p = 1 / q + 1 / r \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r) =     (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) ** 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 : \u211d hp0_lt : 0 < p hpq\u271d : p \u2264 q f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0_lt : 0 < q hpq_eq : \u00acp = q hpq : p < q g : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1 h_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc r : \u211d := p * q / (q - p) hpqr : 1 / p = 1 / q + 1 / r \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r) =     (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / q + 1 / r - 1 / q) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.NullMeasurableSet.image ** \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 : 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 : Set \u03b1 hs : NullMeasurableSet s \u22a2 NullMeasurableSet (f '' s) ** refine' \u27e8toMeasurable \u03bc (f '' toMeasurable (\u03bc.comap f) s), measurableSet_toMeasurable _ _, _\u27e9 ** \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 : 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 : Set \u03b1 hs : NullMeasurableSet s \u22a2 f '' s =\u1da0[ae \u03bc] toMeasurable \u03bc (f '' toMeasurable (comap f \u03bc) s) ** refine' EventuallyEq.trans _ (NullMeasurableSet.toMeasurable_ae_eq _).symm ** case refine'_1 \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 : 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 : Set \u03b1 hs : NullMeasurableSet s \u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s  case refine'_2 \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 : 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 : Set \u03b1 hs : NullMeasurableSet s \u22a2 NullMeasurableSet (f '' toMeasurable (comap f \u03bc) s) ** swap ** case refine'_1 \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 : 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 : Set \u03b1 hs : NullMeasurableSet s \u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s ** have h : toMeasurable (comap f \u03bc) s =\u1d50[comap f \u03bc] s :=\n  @NullMeasurableSet.toMeasurable_ae_eq _ _ (\u03bc.comap f : Measure \u03b1) s hs ** case refine'_1 \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 : 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 : Set \u03b1 hs : NullMeasurableSet s h : toMeasurable (comap f \u03bc) s =\u1da0[ae (comap f \u03bc)] s \u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s ** exact ae_eq_image_of_ae_eq_comap f \u03bc hfi hf h.symm ** case refine'_2 \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 : 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 : Set \u03b1 hs : NullMeasurableSet s \u22a2 NullMeasurableSet (f '' toMeasurable (comap f \u03bc) s) ** exact hf _ (measurableSet_toMeasurable _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Option.join_pmap_eq_pmap_join ** \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 : (a : \u03b1) \u2192 p a \u2192 \u03b2 x : Option (Option \u03b1) H : \u2200 (a : Option \u03b1), a \u2208 x \u2192 \u2200 (a_2 : \u03b1), a_2 \u2208 a \u2192 p a_2 \u22a2 join (pmap (pmap f) x H) = pmap f (join x) (_ : \u2200 (a : \u03b1), a \u2208 join x \u2192 p a) ** rcases x with (_ | _ | x) <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.addBottom_nth_snd ** 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 L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n : \u2115 \u22a2 (ListBlank.nth (addBottom L) n).2 = ListBlank.nth L n ** conv => rhs; rw [\u2190 addBottom_map L, ListBlank.nth_map] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.prod_apply ** \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 \u2191\u2191(Measure.prod \u03bc \u03bd) s = \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc ** simp_rw [Measure.prod, bind_apply hs (Measurable.map_prod_mk_left (\u03bd := \u03bd)),\n  map_apply measurable_prod_mk_left hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.ofReal_condCdf_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 (\u2191(condCdf \u03c1 a) \u2191r)) =\u1d50[Measure.fst \u03c1] preCdf \u03c1 r ** filter_upwards [condCdf_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 : \u2191(condCdf \u03c1 a) \u2191r = ENNReal.toReal (preCdf \u03c1 r a) ha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 \u22a2 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = 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 : \u2191(condCdf \u03c1 a) \u2191r = 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": "ProbabilityTheory.strong_law_aux5 ** \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), (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n ** have A : (\u2211' j : \u2115, \u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi (j : \u211d)}) < \u221e := by\n  convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2\n  ext1 j\n  exact (hident j).measure_mem_eq measurableSet_Ioi ** \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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 B : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n ** filter_upwards [B] 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 A : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 B : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) \u03c9 : \u03a9 h\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) \u22a2 (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n ** convert isLittleO_sum_range_of_tendsto_zero h\u03c9 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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 B : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) \u03c9 : \u03a9 h\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) \u22a2 (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) = fun n =>     \u2211 i in range n, (truncation (X i) (\u2191i) \u03c9 - X i \u03c9) ** ext 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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 B : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) \u03c9 : \u03a9 h\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9 = \u2211 i in range n, (truncation (X i) (\u2191i) \u03c9 - X i \u03c9) ** rw [sum_sub_distrib] ** \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 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 ** convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2 ** case h.e'_3.h.e'_5 \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 j => \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j}) = fun j => \u2191\u2191\u2119 {\u03c9 | X 0 \u03c9 \u2208 Set.Ioi \u2191j} ** ext1 j ** case h.e'_3.h.e'_5.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 j : \u2115 \u22a2 \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} = \u2191\u2191\u2119 {\u03c9 | X 0 \u03c9 \u2208 Set.Ioi \u2191j} ** exact (hident j).measure_mem_eq measurableSet_Ioi ** \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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) ** filter_upwards [ae_eventually_not_mem A.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 A : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n \u22a2 Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) ** apply tendsto_const_nhds.congr' _ ** \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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n \u22a2 (fun x => 0) =\u1da0[atTop] fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9 ** filter_upwards [h\u03c9, Ioi_mem_atTop 0] with n hn npos ** 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 A : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n n : \u2115 hn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n npos : n \u2208 Set.Ioi 0 \u22a2 0 = truncation (X n) (\u2191n) \u03c9 - X n \u03c9 ** simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply] ** 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 A : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n n : \u2115 hn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n npos : n \u2208 Set.Ioi 0 \u22a2 0 = (if -\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n then X n \u03c9 else 0) - X n \u03c9 ** split_ifs with h ** case pos \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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n n : \u2115 hn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n npos : n \u2208 Set.Ioi 0 h : -\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n \u22a2 0 = X n \u03c9 - X n \u03c9 ** exact (sub_self _).symm ** case neg \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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n n : \u2115 hn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n npos : n \u2208 Set.Ioi 0 h : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n) \u22a2 0 = 0 - X n \u03c9 ** have : -(n : \u211d) < X n \u03c9 := by\n  apply lt_of_lt_of_le _ (hnonneg n \u03c9)\n  simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos ** case neg \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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n n : \u2115 hn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n npos : n \u2208 Set.Ioi 0 h : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n) this : -\u2191n < X n \u03c9 \u22a2 0 = 0 - X n \u03c9 ** simp only [this, true_and_iff, not_le] at h ** case neg \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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n n : \u2115 hn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n npos : n \u2208 Set.Ioi 0 this : -\u2191n < X n \u03c9 h : \u2191n < X n \u03c9 \u22a2 0 = 0 - X n \u03c9 ** exact (hn h).elim ** \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 : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n n : \u2115 hn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n npos : n \u2208 Set.Ioi 0 h : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n) \u22a2 -\u2191n < X n \u03c9 ** apply lt_of_lt_of_le _ (hnonneg n \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 A : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n n : \u2115 hn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n npos : n \u2208 Set.Ioi 0 h : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n) \u22a2 -\u2191n < 0 ** simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Egorov.measure_iUnionNotConvergentSeq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n : \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)) \u22a2 \u2191\u2191\u03bc (iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg) \u2264 ENNReal.ofReal \u03b5 ** refine' le_trans (measure_iUnion_le _) (le_trans\n  (ENNReal.tsum_le_tsum <| notConvergentSeqLTIndex_spec (half_pos h\u03b5) hf hg hsm hs hfg) _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n : \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)) \u22a2 \u2211' (a : \u2115), ENNReal.ofReal (\u03b5 / 2 * 2\u207b\u00b9 ^ a) \u2264 ENNReal.ofReal \u03b5 ** simp_rw [ENNReal.ofReal_mul (half_pos h\u03b5).le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n : \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)) \u22a2 \u2211' (a : \u2115), ENNReal.ofReal (\u03b5 / 2) * ENNReal.ofReal (2\u207b\u00b9 ^ a) \u2264 ENNReal.ofReal \u03b5 ** rw [ENNReal.tsum_mul_left, \u2190 ENNReal.ofReal_tsum_of_nonneg, inv_eq_one_div, tsum_geometric_two,\n  \u2190 ENNReal.ofReal_mul (half_pos h\u03b5).le, div_mul_cancel \u03b5 two_ne_zero] ** case hf_nonneg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n : \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)) \u22a2 \u2200 (n : \u2115), 0 \u2264 2\u207b\u00b9 ^ n ** exact fun n => pow_nonneg (by norm_num) _ ** \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 \u2264 2\u207b\u00b9 ** norm_num ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n : \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)) \u22a2 Summable fun i => 2\u207b\u00b9 ^ i ** rw [inv_eq_one_div] ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n : \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)) \u22a2 Summable fun i => (1 / 2) ^ i ** exact summable_geometric_two ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.dvd_of_mul_dvd_mul_right ** i j k : \u2124 k_non_zero : k \u2260 0 H : i * k \u2223 j * k \u22a2 i \u2223 j ** rw [mul_comm i k, mul_comm j k] at H ** i j k : \u2124 k_non_zero : k \u2260 0 H : k * i \u2223 k * j \u22a2 i \u2223 j ** exact dvd_of_mul_dvd_mul_left k_non_zero H ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9 inst\u271d : Countable \u03b9 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 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0) ** cases' isEmpty_or_nonempty \u03b9 with h h ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9 inst\u271d : Countable \u03b9 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 h : Nonempty \u03b9 \u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0) ** rw [\u2190 measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter] ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9 inst\u271d : Countable \u03b9 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 h : Nonempty \u03b9 \u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop     (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 i, s \u2229 notConvergentSeq (fun n x => f n x) (fun x => g x) n i))) ** refine' tendsto_measure_iInter (fun n => hsm.inter <| notConvergentSeq_measurableSet hf hg)\n  (fun k l hkl => Set.inter_subset_inter_right _ <| notConvergentSeq_antitone hkl)\n  \u27e8h.some,\n    (lt_of_le_of_lt (measure_mono <| Set.inter_subset_left _ _) (lt_top_iff_ne_top.2 hs)).ne\u27e9 ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9 inst\u271d : Countable \u03b9 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 h : IsEmpty \u03b9 \u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0) ** have : (fun j => \u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0 := by\n  simp only [eq_iff_true_of_subsingleton] ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9 inst\u271d : Countable \u03b9 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 h : IsEmpty \u03b9 this : (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0 \u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0) ** rw [this] ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9 inst\u271d : Countable \u03b9 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 h : IsEmpty \u03b9 this : (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0 \u22a2 Tendsto (fun j => 0) atTop (\ud835\udcdd 0) ** exact tendsto_const_nhds ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : 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\u00b9 : SemilatticeSup \u03b9 inst\u271d : Countable \u03b9 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 h : IsEmpty \u03b9 \u22a2 (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0 ** simp only [eq_iff_true_of_subsingleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_smul_measure ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (a : \u03b1), f a \u2202c \u2022 \u03bc = c * \u222b\u207b (a : \u03b1), f a \u2202\u03bc ** simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.mul_iSup, smul_eq_mul] ** 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\u00b2 : SemilatticeInf \u03b1 s\u271d : Finset \u03b2 H : Finset.Nonempty s\u271d f\u271d : \u03b2 \u2192 \u03b1 inst\u271d\u00b9 : SemilatticeInf \u03b2 inst\u271d : InfHomClass F \u03b1 \u03b2 f : F s : Finset \u03b9 hs : Finset.Nonempty s g : \u03b9 \u2192 \u03b1 \u22a2 \u2191f (inf' s hs g) = inf' s hs (\u2191f \u2218 g) ** refine' hs.cons_induction _ _ <;> intros <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.comp_assoc ** \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 : \u03b3 \u2192. \u03b4 g : \u03b2 \u2192. \u03b3 h : \u03b1 \u2192. \u03b2 x\u271d\u00b9 : \u03b1 x\u271d : \u03b4 \u22a2 x\u271d \u2208 comp (comp f g) h x\u271d\u00b9 \u2194 x\u271d \u2208 comp f (comp g h) x\u271d\u00b9 ** simp only [comp_apply, Part.bind_comp] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.ListBlank.cons_bind ** \u0393 : Type u_1 \u0393' : Type u_2 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Inhabited \u0393' a : \u0393 l : ListBlank \u0393 f : \u0393 \u2192 List \u0393' hf : \u2203 n, f default = List.replicate n default \u22a2 bind (cons a l) f hf = append (f a) (bind l f hf) ** refine' l.inductionOn fun l \u21a6 _ ** \u0393 : Type u_1 \u0393' : Type u_2 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Inhabited \u0393' a : \u0393 l\u271d : ListBlank \u0393 f : \u0393 \u2192 List \u0393' hf : \u2203 n, f default = List.replicate n default l : List \u0393 \u22a2 bind (cons a (Quotient.mk (BlankRel.setoid \u0393) l)) f hf = append (f a) (bind (Quotient.mk (BlankRel.setoid \u0393) l) f hf) ** suffices ((mk l).cons a).bind f hf = ((mk l).bind f hf).append (f a) by exact this ** \u0393 : Type u_1 \u0393' : Type u_2 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Inhabited \u0393' a : \u0393 l\u271d : ListBlank \u0393 f : \u0393 \u2192 List \u0393' hf : \u2203 n, f default = List.replicate n default l : List \u0393 \u22a2 bind (cons a (mk l)) f hf = append (f a) (bind (mk l) f hf) ** simp only [ListBlank.append_mk, ListBlank.bind_mk, ListBlank.cons_mk, List.cons_bind] ** \u0393 : Type u_1 \u0393' : Type u_2 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Inhabited \u0393' a : \u0393 l\u271d : ListBlank \u0393 f : \u0393 \u2192 List \u0393' hf : \u2203 n, f default = List.replicate n default l : List \u0393 this : bind (cons a (mk l)) f hf = append (f a) (bind (mk l) f hf) \u22a2 bind (cons a (Quotient.mk (BlankRel.setoid \u0393) l)) f hf = append (f a) (bind (Quotient.mk (BlankRel.setoid \u0393) l) f hf) ** exact this ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsFundamentalDomain.fundamentalInterior ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Countable G inst\u271d\u2075 : Group G inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : IsFundamentalDomain G s inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableSMul G \u03b1 inst\u271d : SMulInvariantMeasure G \u03b1 \u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 g, g \u2022 x \u2208 fundamentalInterior G s ** simp_rw [ae_iff, not_exists, \u2190 mem_inv_smul_set_iff, setOf_forall, \u2190 compl_setOf,\n  setOf_mem_eq, \u2190 compl_iUnion] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Countable G inst\u271d\u2075 : Group G inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : IsFundamentalDomain G s inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableSMul G \u03b1 inst\u271d : SMulInvariantMeasure G \u03b1 \u03bc \u22a2 \u2191\u2191\u03bc (\u22c3 i, i\u207b\u00b9 \u2022 fundamentalInterior G s)\u1d9c = 0 ** have :\n  ((\u22c3 g : G, g\u207b\u00b9 \u2022 s) \\ \u22c3 g : G, g\u207b\u00b9 \u2022 fundamentalFrontier G s) \u2286\n    \u22c3 g : G, g\u207b\u00b9 \u2022 fundamentalInterior G s := by\n  simp_rw [diff_subset_iff, \u2190 iUnion_union_distrib, \u2190 smul_set_union (\u03b1 := G) (\u03b2 := \u03b1),\n    fundamentalFrontier_union_fundamentalInterior]; rfl ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Countable G inst\u271d\u2075 : Group G inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : IsFundamentalDomain G s inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableSMul G \u03b1 inst\u271d : SMulInvariantMeasure G \u03b1 \u03bc this : (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s \u22a2 \u2191\u2191\u03bc (\u22c3 i, i\u207b\u00b9 \u2022 fundamentalInterior G s)\u1d9c = 0 ** refine' eq_bot_mono (\u03bc.mono <| compl_subset_compl.2 this) _ ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Countable G inst\u271d\u2075 : Group G inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : IsFundamentalDomain G s inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableSMul G \u03b1 inst\u271d : SMulInvariantMeasure G \u03b1 \u03bc this : (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s \u22a2 \u2191\u2191\u03bc ((\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s)\u1d9c = \u22a5 ** simp only [iUnion_inv_smul, compl_sdiff, ENNReal.bot_eq_zero, himp_eq, sup_eq_union,\n  @iUnion_smul_eq_setOf_exists _ _ _ _ s] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Countable G inst\u271d\u2075 : Group G inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : IsFundamentalDomain G s inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableSMul G \u03b1 inst\u271d : SMulInvariantMeasure G \u03b1 \u03bc this : (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s \u22a2 \u2191\u2191\u03bc ((\u22c3 g, g \u2022 fundamentalFrontier G s) \u222a {a | \u2203 g, g \u2022 a \u2208 s}\u1d9c) = 0 ** exact measure_union_null\n  (measure_iUnion_null fun _ => measure_smul_null hs.measure_fundamentalFrontier _) hs.ae_covers ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Countable G inst\u271d\u2075 : Group G inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : IsFundamentalDomain G s inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableSMul G \u03b1 inst\u271d : SMulInvariantMeasure G \u03b1 \u03bc \u22a2 (\u22c3 g, g\u207b\u00b9 \u2022 s) \\ \u22c3 g, g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 fundamentalInterior G s ** simp_rw [diff_subset_iff, \u2190 iUnion_union_distrib, \u2190 smul_set_union (\u03b1 := G) (\u03b2 := \u03b1),\n  fundamentalFrontier_union_fundamentalInterior] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u2076 : Countable G inst\u271d\u2075 : Group G inst\u271d\u2074 : MulAction G \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : IsFundamentalDomain G s inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : MeasurableSMul G \u03b1 inst\u271d : SMulInvariantMeasure G \u03b1 \u03bc \u22a2 \u22c3 g, g\u207b\u00b9 \u2022 s \u2286 \u22c3 g, g\u207b\u00b9 \u2022 s ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.swapAt!_def ** \u03b1 : Type u_1 a : Array \u03b1 i : Nat v : \u03b1 h : i < size a \u22a2 swapAt! a i v = (a[i], set a { val := i, isLt := h } v) ** simp [swapAt!, h] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.insertNth_eq_insertNthTR ** \u22a2 @insertNth = @insertNthTR ** funext \u03b1 f n l ** case h.h.h.h \u03b1 : Type u_1 f : Nat n : \u03b1 l : List \u03b1 \u22a2 insertNth f n l = insertNthTR f n l ** simp [insertNthTR, insertNthTR_go_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.succ_neg_succ ** a : \u2124 \u22a2 succ (-succ a) = -a ** rw [neg_succ, succ_pred] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.restrict_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 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 hs : MeasurableSet s t : Set \u03b1 \u22a2 restrict \u03bc (s \u222a t) + restrict \u03bc (s \u2229 t) = restrict \u03bc s + restrict \u03bc t ** simpa only [union_comm, inter_comm, add_comm] using restrict_union_add_inter t hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_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 \u22a2 \u2203 x, \u222b (a : \u03b1), f a \u2202\u03bc \u2264 f x ** simpa only [average_eq_integral] using exists_average_le (IsProbabilityMeasure.ne_zero \u03bc) hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PartrecToTM2.ret_supports ** S : Finset \u039b' k : Cont' H\u2081 : contSupp k \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret k)) ** have W := fun {q} => trStmts\u2081_self q ** S : Finset \u039b' k : Cont' H\u2081 : contSupp k \u2286 S W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret k)) ** cases k ** case halt S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q H\u2081 : contSupp Cont'.halt \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret Cont'.halt))  case cons\u2081 S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)))  case cons\u2082 S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d : Cont' H\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))  case comp S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))  case fix S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d))) ** case halt => trivial ** case cons\u2081 S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)))  case cons\u2082 S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d : Cont' H\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))  case comp S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))  case fix S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d))) ** case cons\u2081 => rw [contSupp_cons\u2081, Finset.union_subset_iff] at H\u2081; exact fun _ => H\u2081.1 W ** case cons\u2082 S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d : Cont' H\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d)))  case comp S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))  case fix S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d))) ** case cons\u2082 => rw [contSupp_cons\u2082, Finset.union_subset_iff] at H\u2081; exact fun _ => H\u2081.1 W ** case comp S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d)))  case fix S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d))) ** case comp => rw [contSupp_comp] at H\u2081; exact fun _ => H\u2081 (codeSupp_self _ _ W) ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q H\u2081 : contSupp Cont'.halt \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret Cont'.halt)) ** trivial ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d))) ** rw [contSupp_cons\u2081, Finset.union_subset_iff] at H\u2081 ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' 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 a\u271d\u00b9 (Cont'.cons\u2082 a\u271d))))) \u2286       S \u2227     codeSupp a\u271d\u00b9 (Cont'.cons\u2082 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2081 a\u271d\u00b9 a\u271d))) ** exact fun _ => H\u2081.1 W ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d : Cont' H\u2081 : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d))) ** rw [contSupp_cons\u2082, Finset.union_subset_iff] at H\u2081 ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d : Cont' H\u2081 : trStmts\u2081 (head stack (\u039b'.ret a\u271d)) \u2286 S \u2227 contSupp a\u271d \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.cons\u2082 a\u271d))) ** exact fun _ => H\u2081.1 W ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d))) ** rw [contSupp_comp] at H\u2081 ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : codeSupp a\u271d\u00b9 a\u271d \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.comp a\u271d\u00b9 a\u271d))) ** exact fun _ => H\u2081 (codeSupp_self _ _ W) ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d))) ** rw [contSupp_fix] at H\u2081 ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d))) ** have L := @Finset.mem_union_left ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S L : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d))) ** have R := @Finset.mem_union_right ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S L : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t R : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t \u22a2 TM2.SupportsStmt S (tr (\u039b'.ret (Cont'.fix a\u271d\u00b9 a\u271d))) ** intro s ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S L : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t R : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t s : Option \u0393' \u22a2 (fun s => bif natEnd (Option.iget s) then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) s \u2208 S ** dsimp only ** S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S L : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t R : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t s : Option \u0393' \u22a2 (bif natEnd (Option.iget s) then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) \u2208 S ** cases natEnd s.iget ** case false S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S L : \u2200 {\u03b1 : Type ?u.636746} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t R : \u2200 {\u03b1 : Type ?u.636777} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t s : Option \u0393' \u22a2 (bif false then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) \u2208 S ** refine' H\u2081 (R _ <| L _ <| R _ <| R _ <| L _ W) ** case true S : Finset \u039b' W : \u2200 {q : \u039b'}, q \u2208 trStmts\u2081 q a\u271d\u00b9 : Code a\u271d : Cont' H\u2081 : codeSupp a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S L : \u2200 {\u03b1 : Type} [inst : DecidableEq \u03b1] {s : Finset \u03b1} {a : \u03b1} (t : Finset \u03b1), a \u2208 s \u2192 a \u2208 s \u222a t R : \u2200 {\u03b1 : Type} [inst : DecidableEq \u03b1] {t : Finset \u03b1} {a : \u03b1} (s : Finset \u03b1), a \u2208 t \u2192 a \u2208 s \u222a t s : Option \u0393' \u22a2 (bif true then \u039b'.ret a\u271d else \u039b'.clear natEnd main (trNormal a\u271d\u00b9 (Cont'.fix a\u271d\u00b9 a\u271d))) \u2208 S ** exact H\u2081 (R _ <| L _ <| R _ <| R _ <| R _ <| Finset.mem_singleton_self _) ** Qed",
        "informal": ""
    },
    {
        "formal": "List.getLast?_concat ** \u03b1 : Type u_1 a : \u03b1 l : List \u03b1 \u22a2 getLast? (l ++ [a]) = some a ** simp [getLast?_eq_get?] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_trim ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** 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\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm  case neg \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : \u00acCompleteSpace G \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** swap ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** borelize G ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** by_cases hf_int : Integrable f \u03bc ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm  case neg \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : \u00acIntegrable f \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** swap ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** haveI : SeparableSpace (range f \u222a {0} : Set G) := hf.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\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** let f_seq := @SimpleFunc.approxOn G \u03b2 _ _ _ m _ hf.measurable (range f \u222a {0}) 0 (by simp) _ ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** have hf_seq_meas : \u2200 n, StronglyMeasurable[m] (f_seq n) := fun n =>\n  @SimpleFunc.stronglyMeasurable \u03b2 G m _ (f_seq n) ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** have hf_seq_int : \u2200 n, Integrable (f_seq n) \u03bc :=\n  SimpleFunc.integrable_approxOn_range (hf.mono hm).measurable hf_int ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** have hf_seq_int_m : \u2200 n, Integrable (f_seq n) (\u03bc.trim hm) := fun n =>\n  (hf_seq_int n).trim hm (hf_seq_meas n) ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n) \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** have hf_seq_eq : \u2200 n, \u222b x, f_seq n x \u2202\u03bc = \u222b x, f_seq n x \u2202\u03bc.trim hm := fun n =>\n  integral_trim_simpleFunc hm (f_seq n) (hf_seq_int n) ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** have h_lim_1 : atTop.Tendsto (fun n => \u222b x, f_seq n x \u2202\u03bc) (\ud835\udcdd (\u222b x, f x \u2202\u03bc)) := by\n  refine' tendsto_integral_of_L1 f hf_int (eventually_of_forall hf_seq_int) _\n  exact SimpleFunc.tendsto_approxOn_range_L1_nnnorm (hf.mono hm).measurable hf_int ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm h_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc)) \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** have h_lim_2 : atTop.Tendsto (fun n => \u222b x, f_seq n x \u2202\u03bc) (\ud835\udcdd (\u222b x, f x \u2202\u03bc.trim hm)) := by\n  simp_rw [hf_seq_eq]\n  refine' @tendsto_integral_of_L1 \u03b2 G _ _ m (\u03bc.trim hm) _ f (hf_int.trim hm hf) _ _\n    (eventually_of_forall hf_seq_int_m) _\n  exact @SimpleFunc.tendsto_approxOn_range_L1_nnnorm \u03b2 G m _ _ _ f _ _ hf.measurable\n    (hf_int.trim hm hf) ** case pos \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm h_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc)) h_lim_2 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm)) \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** exact tendsto_nhds_unique h_lim_1 h_lim_2 ** case neg \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : \u00acCompleteSpace G \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** 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\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : \u00acIntegrable f \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** have hf_int_m : \u00acIntegrable f (\u03bc.trim hm) := fun hf_int_m =>\n  hf_int (integrable_of_integrable_trim hm hf_int_m) ** case neg \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : \u00acIntegrable f hf_int_m : \u00acIntegrable f \u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = \u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm ** rw [integral_undef hf_int, integral_undef hf_int_m] ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) \u22a2 0 \u2208 range f \u222a {0} ** simp ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm \u22a2 Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc)) ** refine' tendsto_integral_of_L1 f hf_int (eventually_of_forall hf_seq_int) _ ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(f_seq i) x - f x\u2016\u208a \u2202\u03bc) atTop (\ud835\udcdd 0) ** exact SimpleFunc.tendsto_approxOn_range_L1_nnnorm (hf.mono hm).measurable hf_int ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm h_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc)) \u22a2 Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm)) ** simp_rw [hf_seq_eq] ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm h_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc)) \u22a2 Tendsto     (fun n =>       \u222b (x : \u03b2),         \u2191(SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n) x \u2202Measure.trim \u03bc hm)     atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202Measure.trim \u03bc hm)) ** refine' @tendsto_integral_of_L1 \u03b2 G _ _ m (\u03bc.trim hm) _ f (hf_int.trim hm hf) _ _\n  (eventually_of_forall hf_seq_int_m) _ ** \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 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : CompleteSpace F G : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedSpace \u211d G H : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 inst\u271d : NormedAddCommGroup H m m0 : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 hm : m \u2264 m0 f : \u03b2 \u2192 G hf : StronglyMeasurable f hG : CompleteSpace G this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hf_int : Integrable f this : SeparableSpace \u2191(range f \u222a {0}) f_seq : \u2115 \u2192 \u03b2 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) hf_seq_meas : \u2200 (n : \u2115), StronglyMeasurable \u2191(f_seq n) hf_seq_int : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_int_m : \u2200 (n : \u2115), Integrable \u2191(f_seq n) hf_seq_eq : \u2200 (n : \u2115), \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc = \u222b (x : \u03b2), \u2191(f_seq n) x \u2202Measure.trim \u03bc hm h_lim_1 : Tendsto (fun n => \u222b (x : \u03b2), \u2191(f_seq n) x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2), f x \u2202\u03bc)) \u22a2 Tendsto     (fun i =>       \u222b\u207b (x : \u03b2),         \u2191\u2016\u2191(SimpleFunc.approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) x -               f x\u2016\u208a \u2202Measure.trim \u03bc hm)     atTop (\ud835\udcdd 0) ** exact @SimpleFunc.tendsto_approxOn_range_L1_nnnorm \u03b2 G m _ _ _ f _ _ hf.measurable\n  (hf_int.trim hm hf) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () \u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.map X \u03bc, \u2191\u03ba x = \u2191(condDistrib Y X \u03bc) x ** have heq : \u03bc.map X = (\u03bc.map (fun x => (X x, Y x))).fst ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () heq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc) \u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.map X \u03bc, \u2191\u03ba x = \u2191(condDistrib Y X \u03bc) x ** rw [heq, condDistrib] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () heq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc) \u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc),     \u2191\u03ba x = \u2191(Measure.condKernel (Measure.map (fun a => (X a, Y a)) \u03bc)) x ** refine' eq_condKernel_of_measure_eq_compProd _ _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () heq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc) \u22a2 Measure.map (fun a => (X a, Y a)) \u03bc =     \u2191(kernel.const Unit (Measure.fst (Measure.map (fun a => (X a, Y a)) \u03bc)) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () ** convert h\u03ba ** case h.e'_3.h.e'_5.h.e'_7.h.e'_5 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () heq : Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc) \u22a2 Measure.fst (Measure.map (fun a => (X a, Y a)) \u03bc) = Measure.map X \u03bc ** exact heq.symm ** case heq \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () \u22a2 Measure.map X \u03bc = Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc) ** ext s hs ** case heq.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () s : Set \u03b2 hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.map X \u03bc) s = \u2191\u2191(Measure.fst (Measure.map (fun x => (X x, Y x)) \u03bc)) s ** rw [Measure.map_apply hX hs, Measure.fst_apply hs, Measure.map_apply] ** case heq.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () s : Set \u03b2 hs : MeasurableSet s \u22a2 \u2191\u2191\u03bc (X \u207b\u00b9' s) = \u2191\u2191\u03bc ((fun x => (X x, Y x)) \u207b\u00b9' (Prod.fst \u207b\u00b9' s))  case heq.h.hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () s : Set \u03b2 hs : MeasurableSet s \u22a2 Measurable fun x => (X x, Y x)  case heq.h.hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2077 : TopologicalSpace \u03a9 inst\u271d\u2076 : MeasurableSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : BorelSpace \u03a9 inst\u271d\u00b3 : Nonempty \u03a9 inst\u271d\u00b2 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : 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 hX : Measurable X hY : Measurable Y \u03ba : { x // x \u2208 kernel \u03b2 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : Measure.map (fun x => (X x, Y x)) \u03bc = \u2191(kernel.const Unit (Measure.map X \u03bc) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () s : Set \u03b2 hs : MeasurableSet s \u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s) ** exacts [rfl, Measurable.prod hX hY, measurable_fst hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.core_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\u271d : \u03b1 \u2192. \u03b2 f : \u03b1 \u2192 \u03b2 s : Set \u03b2 \u22a2 core (\u2191f) s = f \u207b\u00b9' s ** ext x ** case h \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 \u03b2 x : \u03b1 \u22a2 x \u2208 core (\u2191f) s \u2194 x \u2208 f \u207b\u00b9' s ** simp [core_def] ** Qed",
        "informal": ""
    },
    {
        "formal": "AddCircle.volume_closedBall ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d \u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5)) ** have hT' : |T| = T := abs_eq_self.mpr hT.out.le ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T \u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5)) ** let I := Ioc (-(T / 2)) (T / 2) ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) \u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5)) ** have h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall (0 : \u211d) \u03b5 \u2229 I = Metric.closedBall (0 : \u211d) \u03b5 := by\n  intro h\u03b5\n  rw [inter_eq_left, Real.closedBall_eq_Icc, zero_sub, zero_add]\n  rintro y \u27e8hy\u2081, hy\u2082\u27e9; constructor <;> linarith ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 \u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5)) ** have h\u2082 : (\u2191) \u207b\u00b9' Metric.closedBall (0 : AddCircle T) \u03b5 \u2229 I =\n    if \u03b5 < T / 2 then Metric.closedBall (0 : \u211d) \u03b5 else I := by\n  conv_rhs => rw [\u2190 if_ctx_congr (Iff.rfl : \u03b5 < T / 2 \u2194 \u03b5 < T / 2) h\u2081 fun _ => rfl, \u2190 hT']\n  apply coe_real_preimage_closedBall_inter_eq\n  simpa only [hT', Real.closedBall_eq_Icc, zero_add, zero_sub] using Ioc_subset_Icc_self ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 h\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I \u22a2 \u2191\u2191volume (Metric.closedBall x \u03b5) = ENNReal.ofReal (min T (2 * \u03b5)) ** rw [addHaar_closedBall_center] ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 h\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I \u22a2 \u2191\u2191volume (Metric.closedBall 0 \u03b5) = ENNReal.ofReal (min T (2 * \u03b5)) ** simp only [restrict_apply' measurableSet_Ioc, (by linarith : -(T / 2) + T = T / 2), h\u2082, \u2190\n  (AddCircle.measurePreserving_mk T (-(T / 2))).measure_preimage measurableSet_closedBall] ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 h\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I \u22a2 \u2191\u2191volume (if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else Ioc (-(T / 2)) (T / 2)) = ENNReal.ofReal (min T (2 * \u03b5)) ** by_cases h\u03b5 : \u03b5 < T / 2 ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) \u22a2 \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 ** intro h\u03b5 ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u03b5 : \u03b5 < T / 2 \u22a2 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 ** rw [inter_eq_left, Real.closedBall_eq_Icc, zero_sub, zero_add] ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u03b5 : \u03b5 < T / 2 \u22a2 Icc (-\u03b5) \u03b5 \u2286 I ** rintro y \u27e8hy\u2081, hy\u2082\u27e9 ** case intro T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u03b5 : \u03b5 < T / 2 y : \u211d hy\u2081 : -\u03b5 \u2264 y hy\u2082 : y \u2264 \u03b5 \u22a2 y \u2208 I ** constructor <;> linarith ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 \u22a2 QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I ** conv_rhs => rw [\u2190 if_ctx_congr (Iff.rfl : \u03b5 < T / 2 \u2194 \u03b5 < T / 2) h\u2081 fun _ => rfl, \u2190 hT'] ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 \u22a2 QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < |T| / 2 then Metric.closedBall 0 \u03b5 \u2229 I else I ** apply coe_real_preimage_closedBall_inter_eq ** case hs T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 \u22a2 I \u2286 Metric.closedBall 0 (|T| / 2) ** simpa only [hT', Real.closedBall_eq_Icc, zero_add, zero_sub] using Ioc_subset_Icc_self ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 h\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I \u22a2 -(T / 2) + T = T / 2 ** linarith ** case pos T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 h\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I h\u03b5 : \u03b5 < T / 2 \u22a2 \u2191\u2191volume (if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else Ioc (-(T / 2)) (T / 2)) = ENNReal.ofReal (min T (2 * \u03b5)) ** simp [h\u03b5, min_eq_right (by linarith : 2 * \u03b5 \u2264 T)] ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 h\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I h\u03b5 : \u03b5 < T / 2 \u22a2 2 * \u03b5 \u2264 T ** linarith ** case neg T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 h\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I h\u03b5 : \u00ac\u03b5 < T / 2 \u22a2 \u2191\u2191volume (if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else Ioc (-(T / 2)) (T / 2)) = ENNReal.ofReal (min T (2 * \u03b5)) ** simp [h\u03b5, min_eq_left (by linarith : T \u2264 2 * \u03b5)] ** T : \u211d hT : Fact (0 < T) x : AddCircle T \u03b5 : \u211d hT' : |T| = T I : Set \u211d := Ioc (-(T / 2)) (T / 2) h\u2081 : \u03b5 < T / 2 \u2192 Metric.closedBall 0 \u03b5 \u2229 I = Metric.closedBall 0 \u03b5 h\u2082 : QuotientAddGroup.mk \u207b\u00b9' Metric.closedBall 0 \u03b5 \u2229 I = if \u03b5 < T / 2 then Metric.closedBall 0 \u03b5 else I h\u03b5 : \u00ac\u03b5 < T / 2 \u22a2 T \u2264 2 * \u03b5 ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "Rel.preimage_comp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : Rel \u03b1 \u03b2 s : Rel \u03b2 \u03b3 t : Set \u03b3 \u22a2 preimage (r \u2022 s) t = preimage r (preimage s t) ** simp only [preimage, inv_comp, image_comp] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.any_iff ** \u03b1 : Type u_1 p : \u03b1 \u2192 Bool t : RBNode \u03b1 \u22a2 any p t = true \u2194 Any (fun x => p x = true) t ** induction t <;> simp [*, any, Any, or_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "UFModel.Agrees.set ** \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 n : \u2115 m : Fin n \u2192 \u03b2 H : Agrees arr f m i : Fin (Array.size arr) x : \u03b1 m' : Fin n \u2192 \u03b2 hm\u2081 : \u2200 (j : Fin n), \u2191j \u2260 \u2191i \u2192 m' j = m j hm\u2082 : \u2200 (h : \u2191i < n), f x = m' { val := \u2191i, isLt := h } \u22a2 Agrees (Array.set arr i x) f m' ** cases H ** case mk \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j \u22a2 Agrees (Array.set arr i x) f m' ** refine mk' (by simp) fun j hj\u2081 hj\u2082 \u21a6 ?_ ** case mk \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j j : \u2115 hj\u2081 : j < Array.size (Array.set arr i x) hj\u2082 : j < Array.size arr \u22a2 f (Array.get (Array.set arr i x) { val := j, isLt := hj\u2081 }) = m' { val := j, isLt := hj\u2082 } ** suffices f (Array.set arr i x)[j] = m' \u27e8j, hj\u2082\u27e9 by simp_all [Array.get_set] ** case mk \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j j : \u2115 hj\u2081 : j < Array.size (Array.set arr i x) hj\u2082 : j < Array.size arr \u22a2 f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 } ** by_cases h : i = j ** \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j \u22a2 Array.size arr = Array.size (Array.set arr i x) ** simp ** \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j j : \u2115 hj\u2081 : j < Array.size (Array.set arr i x) hj\u2082 : j < Array.size arr this : f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 } \u22a2 f (Array.get (Array.set arr i x) { val := j, isLt := hj\u2081 }) = m' { val := j, isLt := hj\u2082 } ** simp_all [Array.get_set] ** case pos \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j j : \u2115 hj\u2081 : j < Array.size (Array.set arr i x) hj\u2082 : j < Array.size arr h : \u2191i = j \u22a2 f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 } ** subst h ** case pos \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j hj\u2081 : \u2191i < Array.size (Array.set arr i x) hj\u2082 : \u2191i < Array.size arr \u22a2 f (Array.set arr i x)[\u2191i] = m' { val := \u2191i, isLt := hj\u2082 } ** rw [Array.get_set_eq, \u2190 hm\u2082] ** case neg \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j j : \u2115 hj\u2081 : j < Array.size (Array.set arr i x) hj\u2082 : j < Array.size arr h : \u00ac\u2191i = j \u22a2 f (Array.set arr i x)[j] = m' { val := j, isLt := hj\u2082 } ** rw [arr.get_set_ne _ _ _ h, hm\u2081 \u27e8j, _\u27e9 (Ne.symm h)] ** case neg \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j j : \u2115 hj\u2081 : j < Array.size (Array.set arr i x) hj\u2082 : j < Array.size arr h : \u00ac\u2191i = j \u22a2 f arr[j] = (fun i => f (Array.get arr i)) { val := j, isLt := hj\u2082 }  \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j j : \u2115 hj\u2081 : j < Array.size (Array.set arr i x) hj\u2082 : j < Array.size arr h : \u00ac\u2191i = j \u22a2 j < Array.size arr  \u03b1 : Type u_1 \u03b2 : Sort u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 i : Fin (Array.size arr) x : \u03b1 m' : Fin (Array.size arr) \u2192 \u03b2 hm\u2082 : \u2200 (h : \u2191i < Array.size arr), f x = m' { val := \u2191i, isLt := h } hm\u2081 : \u2200 (j : Fin (Array.size arr)), \u2191j \u2260 \u2191i \u2192 m' j = (fun i => f (Array.get arr i)) j j : \u2115 hj\u2081 : j < Array.size (Array.set arr i x) hj\u2082 : j < Array.size arr h : \u00ac\u2191i = j \u22a2 j < Array.size arr ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.ssubset_iff_exists_subset_erase ** \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 \u22a2 s \u2282 t \u2194 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a ** refine' \u27e8fun h => _, fun \u27e8a, ha, h\u27e9 => ssubset_of_subset_of_ssubset h <| erase_ssubset ha\u27e9 ** \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 h : s \u2282 t \u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a ** obtain \u27e8a, ht, hs\u27e9 := not_subset.1 h.2 ** case intro.intro \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 h : s \u2282 t a : \u03b1 ht : a \u2208 t hs : \u00aca \u2208 s \u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a ** exact \u27e8a, ht, subset_erase.2 \u27e8h.1, hs\u27e9\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.card_Ioi_eq_card_Ici_sub_one ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrderTop \u03b1 a : \u03b1 \u22a2 card (Ioi a) = card (Ici a) - 1 ** rw [Ici_eq_cons_Ioi, card_cons, add_tsub_cancel_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.chaar_sup_eq ** 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** rcases isCompact_isCompact_separated K\u2081.2 K\u2082.2 h with \u27e8U\u2081, U\u2082, h1U\u2081, h1U\u2082, h2U\u2081, h2U\u2082, hU\u27e9 ** case intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** rcases compact_open_separated_mul_right K\u2081.2 h1U\u2081 h2U\u2081 with \u27e8L\u2081, h1L\u2081, h2L\u2081\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 h2L\u2081 : K\u2081.carrier * L\u2081 \u2286 U\u2081 \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** rcases mem_nhds_iff.mp h1L\u2081 with \u27e8V\u2081, h1V\u2081, h2V\u2081, h3V\u2081\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 h2L\u2081 : K\u2081.carrier * L\u2081 \u2286 U\u2081 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** replace h2L\u2081 := Subset.trans (mul_subset_mul_left h1V\u2081) h2L\u2081 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** rcases compact_open_separated_mul_right K\u2082.2 h1U\u2082 h2U\u2082 with \u27e8L\u2082, h1L\u2082, h2L\u2082\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 h2L\u2082 : K\u2082.carrier * L\u2082 \u2286 U\u2082 \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** rcases mem_nhds_iff.mp h1L\u2082 with \u27e8V\u2082, h1V\u2082, h2V\u2082, h3V\u2082\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 h2L\u2082 : K\u2082.carrier * L\u2082 \u2286 U\u2082 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** replace h2L\u2082 := Subset.trans (mul_subset_mul_left h1V\u2082) h2L\u2082 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = 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) ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 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) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** have : Continuous eval :=\n  ((continuous_apply K\u2081).add (continuous_apply K\u2082)).sub (continuous_apply (K\u2081 \u2294 K\u2082)) ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 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) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** rw [eq_comm, \u2190 sub_eq_zero] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 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 + chaar K\u2080 K\u2082 - chaar K\u2080 (K\u2081 \u2294 K\u2082) = 0 ** show chaar K\u2080 \u2208 eval \u207b\u00b9' {(0 : \u211d)} ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 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' {0} ** let V := V\u2081 \u2229 V\u2082 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 \u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0} ** apply\n  mem_of_subset_of_mem _\n    (chaar_mem_clPrehaar K\u2080\n      \u27e8\u27e8V\u207b\u00b9, (h2V\u2081.inter h2V\u2082).preimage continuous_inv\u27e9, by\n        simp only [mem_inv, inv_one, h3V\u2081, h3V\u2082, mem_inter_iff, true_and_iff]\u27e9) ** 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 \u22a2 clPrehaar \u2191K\u2080       { toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },         mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) } \u2286     eval \u207b\u00b9' {0} ** unfold clPrehaar ** 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 \u22a2 closure       (prehaar \u2191K\u2080 ''         {U |           U \u2286               \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },                     mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens \u2227             IsOpen U \u2227 1 \u2208 U}) \u2286     eval \u207b\u00b9' {0} ** rw [IsClosed.closure_subset_iff] ** 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 \u22a2 1 \u2208 { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) }.carrier ** simp only [mem_inv, inv_one, h3V\u2081, h3V\u2082, mem_inter_iff, true_and_iff] ** 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 \u22a2 prehaar \u2191K\u2080 ''       {U |         U \u2286             \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },                   mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens \u2227           IsOpen U \u2227 1 \u2208 U} \u2286     eval \u207b\u00b9' {0} ** rintro _ \u27e8U, \u27e8h1U, h2U, h3U\u27e9, rfl\u27e9 ** case intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0} ** simp only [mem_preimage, sub_eq_zero, mem_singleton_iff] ** case intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082 = prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) ** rw [eq_comm] ** case intro.intro.intro.intro 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) = prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082 ** apply prehaar_sup_eq ** case intro.intro.intro.intro.hU 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.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\u00b3 : Group G inst\u271d\u00b2 : TopologicalSpace G inst\u271d\u00b9 : TopologicalGroup G inst\u271d : T2Space G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Set.Nonempty U ** exact \u27e81, h3U\u27e9 ** case intro.intro.intro.intro.h 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9) ** refine' disjoint_of_subset _ _ hU ** case intro.intro.intro.intro.h.refine'_1 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 K\u2081.carrier * U\u207b\u00b9 \u2286 U\u2081 ** refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L\u2081 ** case intro.intro.intro.intro.h.refine'_1 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 U\u207b\u00b9 \u2286 V\u2081 ** exact Subset.trans (inv_subset.mpr h1U) (inter_subset_left _ _) ** case intro.intro.intro.intro.h.refine'_2 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 K\u2082.carrier * U\u207b\u00b9 \u2286 U\u2082 ** refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L\u2082 ** case intro.intro.intro.intro.h.refine'_2 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 U : Set G h1U :   U \u2286     \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },           mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 U\u207b\u00b9 \u2286 V\u2082 ** exact Subset.trans (inv_subset.mpr h1U) (inter_subset_right _ _) ** 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 \u22a2 IsClosed (eval \u207b\u00b9' {0}) ** apply continuous_iff_isClosed.mp this ** case a 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 K\u2081 K\u2082 : Compacts G h : Disjoint K\u2081.carrier K\u2082.carrier U\u2081 U\u2082 : Set G h1U\u2081 : IsOpen U\u2081 h1U\u2082 : IsOpen U\u2082 h2U\u2081 : K\u2081.carrier \u2286 U\u2081 h2U\u2082 : K\u2082.carrier \u2286 U\u2082 hU : Disjoint U\u2081 U\u2082 L\u2081 : Set G h1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1 V\u2081 : Set G h1V\u2081 : V\u2081 \u2286 L\u2081 h2V\u2081 : IsOpen V\u2081 h3V\u2081 : 1 \u2208 V\u2081 h2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081 L\u2082 : Set G h1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1 V\u2082 : Set G h1V\u2082 : V\u2082 \u2286 L\u2082 h2V\u2082 : IsOpen V\u2082 h3V\u2082 : 1 \u2208 V\u2082 h2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval V : Set G := V\u2081 \u2229 V\u2082 \u22a2 IsClosed {0} ** exact isClosed_singleton ** Qed",
        "informal": ""
    },
    {
        "formal": "Filter.EventuallyEq.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\u271d s' t : Set \u03b1 f g : \u03b1 \u2192 \u03b4 s : Set \u03b1 hfg : f =\u1da0[ae \u03bc] g \u22a2 f =\u1da0[ae (Measure.restrict \u03bc s)] g ** refine' hfg.filter_mono _ ** \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 f g : \u03b1 \u2192 \u03b4 s : Set \u03b1 hfg : f =\u1da0[ae \u03bc] g \u22a2 ae (Measure.restrict \u03bc s) \u2264 ae \u03bc ** rw [Measure.ae_le_iff_absolutelyContinuous] ** \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 f g : \u03b1 \u2192 \u03b4 s : Set \u03b1 hfg : f =\u1da0[ae \u03bc] g \u22a2 Measure.restrict \u03bc s \u226a \u03bc ** exact Measure.absolutelyContinuous_of_le Measure.restrict_le_self ** 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' 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 : { x // x \u2208 Lp E 1 } \u22a2 \u2191(setToL1 hT'') f = \u2191(setToL1 hT) f + \u2191(setToL1 hT') f ** suffices setToL1 hT'' = setToL1 hT + setToL1 hT' by 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' 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 : { x // x \u2208 Lp E 1 } \u22a2 setToL1 hT'' = 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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d 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 : { x // x \u2208 Lp E 1 } \u22a2 ContinuousLinearMap.comp (setToL1 hT + 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' 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\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 hT'') f ** suffices setToL1 hT f + 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' 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\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 hT'') f ** rw [setToL1_eq_setToL1SCLM, setToL1_eq_setToL1SCLM,\n  setToL1SCLM_add_left' hT hT' hT'' 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\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' 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 : { x // x \u2208 Lp E 1 } this : setToL1 hT'' = setToL1 hT + setToL1 hT' \u22a2 \u2191(setToL1 hT'') 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' 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\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 hT'') f \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + 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' 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\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 hT'') f \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f ** congr ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.sub_ediv_of_dvd_sub ** a b c : Int hcab : c \u2223 a - b \u22a2 (a - b) / c = a / c - b / c ** rw [\u2190 Int.add_sub_cancel ((a-b) / c), \u2190 Int.add_ediv_of_dvd_left hcab, Int.sub_add_cancel] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.measure_ext_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 \u03bc \u03bd : Measure \u211d inst\u271d : IsLocallyFiniteMeasure \u03bc h : \u2200 (a b : \u211a), \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b) \u22a2 \u2200 (s : Set \u211d), s \u2208 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b} \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s ** simp only [mem_iUnion, mem_singleton_iff] ** \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 \u03bc \u03bd : Measure \u211d inst\u271d : IsLocallyFiniteMeasure \u03bc h : \u2200 (a b : \u211a), \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b) \u22a2 \u2200 (s : Set \u211d), (\u2203 i i_1 h, s = Ioo \u2191i \u2191i_1) \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s ** rintro _ \u27e8a, b, -, rfl\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 t u : Set \u03b1 \u03bc \u03bd : Measure \u211d inst\u271d : IsLocallyFiniteMeasure \u03bc h : \u2200 (a b : \u211a), \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b) a b : \u211a \u22a2 \u2191\u2191\u03bc (Ioo \u2191a \u2191b) = \u2191\u2191\u03bd (Ioo \u2191a \u2191b) ** apply h ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_emetric_ball ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d r : \u211d\u22650\u221e \u22a2 \u2191\u2191volume (EMetric.ball 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.ball a \u22a4) = 2 * \u22a4 ** rw [Metric.emetric_ball_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.ball 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.ball a \u2191r) = 2 * \u2191r ** rw [Metric.emetric_ball_nnreal, volume_ball, two_mul, \u2190 NNReal.coe_add,\n  ENNReal.ofReal_coe_nnreal, ENNReal.coe_add, two_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.mkRat_add_mkRat ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 \u22a2 mkRat n\u2081 d\u2081 + mkRat n\u2082 d\u2082 = mkRat (n\u2081 * \u2191d\u2082 + n\u2082 * \u2191d\u2081) (d\u2081 * d\u2082) ** rw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_add_normalize, normalize_eq_mkRat] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_integral_condexpL1 ** \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 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 ** simp_rw [condexpL1_eq hf] ** \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 hf : Integrable f hs : MeasurableSet s \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hf)) x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** rw [set_integral_condexpL1Clm (hf.toL1 f) hs] ** \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 hf : Integrable f hs : MeasurableSet s \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(Integrable.toL1 f hf) x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** exact set_integral_congr_ae (hm s hs) (hf.coeFn_toL1.mono fun x hx _ => hx) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condCount_compl ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t\u271d u t : Set \u03a9 hs : Set.Finite s hs' : Set.Nonempty s \u22a2 \u2191\u2191(condCount s) t + \u2191\u2191(condCount s) t\u1d9c = 1 ** rw [\u2190 condCount_union hs disjoint_compl_right, Set.union_compl_self,\n  (condCount_isProbabilityMeasure hs hs').measure_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2.stmts_supportsStmt ** K : Type u_1 inst\u271d\u00b9 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 \u03c3 : Type u_4 inst\u271d : Inhabited \u039b M : \u039b \u2192 Stmt\u2082 S : Finset \u039b q : Stmt\u2082 ss : Supports M S \u22a2 some q \u2208 stmts M S \u2192 SupportsStmt S q ** simp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq,\n  forall_eq', exists_imp, and_imp] ** K : Type u_1 inst\u271d\u00b9 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 \u03c3 : Type u_4 inst\u271d : Inhabited \u039b M : \u039b \u2192 Stmt\u2082 S : Finset \u039b q : Stmt\u2082 ss : Supports M S \u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q ** exact fun l ls h \u21a6 stmts\u2081_supportsStmt_mono h (ss.2 _ ls) ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.AssocList.eraseP_toList ** \u03b1 : Type u_1 \u03b2 : Type u_2 p : \u03b1 \u2192 \u03b2 \u2192 Bool l : AssocList \u03b1 \u03b2 \u22a2 toList (eraseP p l) =     List.eraseP       (fun x =>         match x with         | (a, b) => p a b)       (toList l) ** induction l <;> simp [List.eraseP, cond] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 p : \u03b1 \u2192 \u03b2 \u2192 Bool key\u271d : \u03b1 value\u271d : \u03b2 tail\u271d : AssocList \u03b1 \u03b2 tail_ih\u271d :   toList (eraseP p tail\u271d) =     List.eraseP       (fun x =>         match x with         | (a, b) => p a b)       (toList tail\u271d) \u22a2 toList       (match p key\u271d value\u271d with       | true => tail\u271d       | false => cons key\u271d value\u271d (eraseP p tail\u271d)) =     match p key\u271d value\u271d with     | true => toList tail\u271d     | false => (key\u271d, value\u271d) :: List.eraseP (fun x => p x.fst x.snd) (toList tail\u271d) ** split <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.toList_take ** \u03b1 : Type u \u03b2 : Type v \u03c6 : Type w n\u271d n m : \u2115 v : Vector \u03b1 m \u22a2 toList (take n v) = List.take 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 (take n { val := val\u271d, property := property\u271d }) = List.take n (toList { val := val\u271d, property := property\u271d }) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.prod_set' ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : CommGroup \u03b1 v : Vector \u03b1 n i : Fin n a : \u03b1 \u22a2 List.prod (toList (set v i a)) = List.prod (toList v) * (get v i)\u207b\u00b9 * a ** refine' (List.prod_set' v.toList i a).trans _ ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : CommGroup \u03b1 v : Vector \u03b1 n i : Fin n a : \u03b1 \u22a2 (List.prod (toList v) * if hn : \u2191i < List.length (toList v) then (List.nthLe (toList v) (\u2191i) hn)\u207b\u00b9 * a else 1) =     List.prod (toList v) * (get v i)\u207b\u00b9 * a ** simp [get_eq_get, mul_assoc] ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : CommGroup \u03b1 v : Vector \u03b1 n i : Fin n a : \u03b1 \u22a2 List.nthLe (toList v) \u2191i (_ : \u2191i < List.length (toList v)) =     List.get (toList v) (Fin.cast (_ : n = List.length (toList v)) i) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 K : Type u_5 inst\u271d : Zero \u03b2 r : \u03b2 s : Set \u03b1 f : \u03b1 \u2192\u209b \u03b2 hr : r \u2208 SimpleFunc.range (restrict f s) h0 : r \u2260 0 hs : MeasurableSet s \u22a2 r \u2208 \u2191f '' s ** simpa [mem_restrict_range hs, h0, -mem_range] using hr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 K : Type u_5 inst\u271d : Zero \u03b2 r : \u03b2 s : Set \u03b1 f : \u03b1 \u2192\u209b \u03b2 hr : r \u2208 SimpleFunc.range (restrict f s) h0 : r \u2260 0 hs : \u00acMeasurableSet s \u22a2 r \u2208 \u2191f '' s ** rw [restrict_of_not_measurable hs] at hr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 K : Type u_5 inst\u271d : Zero \u03b2 r : \u03b2 s : Set \u03b1 f : \u03b1 \u2192\u209b \u03b2 hr : r \u2208 SimpleFunc.range 0 h0 : r \u2260 0 hs : \u00acMeasurableSet s \u22a2 r \u2208 \u2191f '' s ** exact (h0 <| eq_zero_of_mem_range_zero hr).elim ** Qed",
        "informal": ""
    },
    {
        "formal": "String.extract_cons_addChar ** c : Char cs : List Char b e : Pos \u22a2 extract { data := c :: cs } (b + c) (e + c) = extract { data := cs } b e ** simp [extract, Nat.add_le_add_iff_right] ** c : Char cs : List Char b e : Pos \u22a2 (if e.byteIdx \u2264 b.byteIdx then \"\" else { data := extract.go\u2081 (c :: cs) 0 (b + c) (e + c) }) =     if b.byteIdx \u2265 e.byteIdx then \"\" else { data := extract.go\u2081 cs 0 b e } ** split <;> [rfl; rw [extract.go\u2081_cons_addChar]] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.of_nat_cast ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 n : \u2115 \u22a2 \u2191\u2191n = \u2191n ** rw [\u2190 Int.cast_ofNat, of_int_cast, Int.cast_ofNat] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurable_of_pdf_ne_zero ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 E h : pdf X \u2119 \u2260 0 \u22a2 Measurable X ** by_contra hX ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 E h : pdf X \u2119 \u2260 0 hX : \u00acMeasurable X \u22a2 False ** exact h (pdf_eq_zero_of_not_measurable hX) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mul_integral_upcrossingsBefore_le_integral_pos_part_aux ** \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 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc 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 ** refine' le_trans (le_of_eq _)\n  (integral_mul_upcrossingsBefore_le_integral (hf.sub_martingale (martingale_const _ _ _)).pos\n    (fun \u03c9 => LatticeOrderedGroup.pos_nonneg _)\n    (fun \u03c9 => LatticeOrderedGroup.pos_nonneg _) (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 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hab : a < b \u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc =     (b - a - 0) * \u222b (x : \u03a9), \u2191(upcrossingsBefore 0 (b - a) (f - fun x x => a)\u207a N x) \u2202\u03bc ** simp_rw [sub_zero, \u2190 upcrossingsBefore_pos_eq 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 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hab : a < b \u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N x) \u2202\u03bc =     (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore 0 (b - a) (f - fun x x => a)\u207a N x) \u2202\u03bc ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id ** \u03a9 : Type u_1 F : Type u_2 m m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F inst\u271d : TopologicalSpace F hm : m \u2264 m\u03a9 hf : AEStronglyMeasurable f \u03bc \u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, id \u03c9)) \u03bc) ** rw [\u2190 aestronglyMeasurable_comp_snd_map_prod_mk_iff (measurable_id'' hm)] at hf ** \u03a9 : Type u_1 F : Type u_2 m m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F inst\u271d : TopologicalSpace F hm : m \u2264 m\u03a9 hf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, \u03c9)) \u03bc) \u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, id \u03c9)) \u03bc) ** simp_rw [id.def] at hf \u22a2 ** \u03a9 : Type u_1 F : Type u_2 m m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F inst\u271d : TopologicalSpace F hm : m \u2264 m\u03a9 hf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (\u03c9, \u03c9)) \u03bc) \u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (\u03c9, \u03c9)) \u03bc) ** exact hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Function.Periodic.tendsto_atTop_intervalIntegral_of_pos ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d g : \u211d \u2192 \u211d hg : Periodic g T h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082 h\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x hT : 0 < T \u22a2 Tendsto (fun t => \u222b (x : \u211d) in 0 ..t, g x) atTop atTop ** apply tendsto_atTop_mono (hg.sInf_add_zsmul_le_integral_of_pos h_int hT) ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d g : \u211d \u2192 \u211d hg : Periodic g T h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082 h\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x hT : 0 < T \u22a2 Tendsto (fun n => sInf ((fun t => \u222b (x : \u211d) in 0 ..t, g x) '' Icc 0 T) + \u230an / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atTop     atTop ** apply atTop.tendsto_atTop_add_const_left (sInf <| (fun t => \u222b x in (0)..t, g x) '' Icc 0 T) ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d g : \u211d \u2192 \u211d hg : Periodic g T h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082 h\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x hT : 0 < T \u22a2 Tendsto (fun x => \u230ax / T\u230b \u2022 \u222b (x : \u211d) in 0 ..T, g x) atTop atTop ** apply Tendsto.atTop_zsmul_const h\u2080 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d g : \u211d \u2192 \u211d hg : Periodic g T h_int : \u2200 (t\u2081 t\u2082 : \u211d), IntervalIntegrable g volume t\u2081 t\u2082 h\u2080 : 0 < \u222b (x : \u211d) in 0 ..T, g x hT : 0 < T \u22a2 Tendsto (fun x => \u230ax / T\u230b) atTop atTop ** exact tendsto_floor_atTop.comp (tendsto_id.atTop_mul_const (inv_pos.mpr hT)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.map_linearMap_volume_pi_eq_smul_volume_pi ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 \u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume ** classical\n  let M := LinearMap.toMatrix' f\n  have A : LinearMap.det f = det M := by simp only [LinearMap.det_toMatrix']\n  have B : f = toLin' M := by simp only [toLin'_toMatrix']\n  rw [A, B]\n  apply map_matrix_volume_pi_eq_smul_volume_pi\n  rwa [A] at hf ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 \u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume ** let M := LinearMap.toMatrix' f ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 M : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f \u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume ** have A : LinearMap.det f = det M := by simp only [LinearMap.det_toMatrix'] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 M : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f A : \u2191LinearMap.det f = det M \u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume ** have B : f = toLin' M := by simp only [toLin'_toMatrix'] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 M : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f A : \u2191LinearMap.det f = det M B : f = \u2191toLin' M \u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume ** rw [A, B] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 M : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f A : \u2191LinearMap.det f = det M B : f = \u2191toLin' M \u22a2 Measure.map (\u2191(\u2191toLin' M)) volume = ofReal |(det M)\u207b\u00b9| \u2022 volume ** apply map_matrix_volume_pi_eq_smul_volume_pi ** case hM \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 M : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f A : \u2191LinearMap.det f = det M B : f = \u2191toLin' M \u22a2 det M \u2260 0 ** rwa [A] at hf ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 M : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f \u22a2 \u2191LinearMap.det f = det M ** simp only [LinearMap.det_toMatrix'] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 M : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f A : \u2191LinearMap.det f = det M \u22a2 f = \u2191toLin' M ** simp only [toLin'_toMatrix'] ** Qed",
        "informal": ""
    },
    {
        "formal": "RegularExpression.char_rmatch_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a : \u03b1 x : List \u03b1 \u22a2 rmatch (char a) x = true \u2194 x = [a] ** cases' x with _ x ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d : \u03b1 x : List \u03b1 \u22a2 rmatch (char a) (head\u271d :: x) = true \u2194 head\u271d :: x = [a] ** cases' x with head tail ** case nil \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a : \u03b1 \u22a2 rmatch (char a) [] = true \u2194 [] = [a] ** exact of_decide_eq_true rfl ** case cons.nil \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d : \u03b1 \u22a2 rmatch (char a) [head\u271d] = true \u2194 [head\u271d] = [a] ** rw [rmatch, deriv] ** case cons.nil \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d : \u03b1 \u22a2 rmatch (if a = head\u271d then 1 else 0) [] = true \u2194 [head\u271d] = [a] ** split_ifs ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d : \u03b1 h\u271d : a = head\u271d \u22a2 rmatch 1 [] = true \u2194 [head\u271d] = [a] ** tauto ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d : \u03b1 h\u271d : \u00aca = head\u271d \u22a2 rmatch 0 [] = true \u2194 [head\u271d] = [a] ** simp [List.singleton_inj] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d : \u03b1 h\u271d : \u00aca = head\u271d \u22a2 \u00achead\u271d = a ** tauto ** case cons.cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d head : \u03b1 tail : List \u03b1 \u22a2 rmatch (char a) (head\u271d :: head :: tail) = true \u2194 head\u271d :: head :: tail = [a] ** rw [rmatch, rmatch, deriv] ** case cons.cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d head : \u03b1 tail : List \u03b1 \u22a2 rmatch (deriv (if a = head\u271d then 1 else 0) head) tail = true \u2194 head\u271d :: head :: tail = [a] ** split_ifs with h ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d head : \u03b1 tail : List \u03b1 h : a = head\u271d \u22a2 rmatch (deriv 1 head) tail = true \u2194 head\u271d :: head :: tail = [a] ** simp only [deriv_one, zero_rmatch, cons.injEq, and_false] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b a head\u271d head : \u03b1 tail : List \u03b1 h : \u00aca = head\u271d \u22a2 rmatch (deriv 0 head) tail = true \u2194 head\u271d :: head :: tail = [a] ** simp only [deriv_zero, zero_rmatch, cons.injEq, and_false] ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.size_uset ** \u03b1 : Type u_1 a : Array \u03b1 v : \u03b1 i : USize h : USize.toNat i < size a \u22a2 size (uset a i v h) = size a ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.setToSimpleFunc_congr ** \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 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 f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f h : \u2191f =\u1d50[\u03bc] \u2191g \u22a2 setToSimpleFunc T f = setToSimpleFunc T g ** refine' setToSimpleFunc_congr' T h_add hf ((integrable_congr h).mp hf) _ ** \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 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 f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f h : \u2191f =\u1d50[\u03bc] \u2191g \u22a2 \u2200 (x y : E), x \u2260 y \u2192 T (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 ** refine' fun x y hxy => h_zero _ ((measurableSet_fiber f x).inter (measurableSet_fiber g y)) _ ** \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 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 f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f h : \u2191f =\u1d50[\u03bc] \u2191g x y : E hxy : x \u2260 y \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 ** rw [EventuallyEq, ae_iff] at 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 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 f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f h : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0 x y : E hxy : x \u2260 y \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y}) = 0 ** refine' measure_mono_null (fun z => _) 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 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 f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f h : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0 x y : E hxy : x \u2260 y z : \u03b1 \u22a2 z \u2208 \u2191f \u207b\u00b9' {x} \u2229 \u2191g \u207b\u00b9' {y} \u2192 z \u2208 {a | \u00ac\u2191f a = \u2191g a} ** simp_rw [Set.mem_inter_iff, Set.mem_setOf_eq, Set.mem_preimage, Set.mem_singleton_iff] ** \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 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 f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f h : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0 x y : E hxy : x \u2260 y z : \u03b1 \u22a2 \u2191f z = x \u2227 \u2191g z = y \u2192 \u00ac\u2191f z = \u2191g z ** intro 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 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 f g : \u03b1 \u2192\u209b E hf : Integrable \u2191f h\u271d : \u2191\u2191\u03bc {a | \u00ac\u2191f a = \u2191g a} = 0 x y : E hxy : x \u2260 y z : \u03b1 h : \u2191f z = x \u2227 \u2191g z = y \u22a2 \u00ac\u2191f z = \u2191g z ** rwa [h.1, h.2] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1to1.stepAux_read ** \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 f : \u0393 \u2192 Stmt Bool \u039b' \u03c3 v : \u03c3 L R : ListBlank \u0393 \u22a2 stepAux (read dec f) v (trTape' enc0 L R) = stepAux (f (ListBlank.head R)) v (trTape' enc0 L R) ** suffices \u2200 f, stepAux (readAux n f) v (trTape' enc0 L R) =\n    stepAux (f (enc R.head)) v (trTape' enc0 (L.cons R.head) R.tail) by\n  rw [read, this, stepAux_move, encdec, trTape'_move_left enc0]\n  simp only [ListBlank.head_cons, ListBlank.cons_head_tail, ListBlank.tail_cons] ** \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 f : \u0393 \u2192 Stmt Bool \u039b' \u03c3 v : \u03c3 L R : ListBlank \u0393 \u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux n f) v (trTape' enc0 L R) =       stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail 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 f : \u0393 \u2192 Stmt Bool \u039b' \u03c3 v : \u03c3 L : ListBlank \u0393 a : \u0393 R : ListBlank \u0393 \u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux n f) v (trTape' enc0 L (ListBlank.cons a R)) =       stepAux (f (enc (ListBlank.head (ListBlank.cons a R)))) v         (trTape' enc0 (ListBlank.cons (ListBlank.head (ListBlank.cons a R)) L) (ListBlank.tail (ListBlank.cons a R))) ** simp only [ListBlank.head_cons, ListBlank.tail_cons, trTape', ListBlank.cons_bind,\n  ListBlank.append_assoc] ** 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 f : \u0393 \u2192 Stmt Bool \u039b' \u03c3 v : \u03c3 L : ListBlank \u0393 a : \u0393 R : ListBlank \u0393 \u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux n f) v         (Tape.mk'           (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))             (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))           (ListBlank.append (Vector.toList (enc a))             (ListBlank.bind R (fun x => Vector.toList (enc x))               (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =       stepAux (f (enc a)) v         (Tape.mk'           (ListBlank.append (List.reverse (Vector.toList (enc a)))             (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))               (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))           (ListBlank.bind R (fun x => Vector.toList (enc x))             (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) ** suffices \u2200 i f L' R' l\u2081 l\u2082 h,\n    stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f \u27e8l\u2082, h\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R') by\n  intro f\n  exact this n f (L.bind (fun x => (enc x).1.reverse) _)\n    (R.bind (fun x => (enc x).1) _) [] _ (enc a).2 ** 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 f : \u0393 \u2192 Stmt Bool \u039b' \u03c3 v : \u03c3 L : ListBlank \u0393 a : \u0393 R : ListBlank \u0393 \u22a2 \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),     stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') ** clear f L a R ** 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 v : \u03c3 \u22a2 \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),     stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') ** intro i f L' R' l\u2081 l\u2082 _ ** 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 v : \u03c3 i : \u2115 f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3 L' R' : ListBlank Bool l\u2081 l\u2082 : List Bool h\u271d : List.length l\u2082 = i \u22a2 stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =     stepAux (f { val := l\u2082, property := h\u271d }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') ** subst i ** 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 v : \u03c3 L' R' : ListBlank Bool l\u2081 l\u2082 : List Bool f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3 \u22a2 stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =     stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v       (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') ** induction' l\u2082 with a l\u2082 IH generalizing l\u2081 ** case intro.intro.cons \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 v : \u03c3 L' R' : ListBlank Bool l\u2081\u271d l\u2082\u271d : List Bool f\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3 a : Bool l\u2082 : List Bool IH :   \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v         (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') l\u2081 : List Bool f : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3 \u22a2 stepAux (readAux (List.length (a :: l\u2082)) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =     stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v       (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R') ** trans\n  stepAux (readAux l\u2082.length fun v \u21a6 f (a ::\u1d65 v)) v\n    (Tape.mk' ((L'.append l\u2081).cons a) (R'.append l\u2082)) ** \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 v : \u03c3 L' R' : ListBlank Bool l\u2081\u271d l\u2082\u271d : List Bool f\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3 a : Bool l\u2082 : List Bool IH :   \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v         (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') l\u2081 : List Bool f : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3 \u22a2 stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v       (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R')) =     stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v       (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R') ** rw [\u2190 ListBlank.append, 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 v : \u03c3 L' R' : ListBlank Bool l\u2081\u271d l\u2082\u271d : List Bool f\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3 a : Bool l\u2082 : List Bool IH :   \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v         (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') l\u2081 : List Bool f : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3 \u22a2 stepAux (f (a ::\u1d65 { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) })) v       (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 (a :: l\u2081)) L') R') =     stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v       (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R') ** 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 f : \u0393 \u2192 Stmt Bool \u039b' \u03c3 v : \u03c3 L R : ListBlank \u0393 this :   \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux n f) v (trTape' enc0 L R) =       stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R)) \u22a2 stepAux (read dec f) v (trTape' enc0 L R) = stepAux (f (ListBlank.head R)) v (trTape' enc0 L R) ** rw [read, this, stepAux_move, encdec, trTape'_move_left enc0] ** \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 f : \u0393 \u2192 Stmt Bool \u039b' \u03c3 v : \u03c3 L R : ListBlank \u0393 this :   \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux n f) v (trTape' enc0 L R) =       stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R)) \u22a2 stepAux (f (ListBlank.head R)) v       (trTape' enc0 (ListBlank.tail (ListBlank.cons (ListBlank.head R) L))         (ListBlank.cons (ListBlank.head (ListBlank.cons (ListBlank.head R) L)) (ListBlank.tail R))) =     stepAux (f (ListBlank.head R)) v (trTape' enc0 L R) ** simp only [ListBlank.head_cons, ListBlank.cons_head_tail, ListBlank.tail_cons] ** \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 f : \u0393 \u2192 Stmt Bool \u039b' \u03c3 v : \u03c3 L : ListBlank \u0393 a : \u0393 R : ListBlank \u0393 this :   \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),     stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') \u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux n f) v         (Tape.mk'           (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))             (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))           (ListBlank.append (Vector.toList (enc a))             (ListBlank.bind R (fun x => Vector.toList (enc x))               (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =       stepAux (f (enc a)) v         (Tape.mk'           (ListBlank.append (List.reverse (Vector.toList (enc a)))             (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))               (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))           (ListBlank.bind R (fun x => Vector.toList (enc x))             (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) ** intro f ** \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 f\u271d : \u0393 \u2192 Stmt Bool \u039b' \u03c3 v : \u03c3 L : ListBlank \u0393 a : \u0393 R : ListBlank \u0393 this :   \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),     stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3 \u22a2 stepAux (readAux n f) v       (Tape.mk'         (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))           (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))         (ListBlank.append (Vector.toList (enc a))           (ListBlank.bind R (fun x => Vector.toList (enc x))             (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =     stepAux (f (enc a)) v       (Tape.mk'         (ListBlank.append (List.reverse (Vector.toList (enc a)))           (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))             (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))         (ListBlank.bind R (fun x => Vector.toList (enc x))           (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) ** exact this n f (L.bind (fun x => (enc x).1.reverse) _)\n  (R.bind (fun x => (enc x).1) _) [] _ (enc a).2 ** case intro.intro.nil \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 v : \u03c3 L' R' : ListBlank Bool l\u2081\u271d l\u2082 : List Bool f\u271d : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3 l\u2081 : List Bool f : Vector Bool (List.length []) \u2192 Stmt Bool \u039b' \u03c3 \u22a2 stepAux (readAux (List.length []) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append [] R')) =     stepAux (f { val := [], property := (_ : List.length [] = List.length []) }) v       (Tape.mk' (ListBlank.append (List.reverseAux [] l\u2081) L') R') ** 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 v : \u03c3 L' R' : ListBlank Bool l\u2081\u271d l\u2082\u271d : List Bool f\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3 a : Bool l\u2082 : List Bool IH :   \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v         (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') l\u2081 : List Bool f : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3 \u22a2 stepAux (readAux (List.length (a :: l\u2082)) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =     stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v       (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R')) ** dsimp [readAux, 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 v : \u03c3 L' R' : ListBlank Bool l\u2081\u271d l\u2082\u271d : List Bool f\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3 a : Bool l\u2082 : List Bool IH :   \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v         (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') l\u2081 : List Bool f : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3 \u22a2 (bif ListBlank.head (ListBlank.cons a (ListBlank.append l\u2082 R')) then       stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v         (Tape.move Dir.right (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.cons a (ListBlank.append l\u2082 R'))))     else       stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v         (Tape.move Dir.right (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.cons a (ListBlank.append l\u2082 R'))))) =     stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v       (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R')) ** simp only [ListBlank.head_cons, Tape.move_right_mk', ListBlank.tail_cons] ** \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 v : \u03c3 L' R' : ListBlank Bool l\u2081\u271d l\u2082\u271d : List Bool f\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3 a : Bool l\u2082 : List Bool IH :   \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),     stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =       stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v         (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R') l\u2081 : List Bool f : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3 \u22a2 (bif a then       stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v         (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))     else       stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v         (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))) =     stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v       (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R')) ** cases a <;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_le_one_iff_subset_singleton ** \u03b1 : Type u_1 s t : Set \u03b1 inst\u271d : Nonempty \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard s \u2264 1 \u2194 \u2203 x, s \u2286 {x} ** simp_rw [ncard_eq_toFinset_card _ hs, Finset.card_le_one_iff_subset_singleton,\n  Finite.toFinset_subset, Finset.coe_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.PairingHeapImp.Heap.WF.deleteMin ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool a : \u03b1 s' s : Heap \u03b1 h : WF le s eq : Heap.deleteMin le s = some (a, s') \u22a2 WF le s' ** cases h with cases eq | node h => exact Heap.WF.combine h ** case node.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool a : \u03b1 c\u271d : Heap \u03b1 h : NodeWF le a c\u271d \u22a2 WF le (Heap.combine le c\u271d) ** exact Heap.WF.combine h ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_iterate_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f\u271d : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b1 n : \u2115 \u22a2 preimage f^[n] = (preimage f)^[n] ** induction' n with n ih ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f\u271d : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b1 n : \u2115 ih : preimage f^[n] = (preimage f)^[n] \u22a2 preimage f^[Nat.succ n] = (preimage f)^[Nat.succ n] ** rw [iterate_succ, iterate_succ', Set.preimage_comp_eq, ih] ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f\u271d : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b1 \u22a2 preimage f^[Nat.zero] = (preimage f)^[Nat.zero] ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_diff ** \u03b1 : Type u_1 s t : Set \u03b1 h : s \u2286 t ht : autoParam (Set.Finite t) _auto\u271d \u22a2 ncard (t \\ s) = ncard t - ncard s ** rw [\u2190 ncard_diff_add_ncard_of_subset h ht, add_tsub_cancel_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 \u22a2 IsSFiniteKernel (withDensity \u03ba 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) \u22a2 IsSFiniteKernel (withDensity \u03ba f)  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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : \u00acMeasurable (Function.uncurry f) \u22a2 IsSFiniteKernel (withDensity \u03ba f) ** swap ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) \u22a2 IsSFiniteKernel (withDensity \u03ba f) ** let fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (n + 1) - min (f a b) n ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n \u22a2 IsSFiniteKernel (withDensity \u03ba f) ** have h_zero : \u2200 a b n, \u2308(f a b).toReal\u2309\u208a \u2264 n \u2192 fs n a b = 0 := by\n  intro a b n hn\n  suffices min (f a b) (n + 1) = f a b \u2227 min (f a b) n = f a b by\n    simp_rw [this.1, this.2, tsub_self (f a b)]\n  exact \u27e8min_eq_left ((h_le a b n hn).trans (le_add_of_nonneg_right zero_le_one)),\n    min_eq_left (h_le a b n hn)\u27e9 ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n \u22a2 IsSFiniteKernel (withDensity \u03ba f) ** rw [hf_eq_tsum, withDensity_tsum _ fun n : \u2115 => _] ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n \u22a2 IsSFiniteKernel (kernel.sum fun n => withDensity \u03ba (fs n))  \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n \u22a2 \u2200 (n : \u2115), Measurable (Function.uncurry (fs n)) ** swap ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n \u22a2 IsSFiniteKernel (kernel.sum fun n => withDensity \u03ba (fs n)) ** refine' isSFiniteKernel_sum fun n => _ ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n n : \u2115 \u22a2 IsSFiniteKernel (withDensity \u03ba (fs n)) ** suffices IsFiniteKernel (withDensity \u03ba (fs n)) by haveI := this; infer_instance ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n n : \u2115 \u22a2 IsFiniteKernel (withDensity \u03ba (fs n)) ** refine' isFiniteKernel_withDensity_of_bounded _ (ENNReal.coe_ne_top : \u2191n + 1 \u2260 \u221e) fun a b => _ ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n n : \u2115 a : \u03b1 b : \u03b2 \u22a2 fs n a b \u2264 \u2191((fun x x_1 => x + x_1) (\u2191n) 1) ** norm_cast ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n n : \u2115 a : \u03b1 b : \u03b2 \u22a2 fs n a b \u2264 \u2191(n + 1) ** calc\n  fs n a b \u2264 min (f a b) (n + 1) := tsub_le_self\n  _ \u2264 n + 1 := (min_le_right _ _)\n  _ = \u2191(n + 1) := by norm_cast ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : \u00acMeasurable (Function.uncurry f) \u22a2 IsSFiniteKernel (withDensity \u03ba f) ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : \u00acMeasurable (Function.uncurry f) \u22a2 IsSFiniteKernel 0 ** infer_instance ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n \u22a2 \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n ** intro a b n hn ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n a : \u03b1 b : \u03b2 n : \u2115 hn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u22a2 f a b \u2264 \u2191n ** have : (f a b).toReal \u2264 n := Nat.le_of_ceil_le hn ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n a : \u03b1 b : \u03b2 n : \u2115 hn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n this : ENNReal.toReal (f a b) \u2264 \u2191n \u22a2 f a b \u2264 \u2191n ** rw [\u2190 ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n a : \u03b1 b : \u03b2 n : \u2115 hn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n this : f a b \u2264 ENNReal.ofReal \u2191n \u22a2 f a b \u2264 \u2191n ** refine' this.trans (le_of_eq _) ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n a : \u03b1 b : \u03b2 n : \u2115 hn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n this : f a b \u2264 ENNReal.ofReal \u2191n \u22a2 ENNReal.ofReal \u2191n = \u2191n ** rw [ENNReal.ofReal_coe_nat] ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n a : \u03b1 b : \u03b2 n : \u2115 hn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n this : ENNReal.toReal (f a b) \u2264 \u2191n \u22a2 0 \u2264 \u2191n ** norm_cast ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n a : \u03b1 b : \u03b2 n : \u2115 hn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n this : ENNReal.toReal (f a b) \u2264 \u2191n \u22a2 0 \u2264 n ** exact zero_le _ ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n \u22a2 \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 ** intro a b n hn ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n a : \u03b1 b : \u03b2 n : \u2115 hn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u22a2 fs n a b = 0 ** suffices min (f a b) (n + 1) = f a b \u2227 min (f a b) n = f a b by\n  simp_rw [this.1, this.2, tsub_self (f a b)] ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n a : \u03b1 b : \u03b2 n : \u2115 hn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u22a2 min (f a b) (\u2191n + 1) = f a b \u2227 min (f a b) \u2191n = f a b ** exact \u27e8min_eq_left ((h_le a b n hn).trans (le_add_of_nonneg_right zero_le_one)),\n  min_eq_left (h_le a b n hn)\u27e9 ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n a : \u03b1 b : \u03b2 n : \u2115 hn : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n this : min (f a b) (\u2191n + 1) = f a b \u2227 min (f a b) \u2191n = f a b \u22a2 fs n a b = 0 ** simp_rw [this.1, this.2, tsub_self (f a b)] ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 \u22a2 f = \u2211' (n : \u2115), fs n ** have h_sum_a : \u2200 a, Summable fun n => fs n a := by\n  refine' fun a => Pi.summable.mpr fun b => _\n  suffices : \u2200 n, n \u2209 Finset.range \u2308(f a b).toReal\u2309\u208a \u2192 fs n a b = 0\n  exact summable_of_ne_finset_zero this\n  intro n hn_not_mem\n  rw [Finset.mem_range, not_lt] at hn_not_mem\n  exact h_zero a b n hn_not_mem ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a \u22a2 f = \u2211' (n : \u2115), fs n ** ext a b : 2 ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 \u22a2 f a b = tsum (fun n => fs n) a b ** rw [tsum_apply (Pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),\n  ENNReal.tsum_eq_liminf_sum_nat] ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 h_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n \u22a2 f a b = Filter.liminf (fun n => \u2211 i in Finset.range n, fs i a b) Filter.atTop ** simp_rw [h_finset_sum] ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 h_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n \u22a2 f a b = Filter.liminf (fun n => min (f a b) \u2191n) Filter.atTop ** refine' (Filter.Tendsto.liminf_eq _).symm ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 h_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n \u22a2 Filter.Tendsto (fun n => min (f a b) \u2191n) Filter.atTop (nhds (f a b)) ** refine' Filter.Tendsto.congr' _ tendsto_const_nhds ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 h_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n \u22a2 (fun x => f a b) =\u1da0[Filter.atTop] fun n => min (f a b) \u2191n ** rw [Filter.EventuallyEq, Filter.eventually_atTop] ** 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 h_finset_sum : \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n \u22a2 \u2203 a_1, \u2200 (b_1 : \u2115), b_1 \u2265 a_1 \u2192 f a b = min (f a b) \u2191b_1 ** exact \u27e8\u2308(f a b).toReal\u2309\u208a, fun n hn => (min_eq_left (h_le a b n hn)).symm\u27e9 ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 \u22a2 \u2200 (a : \u03b1), Summable fun n => fs n a ** refine' fun a => Pi.summable.mpr fun b => _ ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 a : \u03b1 b : \u03b2 \u22a2 Summable fun i => fs i a b ** suffices : \u2200 n, n \u2209 Finset.range \u2308(f a b).toReal\u2309\u208a \u2192 fs n a b = 0 ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 a : \u03b1 b : \u03b2 this : \u2200 (n : \u2115), \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u2192 fs n a b = 0 \u22a2 Summable fun i => fs i a b  case 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 a : \u03b1 b : \u03b2 \u22a2 \u2200 (n : \u2115), \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u2192 fs n a b = 0 ** exact summable_of_ne_finset_zero this ** case 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 a : \u03b1 b : \u03b2 \u22a2 \u2200 (n : \u2115), \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u2192 fs n a b = 0 ** intro n hn_not_mem ** case 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 a : \u03b1 b : \u03b2 n : \u2115 hn_not_mem : \u00acn \u2208 Finset.range \u2308ENNReal.toReal (f a b)\u2309\u208a \u22a2 fs n a b = 0 ** rw [Finset.mem_range, not_lt] at hn_not_mem ** case 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 a : \u03b1 b : \u03b2 n : \u2115 hn_not_mem : \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u22a2 fs n a b = 0 ** exact h_zero a b n hn_not_mem ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 \u22a2 \u2200 (n : \u2115), \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n ** intro n ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 n : \u2115 \u22a2 \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n ** induction' n with n hn ** case succ \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 n : \u2115 hn : \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n \u22a2 \u2211 i in Finset.range (Nat.succ n), fs i a b = min (f a b) \u2191(Nat.succ n) ** rw [Finset.sum_range_succ, hn] ** case succ \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 n : \u2115 hn : \u2211 i in Finset.range n, fs i a b = min (f a b) \u2191n \u22a2 min (f a b) \u2191n + fs n a b = min (f a b) \u2191(Nat.succ n) ** simp ** case zero \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 h_sum_a : \u2200 (a : \u03b1), Summable fun n => fs n a a : \u03b1 b : \u03b2 \u22a2 \u2211 i in Finset.range Nat.zero, fs i a b = min (f a b) \u2191Nat.zero ** simp ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n \u22a2 \u2200 (n : \u2115), Measurable (Function.uncurry (fs n)) ** exact fun _ => (hf.min measurable_const).sub (hf.min measurable_const) ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n n : \u2115 this : IsFiniteKernel (withDensity \u03ba (fs n)) \u22a2 IsSFiniteKernel (withDensity \u03ba (fs n)) ** haveI := 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 : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n n : \u2115 this\u271d this : IsFiniteKernel (withDensity \u03ba (fs n)) \u22a2 IsSFiniteKernel (withDensity \u03ba (fs n)) ** infer_instance ** \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 : IsFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 hf : Measurable (Function.uncurry f) fs : \u2115 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e := fun n a b => min (f a b) (\u2191n + 1) - min (f a b) \u2191n h_le : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 f a b \u2264 \u2191n h_zero : \u2200 (a : \u03b1) (b : \u03b2) (n : \u2115), \u2308ENNReal.toReal (f a b)\u2309\u208a \u2264 n \u2192 fs n a b = 0 hf_eq_tsum : f = \u2211' (n : \u2115), fs n n : \u2115 a : \u03b1 b : \u03b2 \u22a2 \u2191n + 1 = \u2191(n + 1) ** norm_cast ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.Fix.rec_eq ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x : F (\u03b1 ::: Fix F \u03b1) \u22a2 rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x) ** have : recF g \u2218 fixToW = Fix.rec g := by\n  apply funext\n  apply Quotient.ind\n  intro x\n  apply recF_eq_of_wEquiv\n  apply wrepr_equiv ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x : F (\u03b1 ::: Fix F \u03b1) this : recF g \u2218 fixToW = rec g \u22a2 rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x) ** conv =>\n  lhs\n  rw [Fix.rec, Fix.mk]\n  dsimp ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x : F (\u03b1 ::: Fix F \u03b1) this : recF g \u2218 fixToW = rec g \u22a2 recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> repr x)) = g ((TypeVec.id ::: rec g) <$$> x) ** cases' h : repr x with a f ** case mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x : F (\u03b1 ::: Fix F \u03b1) this : recF g \u2218 fixToW = rec g a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 ::: Fix F \u03b1 h : repr x = { fst := a, snd := f } \u22a2 recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =     g ((TypeVec.id ::: rec g) <$$> x) ** rw [MvPFunctor.map_eq, recF_eq', \u2190 MvPFunctor.map_eq, MvPFunctor.wDest'_wMk'] ** case mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x : F (\u03b1 ::: Fix F \u03b1) this : recF g \u2218 fixToW = rec g a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 ::: Fix F \u03b1 h : repr x = { fst := a, snd := f } \u22a2 g (abs ((TypeVec.id ::: recF g) <$$> (TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =     g ((TypeVec.id ::: rec g) <$$> x) ** rw [\u2190 MvPFunctor.comp_map, abs_map, \u2190 h, abs_repr, \u2190 appendFun_comp, id_comp, this] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x : F (\u03b1 ::: Fix F \u03b1) \u22a2 recF g \u2218 fixToW = rec g ** apply funext ** case 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 g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x : F (\u03b1 ::: Fix F \u03b1) \u22a2 \u2200 (x : Fix F \u03b1), (recF g \u2218 fixToW) x = rec g x ** apply Quotient.ind ** case h.a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x : F (\u03b1 ::: Fix F \u03b1) \u22a2 \u2200 (a : MvPFunctor.W (P F) \u03b1), (recF g \u2218 fixToW) (Quotient.mk (wSetoid \u03b1) a) = rec g (Quotient.mk (wSetoid \u03b1) a) ** intro x ** case h.a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x\u271d : F (\u03b1 ::: Fix F \u03b1) x : MvPFunctor.W (P F) \u03b1 \u22a2 (recF g \u2218 fixToW) (Quotient.mk (wSetoid \u03b1) x) = rec g (Quotient.mk (wSetoid \u03b1) x) ** apply recF_eq_of_wEquiv ** case h.a.a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x\u271d : F (\u03b1 ::: Fix F \u03b1) x : MvPFunctor.W (P F) \u03b1 \u22a2 WEquiv (fixToW (Quotient.mk (wSetoid \u03b1) x)) x ** apply wrepr_equiv ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.ext_of_Ioc' ** \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 inst\u271d : NoMinOrder \u03b1 \u03bc \u03bd : Measure \u03b1 h\u03bc : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) \u2260 \u22a4 h : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b) \u22a2 \u03bc = \u03bd ** refine' @ext_of_Ico' \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a _ \u03bc \u03bd _ _ <;> intro a b hab <;> erw [dual_Ico (\u03b1 := \u03b1)] ** case refine'_1 \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 inst\u271d : NoMinOrder \u03b1 \u03bc \u03bd : Measure \u03b1 h\u03bc : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) \u2260 \u22a4 h : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b) a b : \u03b1\u1d52\u1d48 hab : a < b \u22a2 \u2191\u2191\u03bc (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) \u2260 \u22a4  case refine'_2 \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 inst\u271d : NoMinOrder \u03b1 \u03bc \u03bd : Measure \u03b1 h\u03bc : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) \u2260 \u22a4 h : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ioc a b) = \u2191\u2191\u03bd (Ioc a b) a b : \u03b1\u1d52\u1d48 hab : a < b \u22a2 \u2191\u2191\u03bc (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) = \u2191\u2191\u03bd (\u2191OrderDual.ofDual \u207b\u00b9' Ioc b a) ** exacts [h\u03bc hab, h hab] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.induction_dense ** \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g ** rcases eq_or_ne p 0 with (rfl | hp_pos) ** case 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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 \u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g ** suffices H :\n  \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e) (h\u03b4 : \u03b4 \u2260 0), Mem\u2112p f' p \u03bc \u2192 \u2203 g, snorm (\u21d1f' - g) p \u03bc \u2264 \u03b4 \u2227 P g ** 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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 \u22a2 \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g ** apply SimpleFunc.induction ** case 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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E \u03bc : Measure \u03b1 P : (\u03b1 \u2192 E) \u2192 Prop h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_ne_top : 0 \u2260 \u22a4 h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g hf : Mem\u2112p f 0 \u22a2 \u2203 g, snorm (f - g) 0 \u03bc \u2264 \u03b5 \u2227 P g ** rcases h0P (0 : E) MeasurableSet.empty (by simp only [measure_empty, WithTop.zero_lt_top])\n    h\u03b5 with \u27e8g, _, Pg\u27e9 ** case inl.intro.intro \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\u271d : \u03b1 \u2192 E \u03bc : Measure \u03b1 P : (\u03b1 \u2192 E) \u2192 Prop h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_ne_top : 0 \u2260 \u22a4 h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g hf : Mem\u2112p f 0 g : \u03b1 \u2192 E left\u271d : snorm (g - Set.indicator \u2205 fun x => 0) 0 \u03bc \u2264 \u03b5 Pg : P g \u22a2 \u2203 g, snorm (f - g) 0 \u03bc \u2264 \u03b5 \u2227 P g ** exact \u27e8g, by simp only [snorm_exponent_zero, zero_le'], Pg\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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E \u03bc : Measure \u03b1 P : (\u03b1 \u2192 E) \u2192 Prop h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_ne_top : 0 \u2260 \u22a4 h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g hf : Mem\u2112p f 0 \u22a2 \u2191\u2191\u03bc \u2205 < \u22a4 ** simp only [measure_empty, WithTop.zero_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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E \u03bc : Measure \u03b1 P : (\u03b1 \u2192 E) \u2192 Prop h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_ne_top : 0 \u2260 \u22a4 h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g hf : Mem\u2112p f 0 g : \u03b1 \u2192 E left\u271d : snorm (g - Set.indicator \u2205 fun x => 0) 0 \u03bc \u2264 \u03b5 Pg : P g \u22a2 snorm (f - g) 0 \u03bc \u2264 \u03b5 ** simp only [snorm_exponent_zero, zero_le'] ** case 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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 H : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g ** obtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 := exists_Lp_half E \u03bc p h\u03b5 ** case inr.intro.intro \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 H : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5 \u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g ** rcases hf.exists_simpleFunc_snorm_sub_lt hp_ne_top \u03b7pos.ne' with \u27e8f', hf', f'_mem\u27e9 ** case inr.intro.intro.intro.intro \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 H : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5 f' : \u03b1 \u2192\u209b E hf' : snorm (f - \u2191f') p \u03bc < \u03b7 f'_mem : Mem\u2112p (\u2191f') p \u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g ** rcases H f' \u03b7 \u03b7pos.ne' f'_mem with \u27e8g, hg, Pg\u27e9 ** case inr.intro.intro.intro.intro.intro.intro \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 H : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5 f' : \u03b1 \u2192\u209b E hf' : snorm (f - \u2191f') p \u03bc < \u03b7 f'_mem : Mem\u2112p (\u2191f') p g : \u03b1 \u2192 E hg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7 Pg : P g \u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g ** refine' \u27e8g, _, Pg\u27e9 ** case inr.intro.intro.intro.intro.intro.intro \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 H : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5 f' : \u03b1 \u2192\u209b E hf' : snorm (f - \u2191f') p \u03bc < \u03b7 f'_mem : Mem\u2112p (\u2191f') p g : \u03b1 \u2192 E hg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7 Pg : P g \u22a2 snorm (f - g) p \u03bc \u2264 \u03b5 ** convert (h\u03b7 _ _ (hf.aestronglyMeasurable.sub f'.aestronglyMeasurable)\n      (f'.aestronglyMeasurable.sub (h2P g Pg)) hf'.le hg).le using 2 ** case h.e'_3.h.e'_5 \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 H : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5 f' : \u03b1 \u2192\u209b E hf' : snorm (f - \u2191f') p \u03bc < \u03b7 f'_mem : Mem\u2112p (\u2191f') p g : \u03b1 \u2192 E hg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7 Pg : P g \u22a2 f - g = f - \u2191f' + (\u2191f' - g) ** simp only [sub_add_sub_cancel] ** case H.h_ind \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 \u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (\u03b4 : \u211d\u22650\u221e),     \u03b4 \u2260 0 \u2192       Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p \u2192         \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b4 \u2227 P g ** intro c s hs \u03b5 \u03b5pos Hs ** case H.h_ind \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 c : E s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p \u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g ** rcases eq_or_ne c 0 with (rfl | hc) ** case H.h_ind.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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p \u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g ** rcases h0P (0 : E) MeasurableSet.empty (by simp only [measure_empty, WithTop.zero_lt_top])\n    \u03b5pos with \u27e8g, hg, Pg\u27e9 ** case H.h_ind.inl.intro.intro \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p g : \u03b1 \u2192 E hg : snorm (g - Set.indicator \u2205 fun x => 0) p \u03bc \u2264 \u03b5 Pg : P g \u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g ** rw [\u2190 snorm_neg, neg_sub] at hg ** case H.h_ind.inl.intro.intro \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p g : \u03b1 \u2192 E hg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5 Pg : P g \u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g ** refine' \u27e8g, _, Pg\u27e9 ** case H.h_ind.inl.intro.intro \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p g : \u03b1 \u2192 E hg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5 Pg : P g \u22a2 snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 ** convert hg ** case h.e'_3.h.e'_5.h.e'_5 \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p g : \u03b1 \u2192 E hg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5 Pg : P g \u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) = Set.indicator \u2205 fun x => 0 ** ext x ** case h.e'_3.h.e'_5.h.e'_5.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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p g : \u03b1 \u2192 E hg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5 Pg : P g x : \u03b1 \u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) x = Set.indicator \u2205 (fun x => 0) x ** simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_zero,\n  piecewise_eq_indicator, indicator_zero', Pi.zero_apply, indicator_zero] ** \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p \u22a2 \u2191\u2191\u03bc \u2205 < \u22a4 ** simp only [measure_empty, WithTop.zero_lt_top] ** case H.h_ind.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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 c : E s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p hc : c \u2260 0 \u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g ** have : \u03bc s < \u221e := SimpleFunc.measure_lt_top_of_mem\u2112p_indicator hp_pos hp_ne_top hc hs Hs ** case H.h_ind.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\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 c : E s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p hc : c \u2260 0 this : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g ** rcases h0P c hs this \u03b5pos with \u27e8g, hg, Pg\u27e9 ** case H.h_ind.inr.intro.intro \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 c : E s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p hc : c \u2260 0 this : \u2191\u2191\u03bc s < \u22a4 g : \u03b1 \u2192 E hg : snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 Pg : P g \u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g ** rw [\u2190 snorm_neg, neg_sub] at hg ** case H.h_ind.inr.intro.intro \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5\u271d : \u211d\u22650\u221e h\u03b5 : \u03b5\u271d \u2260 0 hp_pos : p \u2260 0 c : E s : Set \u03b1 hs : MeasurableSet s \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 Hs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p hc : c \u2260 0 this : \u2191\u2191\u03bc s < \u22a4 g : \u03b1 \u2192 E hg : snorm ((Set.indicator s fun x => c) - g) p \u03bc \u2264 \u03b5 Pg : P g \u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g ** exact \u27e8g, hg, Pg\u27e9 ** case H.h_add \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\u271d : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 E hf : Mem\u2112p f p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 \u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984,     Disjoint (support \u2191f) (support \u2191g) \u2192       (\u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g) \u2192         (\u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191g) p \u2192 \u2203 g_1, snorm (\u2191g - g_1) p \u03bc \u2264 \u03b4 \u2227 P g_1) \u2192           \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191(f + g)) p \u2192 \u2203 g_1, snorm (\u2191(f + g) - g_1) p \u03bc \u2264 \u03b4 \u2227 P g_1 ** intro f f' hff' hf hf' \u03b4 \u03b4pos int_ff' ** case H.h_add \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\u271d\u00b9 : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f\u271d : \u03b1 \u2192 E hf\u271d : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 f f' : \u03b1 \u2192\u209b E hff' : Disjoint (support \u2191f) (support \u2191f') hf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g hf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b4 : \u211d\u22650\u221e \u03b4pos : \u03b4 \u2260 0 int_ff' : Mem\u2112p (\u2191(f + f')) p \u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g ** obtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 := exists_Lp_half E \u03bc p \u03b4pos ** case H.h_add.intro.intro \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\u271d\u00b9 : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f\u271d : \u03b1 \u2192 E hf\u271d : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 f f' : \u03b1 \u2192\u209b E hff' : Disjoint (support \u2191f) (support \u2191f') hf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g hf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b4 : \u211d\u22650\u221e \u03b4pos : \u03b4 \u2260 0 int_ff' : Mem\u2112p (\u2191(f + f')) p \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4 \u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g ** rw [SimpleFunc.coe_add,\n  mem\u2112p_add_of_disjoint hff' f.stronglyMeasurable f'.stronglyMeasurable] at int_ff' ** case H.h_add.intro.intro \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\u271d\u00b9 : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f\u271d : \u03b1 \u2192 E hf\u271d : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 f f' : \u03b1 \u2192\u209b E hff' : Disjoint (support \u2191f) (support \u2191f') hf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g hf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b4 : \u211d\u22650\u221e \u03b4pos : \u03b4 \u2260 0 int_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4 \u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g ** rcases hf \u03b7 \u03b7pos.ne' int_ff'.1 with \u27e8g, hg, Pg\u27e9 ** case H.h_add.intro.intro.intro.intro \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\u271d\u00b9 : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f\u271d : \u03b1 \u2192 E hf\u271d : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 f f' : \u03b1 \u2192\u209b E hff' : Disjoint (support \u2191f) (support \u2191f') hf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g hf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b4 : \u211d\u22650\u221e \u03b4pos : \u03b4 \u2260 0 int_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4 g : \u03b1 \u2192 E hg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7 Pg : P g \u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g ** rcases hf' \u03b7 \u03b7pos.ne' int_ff'.2 with \u27e8g', hg', Pg'\u27e9 ** case H.h_add.intro.intro.intro.intro.intro.intro \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\u271d\u00b9 : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f\u271d : \u03b1 \u2192 E hf\u271d : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 f f' : \u03b1 \u2192\u209b E hff' : Disjoint (support \u2191f) (support \u2191f') hf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g hf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b4 : \u211d\u22650\u221e \u03b4pos : \u03b4 \u2260 0 int_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4 g : \u03b1 \u2192 E hg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7 Pg : P g g' : \u03b1 \u2192 E hg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7 Pg' : P g' \u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g ** refine' \u27e8g + g', _, h1P g g' Pg Pg'\u27e9 ** case H.h_add.intro.intro.intro.intro.intro.intro \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\u271d\u00b9 : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f\u271d : \u03b1 \u2192 E hf\u271d : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 f f' : \u03b1 \u2192\u209b E hff' : Disjoint (support \u2191f) (support \u2191f') hf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g hf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b4 : \u211d\u22650\u221e \u03b4pos : \u03b4 \u2260 0 int_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4 g : \u03b1 \u2192 E hg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7 Pg : P g g' : \u03b1 \u2192 E hg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7 Pg' : P g' \u22a2 snorm (\u2191(f + f') - (g + g')) p \u03bc \u2264 \u03b4 ** convert (h\u03b7 _ _ (f.aestronglyMeasurable.sub (h2P g Pg))\n      (f'.aestronglyMeasurable.sub (h2P g' Pg')) hg hg').le using 2 ** case h.e'_3.h.e'_5 \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\u271d\u00b9 : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f\u271d : \u03b1 \u2192 E hf\u271d : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 f f' : \u03b1 \u2192\u209b E hff' : Disjoint (support \u2191f) (support \u2191f') hf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g hf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b4 : \u211d\u22650\u221e \u03b4pos : \u03b4 \u2260 0 int_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4 g : \u03b1 \u2192 E hg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7 Pg : P g g' : \u03b1 \u2192 E hg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7 Pg' : P g' \u22a2 \u2191(f + f') - (g + g') = \u2191f - g + (\u2191f' - g') ** rw [SimpleFunc.coe_add] ** case h.e'_3.h.e'_5 \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\u271d\u00b9 : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 hp_ne_top : p \u2260 \u22a4 P : (\u03b1 \u2192 E) \u2192 Prop h0P :   \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g h1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g) h2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc f\u271d : \u03b1 \u2192 E hf\u271d : Mem\u2112p f\u271d p \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 hp_pos : p \u2260 0 f f' : \u03b1 \u2192\u209b E hff' : Disjoint (support \u2191f) (support \u2191f') hf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g hf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g \u03b4 : \u211d\u22650\u221e \u03b4pos : \u03b4 \u2260 0 int_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p \u03b7 : \u211d\u22650\u221e \u03b7pos : 0 < \u03b7 h\u03b7 :   \u2200 (f g : \u03b1 \u2192 E),     AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4 g : \u03b1 \u2192 E hg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7 Pg : P g g' : \u03b1 \u2192 E hg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7 Pg' : P g' \u22a2 \u2191f + \u2191f' - (g + g') = \u2191f - g + (\u2191f' - g') ** abel ** Qed",
        "informal": ""
    },
    {
        "formal": "Language.mul_self_kstar_comm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x : List \u03b1 l : Language \u03b1 \u22a2 l\u2217 * l = l * l\u2217 ** simp only [kstar_eq_iSup_pow, mul_iSup, iSup_mul, \u2190 pow_succ, \u2190 pow_succ'] ** Qed",
        "informal": ""
    },
    {
        "formal": "WType.leftInverse_list ** \u03b3 : Type u f : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3) \u22a2 ofList \u03b3 (toList \u03b3 (mk List\u03b1.nil f)) = mk List\u03b1.nil f ** simp only [toList, ofList, mk.injEq, heq_eq_eq, true_and] ** \u03b3 : Type u f : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3) \u22a2 PEmpty.elim = f ** ext x ** case h \u03b3 : Type u f : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3) x : PEmpty.{u + 1} \u22a2 PEmpty.elim x = f x ** cases x ** \u03b3 : Type u x : \u03b3 f : List\u03b2 \u03b3 (List\u03b1.cons x) \u2192 WType (List\u03b2 \u03b3) \u22a2 ofList \u03b3 (toList \u03b3 (mk (List\u03b1.cons x) f)) = mk (List\u03b1.cons x) f ** simp only [ofList, leftInverse_list (f PUnit.unit), mk.injEq, heq_eq_eq, true_and] ** \u03b3 : Type u x : \u03b3 f : List\u03b2 \u03b3 (List\u03b1.cons x) \u2192 WType (List\u03b2 \u03b3) \u22a2 (fun x => f PUnit.unit) = f ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.of_graph ** \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 \u2115 h\u2081 : PrimrecBounded f h\u2082 : PrimrecRel fun a b => f a = b \u22a2 Primrec f ** rcases h\u2081 with \u27e8g, pg, hg : \u2200 x, f x \u2264 g x\u27e9 ** case intro.intro \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 \u2115 h\u2082 : PrimrecRel fun a b => f a = b g : \u03b1 \u2192 \u2115 pg : Primrec g hg : \u2200 (x : \u03b1), f x \u2264 g x \u22a2 Primrec f ** refine (nat_findGreatest pg h\u2082).of_eq fun n => ?_ ** case intro.intro \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 \u2115 h\u2082 : PrimrecRel fun a b => f a = b g : \u03b1 \u2192 \u2115 pg : Primrec g hg : \u2200 (x : \u03b1), f x \u2264 g x n : \u03b1 \u22a2 Nat.findGreatest (fun b => f n = b) (g n) = f n ** exact (Nat.findGreatest_spec (P := fun b => f n = b) (hg n) rfl).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.isometryEquiv_map_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 \u2243\u1d62 Y \u22a2 \u2191(map \u2191f) (mkMetric m) = mkMetric m ** rw [\u2190 isometryEquiv_comap_mkMetric _ f, map_comap_of_surjective f.surjective] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.support_finSuccEquiv_nonempty ** 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 Finset.Nonempty (Polynomial.support (\u2191(finSuccEquiv R n) f)) ** simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty] ** 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 \u00ac\u2191(finSuccEquiv R n) f = 0 ** refine fun c => h ?_ ** 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 c : \u2191(finSuccEquiv R n) f = 0 \u22a2 f = 0 ** let ii := (finSuccEquiv R n).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 : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R h : f \u2260 0 c : \u2191(finSuccEquiv R n) f = 0 ii : (MvPolynomial (Fin n) R)[X] \u2243\u2090[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n) \u22a2 f = 0 ** calc\n  f = ii (finSuccEquiv R n f) := by\n    simpa only [\u2190 AlgEquiv.invFun_eq_symm] using ((finSuccEquiv R n).left_inv f).symm\n  _ = ii 0 := by rw [c]\n  _ = 0 := by simp ** 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 c : \u2191(finSuccEquiv R n) f = 0 ii : (MvPolynomial (Fin n) R)[X] \u2243\u2090[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n) \u22a2 f = \u2191ii (\u2191(finSuccEquiv R n) f) ** simpa only [\u2190 AlgEquiv.invFun_eq_symm] using ((finSuccEquiv R n).left_inv f).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 : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R h : f \u2260 0 c : \u2191(finSuccEquiv R n) f = 0 ii : (MvPolynomial (Fin n) R)[X] \u2243\u2090[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n) \u22a2 \u2191ii (\u2191(finSuccEquiv R n) f) = \u2191ii 0 ** rw [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 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R h : f \u2260 0 c : \u2191(finSuccEquiv R n) f = 0 ii : (MvPolynomial (Fin n) R)[X] \u2243\u2090[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n) \u22a2 \u2191ii 0 = 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.pred_add_one ** n : Nat i : Fin (n + 2) h : \u2191i < n + 1 \u22a2 pred (i + 1) (_ : i + 1 \u2260 0) = castLT i h ** rw [ext_iff, coe_pred, coe_castLT, val_add, val_one, Nat.mod_eq_of_lt, Nat.add_sub_cancel] ** n : Nat i : Fin (n + 2) h : \u2191i < n + 1 \u22a2 \u2191i + 1 < n + 1 + 1 ** exact Nat.add_lt_add_right h 1 ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.indepFun_iff_indepSet_preimage ** \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 : \u03a9 \u2192 \u03b2 g : \u03a9 \u2192 \u03b2' m\u03b2 : MeasurableSpace \u03b2 m\u03b2' : MeasurableSpace \u03b2' inst\u271d : IsProbabilityMeasure \u03bc hf : Measurable f hg : Measurable g \u22a2 IndepFun f g \u2194 \u2200 (s : Set \u03b2) (t : Set \u03b2'), MeasurableSet s \u2192 MeasurableSet t \u2192 IndepSet (f \u207b\u00b9' s) (g \u207b\u00b9' t) ** simp only [IndepFun, IndepSet, kernel.indepFun_iff_indepSet_preimage hf hg, ae_dirac_eq,\n  Filter.eventually_pure, kernel.const_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.fiber_ncard_ne_zero_iff_mem_image ** \u03b1 : Type u_2 s t : Set \u03b1 \u03b2 : Type u_1 f : \u03b1 \u2192 \u03b2 y : \u03b2 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard {x | x \u2208 s \u2227 f x = y} \u2260 0 \u2194 y \u2208 f '' s ** refine' \u27e8nonempty_of_ncard_ne_zero, _\u27e9 ** \u03b1 : Type u_2 s t : Set \u03b1 \u03b2 : Type u_1 f : \u03b1 \u2192 \u03b2 y : \u03b2 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 y \u2208 f '' s \u2192 ncard {x | x \u2208 s \u2227 f x = y} \u2260 0 ** rintro \u27e8z, hz, rfl\u27e9 ** case intro.intro \u03b1 : Type u_2 s t : Set \u03b1 \u03b2 : Type u_1 f : \u03b1 \u2192 \u03b2 hs : autoParam (Set.Finite s) _auto\u271d z : \u03b1 hz : z \u2208 s \u22a2 ncard {x | x \u2208 s \u2227 f x = f z} \u2260 0 ** exact @ncard_ne_zero_of_mem _ ({ x \u2208 s | f x = f z }) z (mem_sep hz rfl)\n  (hs.subset (sep_subset _ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hausdorffMeasure_homothety_preimage ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u00b2 : EMetricSpace X inst\u271d\u00b9\u00b9 : EMetricSpace Y inst\u271d\u00b9\u2070 : MeasurableSpace X inst\u271d\u2079 : BorelSpace X inst\u271d\u2078 : MeasurableSpace Y inst\u271d\u2077 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 P : Type u_6 inst\u271d\u2076 : NormedField \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : MeasurableSpace P inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor E P inst\u271d : BorelSpace P d : \u211d hd : 0 \u2264 d x : P c : \ud835\udd5c hc : c \u2260 0 s : Set P \u22a2 \u2191\u2191\u03bcH[d] (\u2191(AffineMap.homothety x c) \u207b\u00b9' s) = NNReal.rpow \u2016c\u2016\u208a\u207b\u00b9 d \u2022 \u2191\u2191\u03bcH[d] s ** change \u03bcH[d] (AffineEquiv.homothetyUnitsMulHom x (Units.mk0 c hc) \u207b\u00b9' s) = _ ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u00b2 : EMetricSpace X inst\u271d\u00b9\u00b9 : EMetricSpace Y inst\u271d\u00b9\u2070 : MeasurableSpace X inst\u271d\u2079 : BorelSpace X inst\u271d\u2078 : MeasurableSpace Y inst\u271d\u2077 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 P : Type u_6 inst\u271d\u2076 : NormedField \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : MeasurableSpace P inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor E P inst\u271d : BorelSpace P d : \u211d hd : 0 \u2264 d x : P c : \ud835\udd5c hc : c \u2260 0 s : Set P \u22a2 \u2191\u2191\u03bcH[d] (\u2191(\u2191(AffineEquiv.homothetyUnitsMulHom x) (Units.mk0 c hc)) \u207b\u00b9' s) = NNReal.rpow \u2016c\u2016\u208a\u207b\u00b9 d \u2022 \u2191\u2191\u03bcH[d] s ** rw [\u2190 AffineEquiv.image_symm, AffineEquiv.coe_homothetyUnitsMulHom_apply_symm,\n  hausdorffMeasure_homothety_image hd x (_ : \ud835\udd5c\u02e3).isUnit.ne_zero, Units.val_inv_eq_inv_val,\n  Units.val_mk0, nnnorm_inv] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Content.outerMeasure_preimage ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G f : G \u2243\u209c G h : \u2200 \u2983K : Compacts G\u2984, (fun s => \u2191(toFun \u03bc s)) (Compacts.map \u2191f (_ : Continuous \u2191f) K) = (fun s => \u2191(toFun \u03bc s)) K A : Set G \u22a2 \u2191(Content.outerMeasure \u03bc) (\u2191f \u207b\u00b9' A) = \u2191(Content.outerMeasure \u03bc) A ** refine' inducedOuterMeasure_preimage _ \u03bc.innerContent_iUnion_nat \u03bc.innerContent_mono _\n  (fun _ => f.isOpen_preimage) _ ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G f : G \u2243\u209c G h : \u2200 \u2983K : Compacts G\u2984, (fun s => \u2191(toFun \u03bc s)) (Compacts.map \u2191f (_ : Continuous \u2191f) K) = (fun s => \u2191(toFun \u03bc s)) K A : Set G \u22a2 \u2200 (s : Set G) (hs : IsOpen s),     innerContent \u03bc { carrier := \u2191f.toEquiv \u207b\u00b9' s, is_open' := (_ : IsOpen (\u2191f.toEquiv \u207b\u00b9' s)) } =       innerContent \u03bc { carrier := s, is_open' := hs } ** intro s hs ** G : Type w inst\u271d\u00b9 : TopologicalSpace G \u03bc : Content G inst\u271d : T2Space G f : G \u2243\u209c G h : \u2200 \u2983K : Compacts G\u2984, (fun s => \u2191(toFun \u03bc s)) (Compacts.map \u2191f (_ : Continuous \u2191f) K) = (fun s => \u2191(toFun \u03bc s)) K A s : Set G hs : IsOpen s \u22a2 innerContent \u03bc { carrier := \u2191f.toEquiv \u207b\u00b9' s, is_open' := (_ : IsOpen (\u2191f.toEquiv \u207b\u00b9' s)) } =     innerContent \u03bc { carrier := s, is_open' := hs } ** convert \u03bc.innerContent_comap f h \u27e8s, hs\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.stoppedProcess ** \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 h : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 \u22a2 Submartingale (stoppedProcess f \u03c4) \ud835\udca2 \u03bc ** rw [submartingale_iff_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 h : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 \u22a2 \u2200 (\u03c4_1 \u03c0 : \u03a9 \u2192 \u2115),     IsStoppingTime \ud835\udca2 \u03c4_1 \u2192       IsStoppingTime \ud835\udca2 \u03c0 \u2192         \u03c4_1 \u2264 \u03c0 \u2192           (\u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N) \u2192             \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c4_1 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c0 x \u2202\u03bc ** intro \u03c3 \u03c0 h\u03c3 h\u03c0 h\u03c3_le_\u03c0 h\u03c0_bdd ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0\u271d : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc h : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 \u03c3 \u03c0 : \u03a9 \u2192 \u2115 h\u03c3 : IsStoppingTime \ud835\udca2 \u03c3 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 h\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0 h\u03c0_bdd : \u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N \u22a2 \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c3 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue (stoppedProcess f \u03c4) \u03c0 x \u2202\u03bc ** simp_rw [stoppedValue_stoppedProcess] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0\u271d : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc h : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 \u03c3 \u03c0 : \u03a9 \u2192 \u2115 h\u03c3 : IsStoppingTime \ud835\udca2 \u03c3 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 h\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0 h\u03c0_bdd : \u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N \u22a2 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c0 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc ** obtain \u27e8n, h\u03c0_le_n\u27e9 := h\u03c0_bdd ** case intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0\u271d : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc h : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 \u03c3 \u03c0 : \u03a9 \u2192 \u2115 h\u03c3 : IsStoppingTime \ud835\udca2 \u03c3 h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0 h\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0 n : \u2115 h\u03c0_le_n : \u2200 (x : \u03a9), \u03c0 x \u2264 n \u22a2 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c3 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun \u03c9 => min (\u03c0 \u03c9) (\u03c4 \u03c9)) x \u2202\u03bc ** exact h.expected_stoppedValue_mono (h\u03c3.min h\u03c4) (h\u03c0.min h\u03c4)\n  (fun \u03c9 => min_le_min (h\u03c3_le_\u03c0 \u03c9) le_rfl) fun \u03c9 => (min_le_left _ _).trans (h\u03c0_le_n \u03c9) ** case hadp \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 h : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 \u22a2 Adapted \ud835\udca2 (stoppedProcess f \u03c4) ** exact Adapted.stoppedProcess_of_discrete h.adapted h\u03c4 ** case hint \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 h : Submartingale f \ud835\udca2 \u03bc h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4 \u22a2 \u2200 (i : \u2115), Integrable (stoppedProcess f \u03c4 i) ** exact fun i =>\n  h.integrable_stoppedValue ((isStoppingTime_const _ i).min h\u03c4) fun \u03c9 => min_le_left _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "WithTop.image_coe_Ioi ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ioi a = Ioo \u2191a \u22a4 ** rw [\u2190 preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Primrec'.prec' ** n : \u2115 f g : Vector \u2115 n \u2192 \u2115 h : Vector \u2115 (n + 2) \u2192 \u2115 hf : Primrec' f hg : Primrec' g hh : Primrec' h \u22a2 Primrec' fun v => Nat.rec (g v) (fun y IH => h (y ::\u1d65 IH ::\u1d65 v)) (f v) ** simpa using comp' (prec hg hh) (hf.cons idv) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.compProd_eq_sum_compProd_right ** \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 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n ** by_cases h\u03ba : IsSFiniteKernel \u03ba ** 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 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 h\u03ba : IsSFiniteKernel \u03ba \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n  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 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 h\u03ba : \u00acIsSFiniteKernel \u03ba \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n ** 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 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 h\u03ba : IsSFiniteKernel \u03ba \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n ** 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 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 h\u03ba : IsSFiniteKernel \u03ba \u22a2 (kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n ** simp_rw [compProd_eq_sum_compProd_left \u03ba _] ** 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 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 h\u03ba : IsSFiniteKernel \u03ba \u22a2 (kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) =     kernel.sum fun n => kernel.sum fun n_1 => seq \u03ba n_1 \u2297\u2096 seq \u03b7 n ** rw [kernel.sum_comm] ** 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 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 h\u03ba : \u00acIsSFiniteKernel \u03ba \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => \u03ba \u2297\u2096 seq \u03b7 n ** simp_rw [compProd_of_not_isSFiniteKernel_left _ _ h\u03ba] ** 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 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 h\u03ba : \u00acIsSFiniteKernel \u03ba \u22a2 0 = kernel.sum fun n => 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_integral_eq_of_subset_of_ae_diff_eq_zero ** \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 ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** by_cases h : IntegrableOn f t \u03bc ** case pos \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 ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : IntegrableOn f t \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc  case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : \u00acIntegrableOn f t \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** swap ** case pos \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 ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : IntegrableOn f t \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** let f' := h.1.mk f ** case pos \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 ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : IntegrableOn f t f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** calc\n  \u222b x in t, f x \u2202\u03bc = \u222b x in t, f' x \u2202\u03bc := integral_congr_ae h.1.ae_eq_mk\n  _ = \u222b x in s, f' x \u2202\u03bc := by\n    apply\n      set_integral_eq_of_subset_of_ae_diff_eq_zero_aux hts _ h.1.stronglyMeasurable_mk\n        (h.congr h.1.ae_eq_mk)\n    filter_upwards [h't, ae_imp_of_ae_restrict h.1.ae_eq_mk] with x hx h'x h''x\n    rw [\u2190 h'x h''x.1, hx h''x]\n  _ = \u222b x in s, f x \u2202\u03bc := by\n    apply integral_congr_ae\n    apply ae_restrict_of_ae_restrict_of_subset hts\n    exact h.1.ae_eq_mk.symm ** case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : \u00acIntegrableOn f t \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** have : \u00acIntegrableOn f s \u03bc := fun H => h (H.of_ae_diff_eq_zero ht h't) ** case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : \u00acIntegrableOn f t this : \u00acIntegrableOn f s \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** rw [integral_undef h, integral_undef this] ** \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 ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : IntegrableOn f t f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) \u22a2 \u222b (x : \u03b1) in t, f' x \u2202\u03bc = \u222b (x : \u03b1) in s, f' x \u2202\u03bc ** apply\n  set_integral_eq_of_subset_of_ae_diff_eq_zero_aux hts _ h.1.stronglyMeasurable_mk\n    (h.congr h.1.ae_eq_mk) ** \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 ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : IntegrableOn f t f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x = 0 ** filter_upwards [h't, ae_imp_of_ae_restrict h.1.ae_eq_mk] with x hx h'x h''x ** case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : IntegrableOn f t f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x : \u03b1 hx : x \u2208 t \\ s \u2192 f x = 0 h'x : x \u2208 t \u2192 f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x h''x : x \u2208 t \\ s \u22a2 AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) x = 0 ** rw [\u2190 h'x h''x.1, hx h''x] ** \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 ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : IntegrableOn f t f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) \u22a2 \u222b (x : \u03b1) in s, f' x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** apply integral_congr_ae ** case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : IntegrableOn f t f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) \u22a2 (fun a => f' a) =\u1d50[Measure.restrict \u03bc s] fun a => f a ** apply ae_restrict_of_ae_restrict_of_subset hts ** case 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 g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht : NullMeasurableSet t hts : s \u2286 t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \\ s \u2192 f x = 0 h : IntegrableOn f t f' : \u03b1 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f (Measure.restrict \u03bc t)) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, (fun a => f' a) x = (fun a => f a) x ** exact h.1.ae_eq_mk.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 mb : MeasurableSpace \u03b2 \u03bd : Measure \u03b2 g : \u03b1 \u2192 \u03b2 hg : MeasurePreserving g s : Set \u03b2 hs : MeasurableSet s f : \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 \u222b\u207b (a : \u03b1) in g \u207b\u00b9' s, f (g a) \u2202\u03bc = \u222b\u207b (b : \u03b2) in s, f b \u2202\u03bd ** rw [\u2190 hg.map_eq, set_lintegral_map hs hf hg.measurable] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpIndL1_disjoint_union ** \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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G \u22a2 condexpIndL1 hm \u03bc (s \u222a t) x = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc t x ** have h\u03bcst : \u03bc (s \u222a t) \u2260 \u221e :=\n  ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr \u27e8h\u03bcs, h\u03bct\u27e9))).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\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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 \u22a2 condexpIndL1 hm \u03bc (s \u222a t) x = condexpIndL1 hm \u03bc s x + condexpIndL1 hm \u03bc t x ** rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs h\u03bcs x,\n  condexpIndL1_of_measurableSet_of_measure_ne_top ht h\u03bct x,\n  condexpIndL1_of_measurableSet_of_measure_ne_top (hs.union ht) h\u03bcst 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 \u22a2 condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x = condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct x ** exact condexpIndL1Fin_disjoint_union hs ht h\u03bcs h\u03bct hst x ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.compl_mem_cofinite ** \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 \u22a2 s\u1d9c \u2208 cofinite \u03bc \u2194 \u2191\u2191\u03bc s < \u22a4 ** rw [mem_cofinite, compl_compl] ** Qed",
        "informal": ""
    },
    {
        "formal": "PosNum.minFac_to_nat ** n : PosNum \u22a2 \u2191(minFac n) = Nat.minFac \u2191n ** cases' n with n ** case one  \u22a2 \u2191(minFac one) = Nat.minFac \u2191one ** rfl ** case bit1 n : PosNum \u22a2 \u2191(minFac (bit1 n)) = Nat.minFac \u2191(bit1 n) ** rw [minFac, Nat.minFac_eq, if_neg] ** case bit1 n : PosNum \u22a2 \u2191(match bit1 n with       | one => 1       | bit0 a => 2       | bit1 n => minFacAux (bit1 n) (\u2191n) 1) =     Nat.minFacAux (\u2191(bit1 n)) 3  case bit1.hnc n : PosNum \u22a2 \u00ac2 \u2223 \u2191(bit1 n) ** swap ** case bit1 n : PosNum \u22a2 \u2191(match bit1 n with       | one => 1       | bit0 a => 2       | bit1 n => minFacAux (bit1 n) (\u2191n) 1) =     Nat.minFacAux (\u2191(bit1 n)) 3 ** rw [minFacAux_to_nat] ** case bit1 n : PosNum \u22a2 Nat.sqrt \u2191(bit1 n) < \u2191n + \u2191(bit1 1) ** simp only [cast_one, cast_bit1] ** case bit1 n : PosNum \u22a2 Nat.sqrt (_root_.bit1 \u2191n) < \u2191n + _root_.bit1 1 ** unfold _root_.bit1 _root_.bit0 ** case bit1 n : PosNum \u22a2 Nat.sqrt (\u2191n + \u2191n + 1) < \u2191n + (1 + 1 + 1) ** rw [Nat.sqrt_lt] ** case bit1 n : PosNum \u22a2 \u2191n + \u2191n + 1 < (\u2191n + (1 + 1 + 1)) * (\u2191n + (1 + 1 + 1)) ** calc\n  (n : \u2115) + (n : \u2115) + 1 \u2264 (n : \u2115) + (n : \u2115) + (n : \u2115) := by simp\n  _ = (n : \u2115) * (1 + 1 + 1) := by simp only [mul_add, mul_one]\n  _ < _ := by simp [mul_lt_mul] ** case bit1.hnc n : PosNum \u22a2 \u00ac2 \u2223 \u2191(bit1 n) ** simp ** case bit1 n : PosNum \u22a2 Nat.minFacAux \u2191(bit1 n) \u2191(bit1 1) = Nat.minFacAux (\u2191(bit1 n)) 3 ** rfl ** n : PosNum \u22a2 \u2191n + \u2191n + 1 \u2264 \u2191n + \u2191n + \u2191n ** simp ** n : PosNum \u22a2 \u2191n + \u2191n + \u2191n = \u2191n * (1 + 1 + 1) ** simp only [mul_add, mul_one] ** n : PosNum \u22a2 \u2191n * (1 + 1 + 1) < (\u2191n + (1 + 1 + 1)) * (\u2191n + (1 + 1 + 1)) ** simp [mul_lt_mul] ** case bit0 a\u271d : PosNum \u22a2 \u2191(minFac (bit0 a\u271d)) = Nat.minFac \u2191(bit0 a\u271d) ** rw [minFac, Nat.minFac_eq, if_pos] ** case bit0.hc a\u271d : PosNum \u22a2 2 \u2223 \u2191(bit0 a\u271d) ** simp ** case bit0 a\u271d : PosNum \u22a2 \u2191(match bit0 a\u271d with       | one => 1       | bit0 a => 2       | bit1 n => minFacAux (bit1 n) (\u2191n) 1) =     2 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "List.getLastD_concat ** \u03b1 : Type u_1 a b : \u03b1 l : List \u03b1 \u22a2 getLastD (l ++ [b]) a = b ** rw [getLastD_eq_getLast?, getLast?_concat] ** \u03b1 : Type u_1 a b : \u03b1 l : List \u03b1 \u22a2 Option.getD (some b) a = b ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one_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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \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 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) ** have L' : Tendsto (fun r : \u211d => \u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n  tendsto_addHaar_inter_smul_zero_of_density_zero \u03bc s\u1d9c x L t ht 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\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 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) ** have L'' : Tendsto (fun r : \u211d => \u03bc ({x} + r \u2022 t) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 1) := by\n  apply tendsto_const_nhds.congr' _\n  filter_upwards [self_mem_nhdsWithin]\n  rintro r (rpos : 0 < r)\n  rw [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne', ENNReal.div_self h't 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) \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 1) ** have := ENNReal.Tendsto.sub L'' L' (Or.inl ENNReal.one_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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) this :   Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd (1 - 0)) \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 1) ** simp only [tsub_zero] at 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 s\u271d s : Set E hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) this :   Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd 1) \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 1) ** apply this.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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) this :   Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd 1) \u22a2 (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) =\u1da0[\ud835\udcdd[Ioi 0] 0]     fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) this :   Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd 1) \u22a2 \u2200 (a : \u211d),     a \u2208 Ioi 0 \u2192       \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) =         \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) this :   Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd 1) r : \u211d rpos : 0 < r \u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) =     \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) ** refine' I ({x} + r \u2022 t) s _ _ hs ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u ** intro u v uzero utop vmeas ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 u v : Set E uzero : \u2191\u2191\u03bc u \u2260 0 utop : \u2191\u2191\u03bc u \u2260 \u22a4 vmeas : MeasurableSet v \u22a2 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u ** simp_rw [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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 u v : Set E uzero : \u2191\u2191\u03bc u \u2260 0 utop : \u2191\u2191\u03bc u \u2260 \u22a4 vmeas : MeasurableSet v \u22a2 \u2191\u2191\u03bc u * (\u2191\u2191\u03bc u)\u207b\u00b9 - \u2191\u2191\u03bc (v\u1d9c \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9 ** rw [\u2190 ENNReal.sub_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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 u v : Set E uzero : \u2191\u2191\u03bc u \u2260 0 utop : \u2191\u2191\u03bc u \u2260 \u22a4 vmeas : MeasurableSet v \u22a2 (\u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u)) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9  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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 u v : Set E uzero : \u2191\u2191\u03bc u \u2260 0 utop : \u2191\u2191\u03bc u \u2260 \u22a4 vmeas : MeasurableSet v \u22a2 0 < \u2191\u2191\u03bc (v\u1d9c \u2229 u) \u2192 \u2191\u2191\u03bc (v\u1d9c \u2229 u) < \u2191\u2191\u03bc u \u2192 (\u2191\u2191\u03bc u)\u207b\u00b9 \u2260 \u22a4 ** swap ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 u v : Set E uzero : \u2191\u2191\u03bc u \u2260 0 utop : \u2191\u2191\u03bc u \u2260 \u22a4 vmeas : MeasurableSet v \u22a2 (\u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u)) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9 ** congr 1 ** case e_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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 u v : Set E uzero : \u2191\u2191\u03bc u \u2260 0 utop : \u2191\u2191\u03bc u \u2260 \u22a4 vmeas : MeasurableSet v \u22a2 \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) = \u2191\u2191\u03bc (v \u2229 u) ** apply\n  ENNReal.sub_eq_of_add_eq (ne_top_of_le_ne_top utop (measure_mono (inter_subset_right _ _))) ** case e_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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 u v : Set E uzero : \u2191\u2191\u03bc u \u2260 0 utop : \u2191\u2191\u03bc u \u2260 \u22a4 vmeas : MeasurableSet v \u22a2 \u2191\u2191\u03bc (v \u2229 u) + \u2191\u2191\u03bc (v\u1d9c \u2229 u) = \u2191\u2191\u03bc u ** rw [inter_comm _ u, inter_comm _ u] ** case e_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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 u v : Set E uzero : \u2191\u2191\u03bc u \u2260 0 utop : \u2191\u2191\u03bc u \u2260 \u22a4 vmeas : MeasurableSet v \u22a2 \u2191\u2191\u03bc (u \u2229 v) + \u2191\u2191\u03bc (u \u2229 v\u1d9c) = \u2191\u2191\u03bc u ** exact measure_inter_add_diff u vmeas ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 u v : Set E uzero : \u2191\u2191\u03bc u \u2260 0 utop : \u2191\u2191\u03bc u \u2260 \u22a4 vmeas : MeasurableSet v \u22a2 0 < \u2191\u2191\u03bc (v\u1d9c \u2229 u) \u2192 \u2191\u2191\u03bc (v\u1d9c \u2229 u) < \u2191\u2191\u03bc u \u2192 (\u2191\u2191\u03bc u)\u207b\u00b9 \u2260 \u22a4 ** simp only [uzero, ENNReal.inv_eq_top, imp_true_iff, Ne.def, not_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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u A : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** have B := ENNReal.Tendsto.sub A h (Or.inl ENNReal.one_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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u A : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) B :   Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (1 - 1)) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** simp only [tsub_self] at B ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u A : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) B :   Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** apply B.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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u A : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) B :   Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 (fun a =>       \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a)) =\u1da0[\ud835\udcdd[Ioi 0] 0]     fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u A : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) B :   Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200 (a : \u211d),     a \u2208 Ioi 0 \u2192       \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a) =         \u2191\u2191\u03bc (s\u1d9c \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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u A : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) B :   Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))     (\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) - \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) =     \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) ** convert I (closedBall x r) s\u1d9c (measure_closedBall_pos \u03bc _ rpos).ne'\n  measure_closedBall_lt_top.ne hs.compl ** case h.e'_2.h.e'_6.h.e'_5.h.e'_3.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 s\u271d s : Set E hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u A : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) B :   Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r \u22a2 s = s\u1d9c\u1d9c ** rw [compl_compl] ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u \u22a2 (fun x => 1) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r) ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u \u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 1 = \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) ** intro r hr ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u r : \u211d hr : r \u2208 Ioi 0 \u22a2 1 = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r) ** rw [div_eq_mul_inv, ENNReal.mul_inv_cancel] ** case h.h0 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u r : \u211d hr : r \u2208 Ioi 0 \u22a2 \u2191\u2191\u03bc (closedBall x r) \u2260 0 ** exact (measure_closedBall_pos \u03bc _ hr).ne' ** case h.ht 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u r : \u211d hr : r \u2208 Ioi 0 \u22a2 \u2191\u2191\u03bc (closedBall x r) \u2260 \u22a4 ** exact measure_closedBall_lt_top.ne ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 (fun x => 1) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t) ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 1 = \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) ** 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r \u22a2 1 = \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t) ** rw [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne', ENNReal.div_self h't h''t] ** case h.refine'_1 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) this :   Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd 1) r : \u211d rpos : 0 < r \u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) \u2260 0 ** simp only [h't, abs_of_nonneg rpos.le, pow_pos rpos, addHaar_smul, image_add_left,\n  ENNReal.ofReal_eq_zero, not_le, or_false_iff, Ne.def, measure_preimage_add, abs_pow,\n  singleton_add, mul_eq_zero] ** case h.refine'_2 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 hs : MeasurableSet s 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 1) t : Set E ht : MeasurableSet t h't : \u2191\u2191\u03bc t \u2260 0 h''t : \u2191\u2191\u03bc t \u2260 \u22a4 I :   \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u L : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) L'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) this :   Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd 1) r : \u211d rpos : 0 < r \u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) \u2260 \u22a4 ** simp [h''t, ENNReal.ofReal_ne_top, addHaar_smul, image_add_left, ENNReal.mul_eq_top,\n  Ne.def, not_false_iff, measure_preimage_add, singleton_add, and_false_iff, false_and_iff,\n  or_self_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.prehaar_mem_haarProduct ** 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) \u22a2 prehaar (\u2191K\u2080) U \u2208 haarProduct \u2191K\u2080 ** rintro \u27e8K, hK\u27e9 _ ** case mk 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 : Set G hK : IsCompact K a\u271d : { carrier := K, isCompact' := hK } \u2208 univ \u22a2 prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2208     (fun K => Icc 0 \u2191(index \u2191K \u2191K\u2080)) { carrier := K, isCompact' := hK } ** rw [mem_Icc] ** case mk 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 : Set G hK : IsCompact K a\u271d : { carrier := K, isCompact' := hK } \u2208 univ \u22a2 0 \u2264 prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2227     prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2264 \u2191(index \u2191{ carrier := K, isCompact' := hK } \u2191K\u2080) ** exact \u27e8prehaar_nonneg K\u2080 _, prehaar_le_index K\u2080 _ hU\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.smul_inter_ne_empty_iff' ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b9 : Group \u03b1 inst\u271d : MulAction \u03b1 \u03b2 s\u271d t\u271d A B : Set \u03b2 a : \u03b1 x\u271d : \u03b2 s t : Set \u03b1 x : \u03b1 \u22a2 x \u2022 s \u2229 t \u2260 \u2205 \u2194 \u2203 a b, (a \u2208 t \u2227 b \u2208 s) \u2227 a / b = x ** simp_rw [smul_inter_ne_empty_iff, div_eq_mul_inv] ** Qed",
        "informal": ""
    },
    {
        "formal": "WithTop.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_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Countable.biUnion_iff ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x s : Set \u03b1 t : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b2 hs : Set.Countable s \u22a2 Set.Countable (\u22c3 a, \u22c3 (h : a \u2208 s), t a h) \u2194 \u2200 (a : \u03b1) (ha : a \u2208 s), Set.Countable (t a ha) ** haveI := hs.to_subtype ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x s : Set \u03b1 t : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b2 hs : Set.Countable s this : Countable \u2191s \u22a2 Set.Countable (\u22c3 a, \u22c3 (h : a \u2208 s), t a h) \u2194 \u2200 (a : \u03b1) (ha : a \u2208 s), Set.Countable (t a ha) ** rw [biUnion_eq_iUnion, countable_iUnion_iff, SetCoe.forall'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.piPremeasure_pi_eval ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) s : Set ((i : \u03b9) \u2192 \u03b1 i) \u22a2 piPremeasure m (Set.pi univ fun i => eval i '' s) = piPremeasure m s ** simp only [eval, piPremeasure_pi'] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) s : Set ((i : \u03b9) \u2192 \u03b1 i) \u22a2 \u220f i : \u03b9, \u2191(m i) ((fun a => a i) '' s) = piPremeasure m s ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_add ** \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 hf : Integrable f hg : Integrable g \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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 hf : Integrable f hg : Integrable g hm : \u00acm \u2264 m0 \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \u03bc[g|m] ** refine' (condexp_ae_eq_condexpL1 hm _).trans _ ** 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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u2191\u2191(condexpL1 hm \u03bc (f + g)) =\u1d50[\u03bc] \u03bc[f|m] + \u03bc[g|m] ** rw [condexpL1_add hf hg] ** 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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u2191\u2191(condexpL1 hm \u03bc f + condexpL1 hm \u03bc g) =\u1d50[\u03bc] \u03bc[f|m] + \u03bc[g|m] ** exact (coeFn_add _ _).trans\n  ((condexp_ae_eq_condexpL1 hm _).symm.add (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\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 hf : Integrable f hg : Integrable g hm : \u00acm \u2264 m0 \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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 hf : Integrable f hg : Integrable g hm : \u00acm \u2264 m0 \u22a2 0 =\u1d50[\u03bc] 0 + 0 ** simp ** 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 hf : Integrable f hg : Integrable 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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f + g|m] =\u1d50[\u03bc] \u03bc[f|m] + \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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 0 =\u1d50[\u03bc] 0 + 0 ** simp ** 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 hf : Integrable f hg : Integrable g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 0 =\u1d50[\u03bc] 0 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.tr_eval_rev ** \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab : b\u2082 \u2208 eval f\u2082 a\u2082 \u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081 ** cases' mem_eval.1 ab with ab b0 ** case intro \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none \u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081 ** rcases tr_reaches_rev H aa ab with \u27e8c\u2081, c\u2082, bc, cc, ac\u27e9 ** case intro.intro.intro.intro.intro \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none c\u2081 : \u03c3\u2081 c\u2082 : \u03c3\u2082 bc : Reaches f\u2082 b\u2082 c\u2082 cc : tr c\u2081 c\u2082 ac : Reaches f\u2081 a\u2081 c\u2081 \u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081 ** cases (reflTransGen_iff_eq (Option.eq_none_iff_forall_not_mem.1 b0)).1 bc ** case intro.intro.intro.intro.intro.refl \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none c\u2081 : \u03c3\u2081 ac : Reaches f\u2081 a\u2081 c\u2081 bc : Reaches f\u2082 b\u2082 b\u2082 cc : tr c\u2081 b\u2082 \u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081 ** refine' \u27e8_, cc, mem_eval.2 \u27e8ac, _\u27e9\u27e9 ** case intro.intro.intro.intro.intro.refl \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none c\u2081 : \u03c3\u2081 ac : Reaches f\u2081 a\u2081 c\u2081 bc : Reaches f\u2082 b\u2082 b\u2082 cc : tr c\u2081 b\u2082 \u22a2 f\u2081 c\u2081 = none ** have := H cc ** case intro.intro.intro.intro.intro.refl \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none c\u2081 : \u03c3\u2081 ac : Reaches f\u2081 a\u2081 c\u2081 bc : Reaches f\u2082 b\u2082 b\u2082 cc : tr c\u2081 b\u2082 this :   match f\u2081 c\u2081 with   | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1   | none => f\u2082 b\u2082 = none \u22a2 f\u2081 c\u2081 = none ** cases' hfc : f\u2081 c\u2081 with d\u2081 ** case intro.intro.intro.intro.intro.refl.some \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none c\u2081 : \u03c3\u2081 ac : Reaches f\u2081 a\u2081 c\u2081 bc : Reaches f\u2082 b\u2082 b\u2082 cc : tr c\u2081 b\u2082 this :   match f\u2081 c\u2081 with   | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1   | none => f\u2082 b\u2082 = none d\u2081 : \u03c3\u2081 hfc : f\u2081 c\u2081 = some d\u2081 \u22a2 some d\u2081 = none ** rw [hfc] at this ** case intro.intro.intro.intro.intro.refl.some \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none c\u2081 : \u03c3\u2081 ac : Reaches f\u2081 a\u2081 c\u2081 bc : Reaches f\u2082 b\u2082 b\u2082 cc : tr c\u2081 b\u2082 d\u2081 : \u03c3\u2081 this :   match some d\u2081 with   | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1   | none => f\u2082 b\u2082 = none hfc : f\u2081 c\u2081 = some d\u2081 \u22a2 some d\u2081 = none ** rcases this with \u27e8d\u2082, _, bd\u27e9 ** case intro.intro.intro.intro.intro.refl.some.intro.intro \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none c\u2081 : \u03c3\u2081 ac : Reaches f\u2081 a\u2081 c\u2081 bc : Reaches f\u2082 b\u2082 b\u2082 cc : tr c\u2081 b\u2082 d\u2081 : \u03c3\u2081 hfc : f\u2081 c\u2081 = some d\u2081 d\u2082 : \u03c3\u2082 left\u271d : tr d\u2081 d\u2082 bd : Reaches\u2081 f\u2082 b\u2082 d\u2082 \u22a2 some d\u2081 = none ** rcases TransGen.head'_iff.1 bd with \u27e8e, h, _\u27e9 ** case intro.intro.intro.intro.intro.refl.some.intro.intro.intro.intro \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none c\u2081 : \u03c3\u2081 ac : Reaches f\u2081 a\u2081 c\u2081 bc : Reaches f\u2082 b\u2082 b\u2082 cc : tr c\u2081 b\u2082 d\u2081 : \u03c3\u2081 hfc : f\u2081 c\u2081 = some d\u2081 d\u2082 : \u03c3\u2082 left\u271d : tr d\u2081 d\u2082 bd : Reaches\u2081 f\u2082 b\u2082 d\u2082 e : \u03c3\u2082 h : e \u2208 f\u2082 b\u2082 right\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) e d\u2082 \u22a2 some d\u2081 = none ** cases b0.symm.trans h ** case intro.intro.intro.intro.intro.refl.none \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 b\u2082 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 b0 : f\u2082 b\u2082 = none c\u2081 : \u03c3\u2081 ac : Reaches f\u2081 a\u2081 c\u2081 bc : Reaches f\u2082 b\u2082 b\u2082 cc : tr c\u2081 b\u2082 this :   match f\u2081 c\u2081 with   | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1   | none => f\u2082 b\u2082 = none hfc : f\u2081 c\u2081 = none \u22a2 none = none ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Martingale.ae_not_tendsto_atTop_atTop ** \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 : Martingale 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, \u00acTendsto (fun n => f n \u03c9) atTop atTop ** filter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with \u03c9 h\u03c9 htop using\n  unbounded_of_tendsto_atTop htop (h\u03c9.2 <| bddBelow_range_of_tendsto_atTop_atTop htop) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2082 ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i) ** rcases I.exists_seq_mono_tendsto with \u27e8J, hJ_sub, hJl, hJu\u27e9 ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i) ** have hJ_sub' : \u2200 k, Box.Icc (J k) \u2286 Box.Icc I := fun k => (hJ_sub k).trans I.Ioo_subset_Icc ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i) ** have hJ_le : \u2200 k, J k \u2264 I := fun k => Box.le_iff_Icc.2 (hJ_sub' k) ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i) ** have HcJ : \u2200 k, ContinuousOn f (Box.Icc (J k)) := fun k => Hc.mono (hJ_sub' k) ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i) ** have HdJ : \u2200 (k), \u2200 x \u2208 (Box.Icc (J k)) \\ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x :=\n  fun k x hx => (Hd x \u27e8hJ_sub k hx.1, hx.2\u27e9).hasFDerivWithinAt ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i) ** have HJ_eq := fun k =>\n  integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2081 (J k) f f' s hs (HcJ k) (HdJ k)\n    (HiJ k) ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i) ** have hI_tendsto :\n  Tendsto (fun k => \u222b x in Box.Icc (J k), \u2211 i, f' x (e i) i) atTop\n    (\ud835\udcdd (\u222b x in Box.Icc I, \u2211 i, f' x (e i) i)) := by\n  simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _)] at Hi \u22a2\n  rw [\u2190 Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi \u22a2\n  exact tendsto_set_integral_of_monotone (fun k => (J k).measurableSet_Ioo)\n    (Box.Ioo.comp J).monotone Hi ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) hI_tendsto :   Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop     (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i) ** refine' tendsto_nhds_unique_of_eventuallyEq hI_tendsto _ (eventually_of_forall HJ_eq) ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) hI_tendsto :   Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop     (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)) \u22a2 Tendsto     (fun x =>       \u2211 i : Fin (n + 1),         ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -           \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))     atTop     (\ud835\udcdd       (\u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i))) ** clear hI_tendsto ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) \u22a2 Tendsto     (fun x =>       \u2211 i : Fin (n + 1),         ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -           \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))     atTop     (\ud835\udcdd       (\u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i))) ** rw [tendsto_pi_nhds] at hJl hJu ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) \u22a2 Tendsto     (fun x =>       \u2211 i : Fin (n + 1),         ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -           \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))     atTop     (\ud835\udcdd       (\u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i))) ** suffices \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d), (\u2200 k, c k \u2208 Icc (I.lower i) (I.upper i)) \u2192\n  Tendsto c atTop (\ud835\udcdd d) \u2192\n    Tendsto (fun k => \u222b x in Box.Icc ((J k).face i), f (i.insertNth (c k) x) i) atTop\n      (\ud835\udcdd <| \u222b x in Box.Icc (I.face i), f (i.insertNth d x) i) by\n  rw [Box.Icc_eq_pi] at hJ_sub'\n  refine' tendsto_finset_sum _ fun i _ => (this _ _ _ _ (hJu _)).sub (this _ _ _ _ (hJl _))\n  exacts [fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k =>\n    hJ_sub' k (J k).lower_mem_Icc _ trivial] ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) \u22a2 \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),     (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192       Tendsto c atTop (\ud835\udcdd d) \u2192         Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop           (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) ** intro i c d hc hcd ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) \u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) ** have hd : d \u2208 Icc (I.lower i) (I.upper i) :=\n  isClosed_Icc.mem_of_tendsto hcd (eventually_of_forall hc) ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) \u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) ** have Hic : \u2200 k, IntegrableOn (fun x => f (i.insertNth (c k) x) i) (Box.Icc (I.face i)) := fun k =>\n  (Box.continuousOn_face_Icc ((continuous_apply i).comp_continuousOn Hc) (hc k)).integrableOn_Icc ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) \u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) ** have Hid : IntegrableOn (fun x => f (i.insertNth d x) i) (Box.Icc (I.face i)) :=\n  (Box.continuousOn_face_Icc ((continuous_apply i).comp_continuousOn Hc) hd).integrableOn_Icc ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) \u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) ** have H :\n  Tendsto (fun k => \u222b x in Box.Icc ((J k).face i), f (i.insertNth d x) i) atTop\n    (\ud835\udcdd <| \u222b x in Box.Icc (I.face i), f (i.insertNth d x) i) := by\n  have hIoo : (\u22c3 k, Box.Ioo ((J k).face i)) = Box.Ioo (I.face i) :=\n    Box.iUnion_Ioo_of_tendsto ((Box.monotone_face i).comp J.monotone)\n      (tendsto_pi_nhds.2 fun _ => hJl _) (tendsto_pi_nhds.2 fun _ => hJu _)\n  simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _), \u2190 hIoo] at Hid \u22a2\n  exact tendsto_set_integral_of_monotone (fun k => ((J k).face i).measurableSet_Ioo)\n    (Box.Ioo.monotone.comp ((Box.monotone_face i).comp J.monotone)) Hid ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) ** refine' H.congr_dist (Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun \u03b5 \u03b5pos => _) ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 \u2200\u1da0 (x : \u2115) in atTop,     dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)         (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208       Metric.closedBall 0 \u03b5 ** have hvol_pos : \u2200 J : Box (Fin n), 0 < \u220f j, (J.upper j - J.lower j) := fun J =>\n  prod_pos fun j hj => sub_pos.2 <| J.lower_lt_upper _ ** case intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u22a2 \u2200\u1da0 (x : \u2115) in atTop,     dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)         (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208       Metric.closedBall 0 \u03b5 ** rcases Metric.uniformContinuousOn_iff_le.1 (I.isCompact_Icc.uniformContinuousOn_of_continuous Hc)\n    (\u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j)) (div_pos \u03b5pos (hvol_pos (I.face i)))\n  with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 ** case intro.intro.intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) \u22a2 \u2200\u1da0 (x : \u2115) in atTop,     dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)         (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208       Metric.closedBall 0 \u03b5 ** refine' (hcd.eventually (Metric.ball_mem_nhds _ \u03b4pos)).mono fun k hk => _ ** case intro.intro.intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 \u22a2 dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i)       (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) \u2208     Metric.closedBall 0 \u03b5 ** have Hsub : Box.Icc ((J k).face i) \u2286 Box.Icc (I.face i) :=\n  Box.le_iff_Icc.1 (Box.face_mono (hJ_le _) i) ** case intro.intro.intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) \u22a2 dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i)       (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) \u2208     Metric.closedBall 0 \u03b5 ** rw [mem_closedBall_zero_iff, Real.norm_eq_abs, abs_of_nonneg dist_nonneg, dist_eq_norm,\n  \u2190 integral_sub (Hid.mono_set Hsub) ((Hic _).mono_set Hsub)] ** case intro.intro.intro.intro.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) \u22a2 \u2016\u222b (a : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d a) i - f (Fin.insertNth i (c k) a) i\u2016 \u2264 \u03b5 ** calc\n  \u2016\u222b x in Box.Icc ((J k).face i), f (i.insertNth d x) i - f (i.insertNth (c k) x) i\u2016 \u2264\n      (\u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j)) *\n        (volume (Box.Icc ((J k).face i))).toReal := by\n    refine norm_set_integral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _)\n      ((J k).face i).measurableSet_Icc fun x hx => ?_\n    rw [\u2190 dist_eq_norm]\n    calc\n      dist (f (i.insertNth d x) i) (f (i.insertNth (c k) x) i) \u2264\n          dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) :=\n        dist_le_pi_dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) i\n      _ \u2264 \u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j) :=\n        h\u03b4 _ (I.mapsTo_insertNth_face_Icc hd <| Hsub hx) _\n          (I.mapsTo_insertNth_face_Icc (hc _) <| Hsub hx) ?_\n    rw [Fin.dist_insertNth_insertNth, dist_self, dist_comm]\n    exact max_le hk.le \u03b4pos.lt.le\n  _ \u2264 \u03b5 := by\n    rw [Box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper,\n      \u2190 le_div_iff (hvol_pos _)]\n    refine' div_le_div_of_le_left \u03b5pos.le (hvol_pos _)\n      (prod_le_prod (fun j _ => _) fun j _ => _)\n    exacts [sub_nonneg.2 (Box.lower_le_upper _ _),\n      sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)] ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) \u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop     (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)) ** simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _)] at Hi \u22a2 ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) Hi : Integrable fun x => \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 \u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo (\u2191J k), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1) atTop     (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1)) ** rw [\u2190 Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi \u22a2 ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) hJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) Hi : Integrable fun x => \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 \u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo (\u2191J k), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1) atTop     (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u22c3 n_1, \u2191Box.Ioo (\u2191J n_1), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1)) ** exact tendsto_set_integral_of_monotone (fun k => (J k).measurableSet_Ioo)\n  (Box.Ioo.comp J).monotone Hi ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) this :   \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),     (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192       Tendsto c atTop (\ud835\udcdd d) \u2192         Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop           (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u22a2 Tendsto     (fun x =>       \u2211 i : Fin (n + 1),         ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -           \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))     atTop     (\ud835\udcdd       (\u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i))) ** rw [Box.Icc_eq_pi] at hJ_sub' ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i) hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) this :   \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),     (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192       Tendsto c atTop (\ud835\udcdd d) \u2192         Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop           (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u22a2 Tendsto     (fun x =>       \u2211 i : Fin (n + 1),         ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -           \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))     atTop     (\ud835\udcdd       (\u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i))) ** refine' tendsto_finset_sum _ fun i _ => (this _ _ _ _ (hJu _)).sub (this _ _ _ _ (hJl _)) ** case refine'_1 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i) hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) this :   \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),     (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192       Tendsto c atTop (\ud835\udcdd d) \u2192         Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop           (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) i : Fin (n + 1) x\u271d : i \u2208 Finset.univ \u22a2 \u2200 (k : \u2115), (Box.upper \u2218 \u2191J) k i \u2208 Set.Icc (Box.lower I i) (Box.upper I i)  case refine'_2 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i) hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) this :   \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),     (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192       Tendsto c atTop (\ud835\udcdd d) \u2192         Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop           (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) i : Fin (n + 1) x\u271d : i \u2208 Finset.univ \u22a2 \u2200 (k : \u2115), Box.lower (\u2191J k) i \u2208 Set.Icc (Box.lower I i) (Box.upper I i) ** exacts [fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k =>\n  hJ_sub' k (J k).lower_mem_Icc _ trivial] ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) \u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) ** have hIoo : (\u22c3 k, Box.Ioo ((J k).face i)) = Box.Ioo (I.face i) :=\n  Box.iUnion_Ioo_of_tendsto ((Box.monotone_face i).comp J.monotone)\n    (tendsto_pi_nhds.2 fun _ => hJl _) (tendsto_pi_nhds.2 fun _ => hJu _) ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) hIoo : \u22c3 k, \u2191Box.Ioo (Box.face (\u2191J k) i) = \u2191Box.Ioo (Box.face I i) \u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) ** simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _), \u2190 hIoo] at Hid \u22a2 ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) hIoo : \u22c3 k, \u2191Box.Ioo (Box.face (\u2191J k) i) = \u2191Box.Ioo (Box.face I i) Hid : Integrable fun x => f (Fin.insertNth i d x) i \u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Ioo (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u22c3 k, \u2191Box.Ioo (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i)) ** exact tendsto_set_integral_of_monotone (fun k => ((J k).face i).measurableSet_Ioo)\n  (Box.Ioo.monotone.comp ((Box.monotone_face i).comp J.monotone)) Hid ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) \u22a2 \u2016\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i - f (Fin.insertNth i (c k) x) i\u2016 \u2264     (\u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)) *       ENNReal.toReal (\u2191\u2191volume (\u2191Box.Icc (Box.face (\u2191J k) i))) ** refine norm_set_integral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _)\n  ((J k).face i).measurableSet_Icc fun x hx => ?_ ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) x : Fin n \u2192 \u211d hx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i) \u22a2 \u2016f (Fin.insertNth i d x) i - f (Fin.insertNth i (c k) x) i\u2016 \u2264     \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) ** rw [\u2190 dist_eq_norm] ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) x : Fin n \u2192 \u211d hx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i) \u22a2 dist (f (Fin.insertNth i d x) i) (f (Fin.insertNth i (c k) x) i) \u2264     \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) ** calc\n  dist (f (i.insertNth d x) i) (f (i.insertNth (c k) x) i) \u2264\n      dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) :=\n    dist_le_pi_dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) i\n  _ \u2264 \u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j) :=\n    h\u03b4 _ (I.mapsTo_insertNth_face_Icc hd <| Hsub hx) _\n      (I.mapsTo_insertNth_face_Icc (hc _) <| Hsub hx) ?_ ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) x : Fin n \u2192 \u211d hx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i) \u22a2 dist (Fin.insertNth i d x) (Fin.insertNth i (c k) x) \u2264 \u03b4 ** rw [Fin.dist_insertNth_insertNth, dist_self, dist_comm] ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) x : Fin n \u2192 \u211d hx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i) \u22a2 max (dist (c k) d) 0 \u2264 \u03b4 ** exact max_le hk.le \u03b4pos.lt.le ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) \u22a2 (\u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)) *       ENNReal.toReal (\u2191\u2191volume (\u2191Box.Icc (Box.face (\u2191J k) i))) \u2264     \u03b5 ** rw [Box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper,\n  \u2190 le_div_iff (hvol_pos _)] ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) \u22a2 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) \u2264     \u03b5 / \u220f j : Fin n, (Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j) ** refine' div_le_div_of_le_left \u03b5pos.le (hvol_pos _)\n  (prod_le_prod (fun j _ => _) fun j _ => _) ** case refine'_1 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) j : Fin n x\u271d : j \u2208 Finset.univ \u22a2 0 \u2264 Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j  case refine'_2 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) J : \u2115 \u2192o Box (Fin (n + 1)) hJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I hJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x)) hJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x)) hJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I hJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I HcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k)) HdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x HiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k)) HJ_eq :   \u2200 (k : \u2115),     \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =       \u2211 i : Fin (n + 1),         ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -           \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i) i : Fin (n + 1) c : \u2115 \u2192 \u211d d : \u211d hc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i) hcd : Tendsto c atTop (\ud835\udcdd d) hd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i) Hic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i)) Hid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i)) H :   Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop     (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i)) \u03b5 : \u211d \u03b5pos : 0 < \u03b5 hvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j) \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 :   \u2200 (x : Fin (n + 1) \u2192 \u211d),     x \u2208 \u2191Box.Icc I \u2192       \u2200 (y : Fin (n + 1) \u2192 \u211d),         y \u2208 \u2191Box.Icc I \u2192           dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) k : \u2115 hk : dist (c k) d < \u03b4 Hsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i) j : Fin n x\u271d : j \u2208 Finset.univ \u22a2 Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j \u2264     Box.upper (Box.face I i) j - Box.lower (Box.face I i) j ** exacts [sub_nonneg.2 (Box.lower_le_upper _ _),\n  sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.zneg_pred ** \u03b1 : Type u_1 n : ZNum \u22a2 -pred n = succ (-n) ** rw [\u2190 zneg_zneg (succ (-n)), zneg_succ, zneg_zneg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.totalDegree_sub ** R : Type u S : Type v \u03c3 : Type u_1 a\u271d a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q a b : MvPolynomial \u03c3 R \u22a2 totalDegree (a - b) = totalDegree (a + -b) ** rw [sub_eq_add_neg] ** R : Type u S : Type v \u03c3 : Type u_1 a\u271d a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q a b : MvPolynomial \u03c3 R \u22a2 max (totalDegree a) (totalDegree (-b)) = max (totalDegree a) (totalDegree b) ** rw [totalDegree_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.tendsto_Icc_vitaliFamily_left ** x : \u211d \u22a2 Tendsto (fun y => Icc y x) (\ud835\udcdd[Iio x] x) (VitaliFamily.filterAt (vitaliFamily volume 1) x) ** refine' (VitaliFamily.tendsto_filterAt_iff _).2 \u27e8_, _\u27e9 ** case refine'_1 x : \u211d \u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2208 VitaliFamily.setsAt (vitaliFamily volume 1) x ** filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy ** case refine'_2 x : \u211d \u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2286 Metric.closedBall x \u03b5 ** intro \u03b5 \u03b5pos ** case refine'_2 x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2286 Metric.closedBall x \u03b5 ** have : x \u2208 Ioc (x - \u03b5) x := \u27e8by linarith, le_refl _\u27e9 ** case refine'_2 x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 this : x \u2208 Ioc (x - \u03b5) x \u22a2 \u2200\u1da0 (i : \u211d) in \ud835\udcdd[Iio x] x, Icc i x \u2286 Metric.closedBall x \u03b5 ** filter_upwards [Icc_mem_nhdsWithin_Iio this] with y hy ** case h x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 this : x \u2208 Ioc (x - \u03b5) x y : \u211d hy : y \u2208 Icc (x - \u03b5) x \u22a2 Icc y x \u2286 Metric.closedBall x \u03b5 ** rw [closedBall_eq_Icc] ** case h x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 this : x \u2208 Ioc (x - \u03b5) x y : \u211d hy : y \u2208 Icc (x - \u03b5) x \u22a2 Icc y x \u2286 Icc (x - \u03b5) (x + \u03b5) ** exact Icc_subset_Icc hy.1 (by linarith) ** x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 x - \u03b5 < x ** linarith ** x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 this : x \u2208 Ioc (x - \u03b5) x y : \u211d hy : y \u2208 Icc (x - \u03b5) x \u22a2 x \u2264 x + \u03b5 ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.sub_add_min_cancel ** n m : Nat \u22a2 n - m + min n m = n ** rw [sub_eq_sub_min, Nat.sub_add_cancel (Nat.min_le_left n m)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Computability.decode_encodeNum ** \u22a2 \u2200 (n : Num), decodeNum (encodeNum n) = n ** intro n ** n : Num \u22a2 decodeNum (encodeNum n) = n ** cases' n with n <;> unfold encodeNum decodeNum ** case pos n : PosNum \u22a2 (if         (match Num.pos n with           | Num.zero => []           | Num.pos n => encodePosNum n) =           [] then       Num.zero     else       \u2191(decodePosNum           (match Num.pos n with           | Num.zero => []           | Num.pos n => encodePosNum n))) =     Num.pos n ** rw [decode_encodePosNum n] ** case pos n : PosNum \u22a2 (if         (match Num.pos n with           | Num.zero => []           | Num.pos n => encodePosNum n) =           [] then       Num.zero     else \u2191n) =     Num.pos n ** rw [PosNum.cast_to_num] ** case pos n : PosNum \u22a2 (if         (match Num.pos n with           | Num.zero => []           | Num.pos n => encodePosNum n) =           [] then       Num.zero     else Num.pos n) =     Num.pos n ** exact if_neg (encodePosNum_nonempty n) ** case zero  \u22a2 (if         (match Num.zero with           | Num.zero => []           | Num.pos n => encodePosNum n) =           [] then       Num.zero     else       \u2191(decodePosNum           (match Num.zero with           | Num.zero => []           | Num.pos n => encodePosNum n))) =     Num.zero ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Complex.continuous_circleTransformDeriv ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous (circleTransformDeriv R z w f) ** rw [circleTransformDeriv_eq] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w\u271d : \u2102 R : \u211d hR : 0 < R f : \u2102 \u2192 E z w : \u2102 hf : ContinuousOn f (sphere z R) hw : w \u2208 ball z R \u22a2 Continuous fun \u03b8 => (circleMap z R \u03b8 - w)\u207b\u00b9 \u2022 circleTransform R z w f \u03b8 ** exact (continuous_circleMap_inv hw).smul (continuous_circleTransform hR hf hw) ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_piCongrLeft ** \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 f : \u03b4' \u2243 \u03b4 \u22a2 Measurable \u2191(piCongrLeft \u03c0 f) ** rw [measurable_pi_iff] ** \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 f : \u03b4' \u2243 \u03b4 \u22a2 \u2200 (a : \u03b4), Measurable fun x => \u2191(piCongrLeft \u03c0 f) x a ** intro i ** \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 f : \u03b4' \u2243 \u03b4 i : \u03b4 \u22a2 Measurable fun x => \u2191(piCongrLeft \u03c0 f) x i ** simp_rw [piCongrLeft_apply_eq_cast] ** \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 \u03c0 : \u03b4 \u2192 Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a) inst\u271d : MeasurableSpace \u03b3 f : \u03b4' \u2243 \u03b4 i : \u03b4 \u22a2 Measurable fun x => cast (_ : \u03c0 (\u2191f (\u2191f.symm i)) = \u03c0 i) (x (\u2191f.symm i)) ** exact Measurable.eq_mp \u03c0 (f.apply_symm_apply i) <| measurable_pi_apply <| f.symm i ** Qed",
        "informal": ""
    },
    {
        "formal": "integrableOn_Ici_iff_integrableOn_Ioi ** \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": "Std.HashMap.Imp.Buckets.WF.update ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : USize d : AssocList \u03b1 \u03b2 h : USize.toNat i < Array.size buckets.val H : WF buckets h\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i]) \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d) h\u2082 :   AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i)       buckets.val[i] \u2192     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i) d \u22a2 WF (Buckets.update buckets i d h) ** refine \u27e8fun l hl => ?_, fun i hi p hp => ?_\u27e9 ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : USize d : AssocList \u03b1 \u03b2 h : USize.toNat i < Array.size buckets.val H : WF buckets h\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i]) \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d) h\u2082 :   AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i)       buckets.val[i] \u2192     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i) d inst\u271d\u00b9 : LawfulHashable \u03b1 inst\u271d : PartialEquivBEq \u03b1 l : AssocList \u03b1 \u03b2 hl : l \u2208 (Buckets.update buckets i d h).val.data \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList l) ** exact match List.mem_or_eq_of_mem_set hl with\n| .inl hl => H.1 _ hl\n| .inr rfl => h\u2081 (H.1 _ (Array.getElem_mem_data ..)) ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i\u271d : USize d : AssocList \u03b1 \u03b2 h : USize.toNat i\u271d < Array.size buckets.val H : WF buckets h\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d) h\u2082 :   AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)       buckets.val[i\u271d] \u2192     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d i : Nat hi : i < Array.size (Buckets.update buckets i\u271d d h).val p : \u03b1 \u00d7 \u03b2 hp : p \u2208 AssocList.toList (Buckets.update buckets i\u271d d h).val[i] \u22a2 (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size (Buckets.update buckets i\u271d d h).val) = i) p.fst p.snd ** revert hp ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i\u271d : USize d : AssocList \u03b1 \u03b2 h : USize.toNat i\u271d < Array.size buckets.val H : WF buckets h\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d) h\u2082 :   AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)       buckets.val[i\u271d] \u2192     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d i : Nat hi : i < Array.size (Buckets.update buckets i\u271d d h).val p : \u03b1 \u00d7 \u03b2 \u22a2 p \u2208 AssocList.toList (Buckets.update buckets i\u271d d h).val[i] \u2192     (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size (Buckets.update buckets i\u271d d h).val) = i) p.fst p.snd ** simp [update_data, Array.getElem_eq_data_get, List.get_set] ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i\u271d : USize d : AssocList \u03b1 \u03b2 h : USize.toNat i\u271d < Array.size buckets.val H : WF buckets h\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d) h\u2082 :   AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)       buckets.val[i\u271d] \u2192     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d i : Nat hi : i < Array.size (Buckets.update buckets i\u271d d h).val p : \u03b1 \u00d7 \u03b2 \u22a2 p \u2208       AssocList.toList         (if USize.toNat i\u271d = i then d         else List.get buckets.val.data { val := i, isLt := (_ : i < List.length buckets.val.data) }) \u2192     USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i ** split <;> intro hp ** case refine_2.inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i\u271d : USize d : AssocList \u03b1 \u03b2 h : USize.toNat i\u271d < Array.size buckets.val H : WF buckets h\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d) h\u2082 :   AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)       buckets.val[i\u271d] \u2192     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d i : Nat hi : i < Array.size (Buckets.update buckets i\u271d d h).val p : \u03b1 \u00d7 \u03b2 h\u271d : USize.toNat i\u271d = i hp : p \u2208 AssocList.toList d \u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i ** next eq => exact eq \u25b8 h\u2082 (H.2 _ _) _ hp ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i\u271d : USize d : AssocList \u03b1 \u03b2 h : USize.toNat i\u271d < Array.size buckets.val H : WF buckets h\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d) h\u2082 :   AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)       buckets.val[i\u271d] \u2192     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d i : Nat hi : i < Array.size (Buckets.update buckets i\u271d d h).val p : \u03b1 \u00d7 \u03b2 eq : USize.toNat i\u271d = i hp : p \u2208 AssocList.toList d \u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i ** exact eq \u25b8 h\u2082 (H.2 _ _) _ hp ** case refine_2.inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i\u271d : USize d : AssocList \u03b1 \u03b2 h : USize.toNat i\u271d < Array.size buckets.val H : WF buckets h\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d) h\u2082 :   AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)       buckets.val[i\u271d] \u2192     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d i : Nat hi : i < Array.size (Buckets.update buckets i\u271d d h).val p : \u03b1 \u00d7 \u03b2 h\u271d : \u00acUSize.toNat i\u271d = i hp : p \u2208 AssocList.toList (List.get buckets.val.data { val := i, isLt := (_ : i < List.length buckets.val.data) }) \u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i ** simp at hi ** case refine_2.inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i\u271d : USize d : AssocList \u03b1 \u03b2 h : USize.toNat i\u271d < Array.size buckets.val H : WF buckets h\u2081 :   \u2200 [inst : PartialEquivBEq \u03b1] [inst : LawfulHashable \u03b1],     List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList buckets.val[i\u271d]) \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList d) h\u2082 :   AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d)       buckets.val[i\u271d] \u2192     AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size buckets.val) = USize.toNat i\u271d) d i : Nat hi\u271d : i < Array.size (Buckets.update buckets i\u271d d h).val p : \u03b1 \u00d7 \u03b2 h\u271d : \u00acUSize.toNat i\u271d = i hp : p \u2208 AssocList.toList (List.get buckets.val.data { val := i, isLt := (_ : i < List.length buckets.val.data) }) hi : i < Array.size buckets.val \u22a2 USize.toNat (UInt64.toUSize (hash p.fst) % Array.size buckets.val) = i ** exact H.2 i hi _ hp ** Qed",
        "informal": ""
    },
    {
        "formal": "Measurable.exists_continuous ** \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f \u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1 ** obtain \u27e8b, b_count, -, hb\u27e9 :\n  \u2203 b : Set (Set (range f)), b.Countable \u2227 \u2205 \u2209 b \u2227 IsTopologicalBasis b :=\n  exists_countable_basis (range f) ** case intro.intro.intro \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f b : Set (Set \u2191(range f)) b_count : Set.Countable b hb : IsTopologicalBasis b \u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1 ** haveI : Countable b := b_count.to_subtype ** case intro.intro.intro \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f b : Set (Set \u2191(range f)) b_count : Set.Countable b hb : IsTopologicalBasis b this : Countable \u2191b \u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1 ** have : \u2200 s : b, IsClopenable (rangeFactorization f \u207b\u00b9' s) := fun s \u21a6 by\n  apply MeasurableSet.isClopenable\n  exact hf.subtype_mk (hb.isOpen s.2).measurableSet ** case intro.intro.intro \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f b : Set (Set \u2191(range f)) b_count : Set.Countable b hb : IsTopologicalBasis b this\u271d : Countable \u2191b this : \u2200 (s : \u2191b), IsClopenable (rangeFactorization f \u207b\u00b9' \u2191s) \u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1 ** choose T Tt Tpolish _ Topen using this ** case intro.intro.intro \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f b : Set (Set \u2191(range f)) b_count : Set.Countable b hb : IsTopologicalBasis b this : Countable \u2191b T : \u2191b \u2192 TopologicalSpace \u03b1 Tt : \u2200 (s : \u2191b), T s \u2264 t Tpolish : \u2200 (s : \u2191b), PolishSpace \u03b1 h\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s) Topen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s) \u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1 ** obtain \u27e8t', t'T, t't, t'_polish\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b1, (\u2200 i, t' \u2264 T i) \u2227 t' \u2264 t \u2227 @PolishSpace \u03b1 t' :=\n  exists_polishSpace_forall_le T Tt Tpolish ** case intro.intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f b : Set (Set \u2191(range f)) b_count : Set.Countable b hb : IsTopologicalBasis b this : Countable \u2191b T : \u2191b \u2192 TopologicalSpace \u03b1 Tt : \u2200 (s : \u2191b), T s \u2264 t Tpolish : \u2200 (s : \u2191b), PolishSpace \u03b1 h\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s) Topen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s) t' : TopologicalSpace \u03b1 t'T : \u2200 (i : \u2191b), t' \u2264 T i t't : t' \u2264 t t'_polish : PolishSpace \u03b1 \u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1 ** refine' \u27e8t', t't, _, t'_polish\u27e9 ** case intro.intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f b : Set (Set \u2191(range f)) b_count : Set.Countable b hb : IsTopologicalBasis b this : Countable \u2191b T : \u2191b \u2192 TopologicalSpace \u03b1 Tt : \u2200 (s : \u2191b), T s \u2264 t Tpolish : \u2200 (s : \u2191b), PolishSpace \u03b1 h\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s) Topen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s) t' : TopologicalSpace \u03b1 t'T : \u2200 (i : \u2191b), t' \u2264 T i t't : t' \u2264 t t'_polish : PolishSpace \u03b1 \u22a2 Continuous f ** have : Continuous[t', _] (rangeFactorization f) :=\n  hb.continuous _ fun s hs => t'T \u27e8s, hs\u27e9 _ (Topen \u27e8s, hs\u27e9) ** case intro.intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f b : Set (Set \u2191(range f)) b_count : Set.Countable b hb : IsTopologicalBasis b this\u271d : Countable \u2191b T : \u2191b \u2192 TopologicalSpace \u03b1 Tt : \u2200 (s : \u2191b), T s \u2264 t Tpolish : \u2200 (s : \u2191b), PolishSpace \u03b1 h\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s) Topen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s) t' : TopologicalSpace \u03b1 t'T : \u2200 (i : \u2191b), t' \u2264 T i t't : t' \u2264 t t'_polish : PolishSpace \u03b1 this : Continuous (rangeFactorization f) \u22a2 Continuous f ** exact continuous_subtype_val.comp this ** \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f b : Set (Set \u2191(range f)) b_count : Set.Countable b hb : IsTopologicalBasis b this : Countable \u2191b s : \u2191b \u22a2 IsClopenable (rangeFactorization f \u207b\u00b9' \u2191s) ** apply MeasurableSet.isClopenable ** case hs \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 \u03b2 : Type u_4 t : TopologicalSpace \u03b1 inst\u271d\u2075 : PolishSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : OpensMeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : SecondCountableTopology \u2191(range f) hf : Measurable f b : Set (Set \u2191(range f)) b_count : Set.Countable b hb : IsTopologicalBasis b this : Countable \u2191b s : \u2191b \u22a2 MeasurableSet (rangeFactorization f \u207b\u00b9' \u2191s) ** exact hf.subtype_mk (hb.isOpen s.2).measurableSet ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.inv_coe_unit ** n : \u2115 u : (ZMod n)\u02e3 \u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9 ** have := congr_arg ((\u2191) : \u2115 \u2192 ZMod n) (val_coe_unit_coprime u) ** n : \u2115 u : (ZMod n)\u02e3 this : \u2191(Nat.gcd (val \u2191u) n) = \u21911 \u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9 ** rw [\u2190 mul_inv_eq_gcd, Nat.cast_one] at this ** n : \u2115 u : (ZMod n)\u02e3 this : \u2191u * (\u2191u)\u207b\u00b9 = 1 \u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9 ** let u' : (ZMod n)\u02e3 := \u27e8u, (u : ZMod n)\u207b\u00b9, this, by rwa [mul_comm]\u27e9 ** n : \u2115 u : (ZMod n)\u02e3 this : \u2191u * (\u2191u)\u207b\u00b9 = 1 u' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) } \u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9 ** have h : u = u' := by\n  apply Units.ext\n  rfl ** n : \u2115 u : (ZMod n)\u02e3 this : \u2191u * (\u2191u)\u207b\u00b9 = 1 u' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) } h : u = u' \u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9 ** rw [h] ** n : \u2115 u : (ZMod n)\u02e3 this : \u2191u * (\u2191u)\u207b\u00b9 = 1 u' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) } h : u = u' \u22a2 (\u2191u')\u207b\u00b9 = \u2191u'\u207b\u00b9 ** rfl ** n : \u2115 u : (ZMod n)\u02e3 this : \u2191u * (\u2191u)\u207b\u00b9 = 1 \u22a2 (\u2191u)\u207b\u00b9 * \u2191u = 1 ** rwa [mul_comm] ** n : \u2115 u : (ZMod n)\u02e3 this : \u2191u * (\u2191u)\u207b\u00b9 = 1 u' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) } \u22a2 u = u' ** apply Units.ext ** case a n : \u2115 u : (ZMod n)\u02e3 this : \u2191u * (\u2191u)\u207b\u00b9 = 1 u' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) } \u22a2 \u2191u = \u2191u' ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.totalVariation_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 \u22a2 totalVariation 0 = 0 ** simp [totalVariation, toJordanDecomposition_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_null_frontier_thickening ** \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b \u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 ** have mbles : \u2200 r : \u211d, MeasurableSet (frontier (Metric.thickening r s)) :=\n  fun r => isClosed_frontier.measurableSet ** \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b mbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s)) \u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 ** have disjs := Metric.frontier_thickening_disjoint s ** \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b mbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s)) disjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s)) \u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 ** have key := Measure.countable_meas_pos_of_disjoint_iUnion (\u03bc := \u03bc) mbles disjs ** \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b mbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s)) disjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s)) key : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))} \u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 ** have aux := measure_diff_null (s\u2081 := Ioo a b) (Set.Countable.measure_zero key volume) ** \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b mbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s)) disjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s)) key : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))} aux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b) \u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 ** have len_pos : 0 < ENNReal.ofReal (b - a) := by simp only [hab, ENNReal.ofReal_pos, sub_pos] ** \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b mbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s)) disjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s)) key : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))} aux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b) len_pos : 0 < ENNReal.ofReal (b - a) \u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 ** rw [\u2190 Real.volume_Ioo, \u2190 aux] at len_pos ** \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b mbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s)) disjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s)) key : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))} aux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b) len_pos : 0 < \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) \u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 ** rcases nonempty_of_measure_ne_zero len_pos.ne.symm with \u27e8r, \u27e8r_in_Ioo, hr\u27e9\u27e9 ** case intro.intro \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b mbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s)) disjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s)) key : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))} aux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b) len_pos : 0 < \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) r : \u211d r_in_Ioo : r \u2208 Ioo a b hr : \u00acr \u2208 {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))} \u22a2 \u2203 r, r \u2208 Ioo a b \u2227 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 ** refine' \u27e8r, r_in_Ioo, _\u27e9 ** case intro.intro \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b mbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s)) disjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s)) key : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))} aux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b) len_pos : 0 < \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) r : \u211d r_in_Ioo : r \u2208 Ioo a b hr : \u00acr \u2208 {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))} \u22a2 \u2191\u2191\u03bc (frontier (Metric.thickening r s)) = 0 ** simpa only [mem_setOf_eq, not_lt, le_zero_iff] using hr ** \u03a9 : Type u_1 inst\u271d\u00b3 : PseudoEMetricSpace \u03a9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : SigmaFinite \u03bc s : Set \u03a9 a b : \u211d hab : a < b mbles : \u2200 (r : \u211d), MeasurableSet (frontier (Metric.thickening r s)) disjs : Pairwise (Disjoint on fun r => frontier (Metric.thickening r s)) key : Set.Countable {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))} aux : \u2191\u2191volume (Ioo a b \\ {i | 0 < \u2191\u2191\u03bc (frontier (Metric.thickening i s))}) = \u2191\u2191volume (Ioo a b) \u22a2 0 < ENNReal.ofReal (b - a) ** simp only [hab, ENNReal.ofReal_pos, sub_pos] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.trim_mkMetric ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e \u22a2 trim (mkMetric m) = mkMetric m ** simp only [mkMetric, mkMetric'.eq_iSup_nat, trim_iSup] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e \u22a2 \u2a06 i, trim (mkMetric'.pre (fun s => m (diam s)) (\u2191i)\u207b\u00b9) = \u2a06 n, mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9 ** congr 1 with n : 1 ** case e_s.h \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e n : \u2115 \u22a2 trim (mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9) = mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9 ** refine' mkMetric'.trim_pre _ (fun s => _) _ ** case e_s.h \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e n : \u2115 s : Set X \u22a2 m (diam (closure s)) = m (diam s) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.isOrdered_iff ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering inst\u271d : TransCmp cmp t : RBNode \u03b1 \u22a2 isOrdered cmp t none none = true \u2194 Ordered cmp t ** simp [isOrdered_iff'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PartrecToTM2.tr_eval ** c : Code v : List \u2115 \u22a2 eval (TM2.step tr) (init c v) = halt <$> Code.eval c v ** obtain \u27e8i, h\u2081, h\u2082\u27e9 := tr_init c v ** case intro.intro c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i \u22a2 eval (TM2.step tr) (init c v) = halt <$> Code.eval c v ** refine' Part.ext fun x => _ ** case intro.intro c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i x : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393') \u22a2 x \u2208 eval (TM2.step tr) (init c v) \u2194 x \u2208 halt <$> Code.eval c v ** rw [reaches_eval h\u2082.to_reflTransGen] ** case intro.intro c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i x : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393') \u22a2 x \u2208 eval (fun a => TM2.step tr a) i \u2194 x \u2208 halt <$> Code.eval c v ** simp [-TM2.step] ** case intro.intro c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i x : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393') \u22a2 x \u2208 eval (fun a => TM2.step tr a) i \u2194 \u2203 a, a \u2208 Code.eval c v \u2227 halt a = x ** refine' \u27e8fun h => _, _\u27e9 ** case intro.intro.refine'_1 c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i x : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393') h : x \u2208 eval (fun a => TM2.step tr a) i \u22a2 \u2203 a, a \u2208 Code.eval c v \u2227 halt a = x ** obtain \u27e8c, hc\u2081, hc\u2082\u27e9 := tr_eval_rev tr_respects h\u2081 h ** case intro.intro.refine'_1.intro.intro c\u271d : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c\u271d Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c\u271d v) i x : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393') h : x \u2208 eval (fun a => TM2.step tr a) i c : Cfg hc\u2081 : TrCfg c x hc\u2082 : c \u2208 eval step (stepNormal c\u271d Cont.halt v) \u22a2 \u2203 a, a \u2208 Code.eval c\u271d v \u2227 halt a = x ** simp [stepNormal_eval] at hc\u2082 ** case intro.intro.refine'_1.intro.intro c\u271d : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c\u271d Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c\u271d v) i x : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393') h : x \u2208 eval (fun a => TM2.step tr a) i c : Cfg hc\u2081 : TrCfg c x hc\u2082 : \u2203 a, a \u2208 Code.eval c\u271d v \u2227 Cfg.halt a = c \u22a2 \u2203 a, a \u2208 Code.eval c\u271d v \u2227 halt a = x ** obtain \u27e8v', hv, rfl\u27e9 := hc\u2082 ** case intro.intro.refine'_1.intro.intro.intro.intro c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i x : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393') h : x \u2208 eval (fun a => TM2.step tr a) i v' : List \u2115 hv : v' \u2208 Code.eval c v hc\u2081 : TrCfg (Cfg.halt v') x \u22a2 \u2203 a, a \u2208 Code.eval c v \u2227 halt a = x ** exact \u27e8_, hv, hc\u2081.symm\u27e9 ** case intro.intro.refine'_2 c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i x : TM2.Cfg (fun x => \u0393') \u039b' (Option \u0393') \u22a2 (\u2203 a, a \u2208 Code.eval c v \u2227 halt a = x) \u2192 x \u2208 eval (fun a => TM2.step tr a) i ** rintro \u27e8v', hv, rfl\u27e9 ** case intro.intro.refine'_2.intro.intro c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i v' : List \u2115 hv : v' \u2208 Code.eval c v \u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i ** have := Turing.tr_eval (b\u2081 := Cfg.halt v') tr_respects h\u2081 ** case intro.intro.refine'_2.intro.intro c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i v' : List \u2115 hv : v' \u2208 Code.eval c v this : Cfg.halt v' \u2208 eval step (stepNormal c Cont.halt v) \u2192 \u2203 b\u2082, TrCfg (Cfg.halt v') b\u2082 \u2227 b\u2082 \u2208 eval (TM2.step tr) i \u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i ** simp [stepNormal_eval, -TM2.step] at this ** case intro.intro.refine'_2.intro.intro c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i v' : List \u2115 hv : v' \u2208 Code.eval c v this : v' \u2208 Code.eval c v \u2192 \u2203 b\u2082, TrCfg (Cfg.halt v') b\u2082 \u2227 b\u2082 \u2208 eval (TM2.step tr) i \u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i ** obtain \u27e8_, \u27e8\u27e9, h\u27e9 := this hv ** case intro.intro.refine'_2.intro.intro.intro.intro.refl c : Code v : List \u2115 i : Cfg' h\u2081 : TrCfg (stepNormal c Cont.halt v) i h\u2082 : Reaches\u2081 (TM2.step tr) (init c v) i v' : List \u2115 hv : v' \u2208 Code.eval c v this : v' \u2208 Code.eval c v \u2192 \u2203 b\u2082, TrCfg (Cfg.halt v') b\u2082 \u2227 b\u2082 \u2208 eval (TM2.step tr) i h : halt v' \u2208 eval (TM2.step tr) i \u22a2 halt v' \u2208 eval (fun a => TM2.step tr a) i ** exact h ** Qed",
        "informal": ""
    },
    {
        "formal": "String.length_push ** s : String c : Char \u22a2 length (push s c) = length s + 1 ** rw [push, mk_length, List.length_append, List.length_singleton, Nat.succ.injEq] ** s : String c : Char \u22a2 List.length s.data = length s ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.multiset_prod_mem_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 t : Multiset \u03b9 f : \u03b9 \u2192 Set \u03b1 g : \u03b9 \u2192 \u03b1 hg : \u2200 (i : \u03b9), i \u2208 t \u2192 g i \u2208 f i \u22a2 Multiset.prod (Multiset.map g t) \u2208 Multiset.prod (Multiset.map f t) ** induction t using Quotient.inductionOn ** case h \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 g : \u03b9 \u2192 \u03b1 a\u271d : List \u03b9 hg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i \u22a2 Multiset.prod (Multiset.map g (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2208     Multiset.prod (Multiset.map f (Quotient.mk (List.isSetoid \u03b9) a\u271d)) ** simp_rw [Multiset.quot_mk_to_coe, Multiset.coe_map, Multiset.coe_prod] ** case h \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 g : \u03b9 \u2192 \u03b1 a\u271d : List \u03b9 hg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i \u22a2 List.prod (List.map g a\u271d) \u2208 List.prod (List.map f a\u271d) ** exact list_prod_mem_list_prod _ _ _ hg ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.swap_mem_mulAntidiagonal ** \u03b1 : Type u_1 inst\u271d : OrderedCancelCommMonoid \u03b1 s t : Set \u03b1 hs : Set.IsPwo s ht : Set.IsPwo t a : \u03b1 u : Set \u03b1 hu : Set.IsPwo u x : \u03b1 \u00d7 \u03b1 \u22a2 Prod.swap x \u2208 mulAntidiagonal hs ht a \u2194 x \u2208 mulAntidiagonal ht hs a ** simp only [mem_mulAntidiagonal, Prod.fst_swap, Prod.snd_swap, Set.swap_mem_mulAntidiagonal_aux,\n           Set.mem_mulAntidiagonal] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_mul_upcrossingsBefore_le_integral ** \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 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9 hfzero : 0 \u2264 f 0 hab : a < b \u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264     \u222b (x : \u03a9), Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x \u2202\u03bc ** rw [\u2190 integral_mul_left] ** \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 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9 hfzero : 0 \u2264 f 0 hab : a < b \u22a2 \u222b (a_1 : \u03a9), (b - a) * \u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc \u2264     \u222b (x : \u03a9), Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x \u2202\u03bc ** refine' integral_mono_of_nonneg _ ((hf.sum_upcrossingStrat_mul a b N).integrable N) _ ** 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 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9 hfzero : 0 \u2264 f 0 hab : a < b \u22a2 0 \u2264\u1d50[\u03bc] fun a_1 => (b - a) * \u2191(upcrossingsBefore a b f N a_1) ** exact eventually_of_forall fun \u03c9 => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _) ** 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 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9 hfzero : 0 \u2264 f 0 hab : a < b \u22a2 (fun a_1 => (b - a) * \u2191(upcrossingsBefore a b f N a_1)) \u2264\u1d50[\u03bc] fun x =>     Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x ** refine' eventually_of_forall fun \u03c9 => _ ** 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 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hfN : \u2200 (\u03c9 : \u03a9), a \u2264 f N \u03c9 hfzero : 0 \u2264 f 0 hab : a < b \u03c9 : \u03a9 \u22a2 (fun a_1 => (b - a) * \u2191(upcrossingsBefore a b f N a_1)) \u03c9 \u2264     (fun x => Finset.sum (Finset.range N) (fun k => upcrossingStrat a b f N k * (f (k + 1) - f k)) x) \u03c9 ** simpa using mul_upcrossingsBefore_le (hfN \u03c9) hab ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.pi'_pi ** \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 : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b9 : Encodable \u03b9 inst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) s : (i : \u03b9) \u2192 Set (\u03b1 i) \u22a2 \u2191\u2191(pi' \u03bc) (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** rw [pi'] ** \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 : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b9 : Encodable \u03b9 inst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) s : (i : \u03b9) \u2192 Set (\u03b1 i) \u22a2 \u2191\u2191(map (TProd.elim' (_ : \u2200 (x : \u03b9), x \u2208 sortedUniv \u03b9)) (Measure.tprod (sortedUniv \u03b9) \u03bc)) (Set.pi univ s) =     \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** simp only [TProd.elim'] ** \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 : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b9 : Encodable \u03b9 inst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) s : (i : \u03b9) \u2192 Set (\u03b1 i) \u22a2 \u2191\u2191(map (fun v i => TProd.elim v (_ : i \u2208 sortedUniv \u03b9)) (Measure.tprod (sortedUniv \u03b9) \u03bc)) (Set.pi univ s) =     \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** erw [\u2190 MeasurableEquiv.piMeasurableEquivTProd_symm_apply, MeasurableEquiv.map_apply,\n  MeasurableEquiv.piMeasurableEquivTProd_symm_apply, elim_preimage_pi, tprod_tprod _ \u03bc, \u2190\n  List.prod_toFinset, sortedUniv_toFinset] <;>\nexact sortedUniv_nodup \u03b9 ** 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 hmeas : StronglyMeasurable f \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 2 * ENNReal.ofReal \u03b5 ** obtain \u27e8M, hMpos, hM\u27e9 := hf.snorm_indicator_norm_ge_pos_le \u03bc hmeas h\u03b5 ** case 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 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} 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 2 * ENNReal.ofReal \u03b5 ** obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 :=\n  snorm_indicator_le_of_bound \u03bc (f := { x | \u2016f x\u2016 < M }.indicator f) hp_top h\u03b5 (by\n    intro x\n    rw [norm_indicator_eq_indicator_norm, Set.indicator_apply]\n    split_ifs with h\n    exacts [h, hMpos]) ** \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 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2200 (x : \u03b1), \u2016Set.indicator {x | \u2016f x\u2016 < M} f x\u2016 < ?m.136721 ** intro 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 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 x : \u03b1 \u22a2 \u2016Set.indicator {x | \u2016f x\u2016 < M} f x\u2016 < ?m.136721 ** rw [norm_indicator_eq_indicator_norm, Set.indicator_apply] ** \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 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 x : \u03b1 \u22a2 (if x \u2208 {x | \u2016f x\u2016 < M} then \u2016f x\u2016 else 0) < ?m.136721  \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 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u211d  \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 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u211d ** split_ifs with h ** 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 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 x : \u03b1 h : x \u2208 {x | \u2016f x\u2016 < M} \u22a2 \u2016f x\u2016 < ?m.136721  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 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 x : \u03b1 h : \u00acx \u2208 {x | \u2016f x\u2016 < M} \u22a2 0 < ?m.136721  \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 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u211d  \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 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u211d ** exacts [h, hMpos] ** case 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 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} 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 2 * ENNReal.ofReal \u03b5 ** refine' \u27e8\u03b4, h\u03b4pos, fun s hs h\u03bcs => _\u27e9 ** case 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 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} 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 2 * ENNReal.ofReal \u03b5 ** rw [(_ : f = { x : \u03b1 | M \u2264 \u2016f x\u2016\u208a }.indicator f + { x : \u03b1 | \u2016f x\u2016 < M }.indicator f)] ** case 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 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} 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 (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264     2 * ENNReal.ofReal \u03b5 ** rw [snorm_indicator_eq_snorm_restrict hs] ** case 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 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264     2 * ENNReal.ofReal \u03b5 ** refine' le_trans (snorm_add_le _ _ hp_one) _ ** case intro.intro.intro.intro.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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 AEStronglyMeasurable (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) (Measure.restrict \u03bc s) ** exact StronglyMeasurable.aestronglyMeasurable\n  (hmeas.indicator (measurableSet_le measurable_const hmeas.nnnorm.measurable.subtype_coe)) ** case intro.intro.intro.intro.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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 AEStronglyMeasurable (Set.indicator {x | \u2016f x\u2016 < M} f) (Measure.restrict \u03bc s) ** exact StronglyMeasurable.aestronglyMeasurable\n  (hmeas.indicator (measurableSet_lt hmeas.nnnorm.measurable.subtype_coe measurable_const)) ** case intro.intro.intro.intro.refine'_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 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p (Measure.restrict \u03bc s) +       snorm (Set.indicator {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264     2 * ENNReal.ofReal \u03b5 ** rw [two_mul] ** case intro.intro.intro.intro.refine'_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 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p (Measure.restrict \u03bc s) +       snorm (Set.indicator {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264     ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b5 ** refine' add_le_add (le_trans (snorm_mono_measure _ Measure.restrict_le_self) hM) _ ** case intro.intro.intro.intro.refine'_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 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} 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 {x | \u2016f x\u2016 < M} f) p (Measure.restrict \u03bc s) \u2264 ENNReal.ofReal \u03b5 ** rw [\u2190 snorm_indicator_eq_snorm_restrict hs] ** case intro.intro.intro.intro.refine'_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 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} 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 (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** exact h\u03b4 s hs h\u03bcs ** \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 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 f = Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f ** ext x ** 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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 x : \u03b1 \u22a2 f x = (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) x ** by_cases hx : M \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 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 x : \u03b1 hx : M \u2264 \u2016f x\u2016 \u22a2 f x = (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) x ** rw [Pi.add_apply, Set.indicator_of_mem, Set.indicator_of_not_mem, add_zero] <;> simpa ** 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 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 M : \u211d hMpos : 0 < M hM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5 \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 (Set.indicator {x | \u2016f x\u2016 < M} f)) p \u03bc \u2264 ENNReal.ofReal \u03b5 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 x : \u03b1 hx : \u00acM \u2264 \u2016f x\u2016 \u22a2 f x = (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f + Set.indicator {x | \u2016f x\u2016 < M} f) x ** rw [Pi.add_apply, Set.indicator_of_not_mem, Set.indicator_of_mem, zero_add] <;>\n  simpa using hx ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PartrecToTM2.tr_ret_respects ** k : Cont v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet k v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }         b\u2082 ** induction k generalizing v s ** case halt v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet Cont.halt v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont Cont.halt)), var := s, stk := elim (trList v) [] [] (trContStack Cont.halt) } b\u2082  case cons\u2081 a\u271d\u00b2 : Code a\u271d\u00b9 : List \u2115 a\u271d : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet a\u271d v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 a\u271d\u00b2 a\u271d\u00b9 a\u271d)) }         b\u2082  case cons\u2082 a\u271d\u00b9 : List \u2115 a\u271d : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet a\u271d v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.cons\u2082 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.cons\u2082 a\u271d\u00b9 a\u271d))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 a\u271d\u00b9 a\u271d)) }         b\u2082  case comp a\u271d\u00b9 : Code a\u271d : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet a\u271d v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.comp a\u271d\u00b9 a\u271d) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.comp a\u271d\u00b9 a\u271d))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.comp a\u271d\u00b9 a\u271d)) }         b\u2082  case fix a\u271d\u00b9 : Code a\u271d : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet a\u271d v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }         b\u2082 ** case halt => exact \u27e8_, rfl, TransGen.single rfl\u27e9 ** case cons\u2082 a\u271d\u00b9 : List \u2115 a\u271d : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet a\u271d v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.cons\u2082 a\u271d\u00b9 a\u271d) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.cons\u2082 a\u271d\u00b9 a\u271d))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 a\u271d\u00b9 a\u271d)) }         b\u2082  case comp a\u271d\u00b9 : Code a\u271d : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet a\u271d v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.comp a\u271d\u00b9 a\u271d) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.comp a\u271d\u00b9 a\u271d))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.comp a\u271d\u00b9 a\u271d)) }         b\u2082  case fix a\u271d\u00b9 : Code a\u271d : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet a\u271d v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }         b\u2082 ** case cons\u2082 ns k IH =>\n  obtain \u27e8c, h\u2081, h\u2082\u27e9 := IH (ns.headI :: v) none\n  exact \u27e8c, h\u2081, TransGen.head rfl <| head_stack_ok.trans h\u2082\u27e9 ** case comp a\u271d\u00b9 : Code a\u271d : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet a\u271d v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.comp a\u271d\u00b9 a\u271d) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.comp a\u271d\u00b9 a\u271d))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.comp a\u271d\u00b9 a\u271d)) }         b\u2082  case fix a\u271d\u00b9 : Code a\u271d : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet a\u271d v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr)           { l := some (\u039b'.ret (trCont a\u271d)), var := s, stk := elim (trList v) [] [] (trContStack a\u271d) } b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.fix a\u271d\u00b9 a\u271d) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.fix a\u271d\u00b9 a\u271d))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.fix a\u271d\u00b9 a\u271d)) }         b\u2082 ** case comp f k _ =>\n  obtain \u27e8s', h\u2081, h\u2082\u27e9 := trNormal_respects f k v s\n  exact \u27e8_, h\u2081, TransGen.head rfl h\u2082\u27e9 ** v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet Cont.halt v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont Cont.halt)), var := s, stk := elim (trList v) [] [] (trContStack Cont.halt) } b\u2082 ** exact \u27e8_, rfl, TransGen.single rfl\u27e9 ** fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.cons\u2081 fs as k) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.cons\u2081 fs as k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) }         b\u2082 ** obtain \u27e8s', h\u2081, h\u2082\u27e9 := trNormal_respects fs (Cont.cons\u2082 v k) as none ** case intro.intro fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.cons\u2081 fs as k) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.cons\u2081 fs as k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) }         b\u2082 ** refine' \u27e8s', h\u2081, TransGen.head rfl _\u27e9 ** case intro.intro fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     (TM2.stepAux (tr (\u039b'.ret (trCont (Cont.cons\u2081 fs as k)))) s       (elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k))))     s' ** simp ** case intro.intro fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     {       l :=         some           (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 fs (Cont'.cons\u2082 (trCont k)))))),       var := s, stk := elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) }     s' ** refine' (move\u2082_ok (by decide) _ (splitAtPred_false _)).trans _ ** case intro.intro.refine'_2 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     {       l :=         some           (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main             (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k))))),       var := none,       stk :=         update           (update (elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k))) main (Option.elim none id List.cons []))           aux           (elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) main ++             elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) aux) }     s' ** simp only [TM2.step, Option.mem_def, Option.elim, id_eq, elim_update_main, elim_main, elim_aux,\n  List.append_nil, elim_update_aux] ** case intro.intro.refine'_2 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     {       l :=         some           (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main             (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k))))),       var := none, stk := elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) }     s' ** refine' (move\u2082_ok (by decide) _ _).trans _ ** case intro.intro.refine'_2.refine'_1 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 List \u0393'  case intro.intro.refine'_2.refine'_2 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 Option \u0393'  case intro.intro.refine'_2.refine'_3 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 List \u0393'  case intro.intro.refine'_2.refine'_4 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) rev = []  case intro.intro.refine'_2.refine'_5 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 splitAtPred (fun s => decide (s = \u0393'.cons\u2097)) (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) stack) =     (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3)  case intro.intro.refine'_2.refine'_6 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     { l := some (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k)))), var := none,       stk :=         update           (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack             (Option.elim ?intro.intro.refine'_2.refine'_2 id List.cons ?intro.intro.refine'_2.refine'_3))           main (?intro.intro.refine'_2.refine'_1 ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) }     s' ** pick_goal 4 ** case intro.intro.refine'_2.refine'_1 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 List \u0393'  case intro.intro.refine'_2.refine'_2 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 Option \u0393'  case intro.intro.refine'_2.refine'_3 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 List \u0393'  case intro.intro.refine'_2.refine'_5 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 splitAtPred (fun s => decide (s = \u0393'.cons\u2097)) (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) stack) =     (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3)  case intro.intro.refine'_2.refine'_6 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     { l := some (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k)))), var := none,       stk :=         update           (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack             (Option.elim ?intro.intro.refine'_2.refine'_2 id List.cons ?intro.intro.refine'_2.refine'_3))           main (?intro.intro.refine'_2.refine'_1 ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) }     s' ** pick_goal 4 ** case intro.intro.refine'_2.refine'_6 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     { l := some (move\u2082 (fun x => false) aux stack (trNormal fs (Cont'.cons\u2082 (trCont k)))), var := none,       stk :=         update           (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack             (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))           main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) }     s' ** refine' (move\u2082_ok (by decide) _ (splitAtPred_false _)).trans _ ** case intro.intro.refine'_2.refine'_6.refine'_2 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     { l := some (trNormal fs (Cont'.cons\u2082 (trCont k))), var := none,       stk :=         update           (update             (update               (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack                 (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))               main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main))             aux (Option.elim none id List.cons []))           stack           (update               (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack                 (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))               main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) aux ++             update               (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack                 (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))               main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) stack) }     s' ** simp only [TM2.step, Option.mem_def, Option.elim, elim_update_stack, elim_main,\n  List.append_nil, elim_update_main,  id_eq, elim_update_aux, ne_eq, Function.update_noteq,\n  elim_aux, elim_stack] ** case intro.intro.refine'_2.refine'_6.refine'_2 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 TransGen     (fun a b =>       (match a with         | { l := none, var := var, stk := stk } => none         | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =         some b)     { l := some (trNormal fs (Cont'.cons\u2082 (trCont k))), var := none,       stk := elim (trList as) [] [] (trList v ++ \u0393'.cons\u2097 :: trLList (contStack k)) }     s' ** exact h\u2082 ** fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 main \u2260 rev \u2227 aux \u2260 rev \u2227 main \u2260 aux ** decide ** case intro.intro.refine'_1 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 elim (trList v) [] [] (trContStack (Cont.cons\u2081 fs as k)) rev = [] ** rfl ** fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 stack \u2260 rev \u2227 main \u2260 rev \u2227 stack \u2260 main ** decide ** case intro.intro.refine'_2.refine'_4 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) rev = [] ** rfl ** case intro.intro.refine'_2.refine'_5 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 splitAtPred (fun s => decide (s = \u0393'.cons\u2097)) (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) stack) =     (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3) ** exact\n  splitAtPred_eq _ _ _ (some \u0393'.cons\u2097) _\n    (fun x h => Bool.decide_false (trList_ne_cons\u2097 _ _ h)) \u27e8rfl, rfl\u27e9 ** fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 aux \u2260 rev \u2227 stack \u2260 rev \u2227 aux \u2260 stack ** decide ** case intro.intro.refine'_2.refine'_6.refine'_1 fs : Code as : List \u2115 k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal fs (Cont.cons\u2082 v k) as) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal fs (trCont (Cont.cons\u2082 v k))), var := none,       stk := elim (trList as) [] [] (trContStack (Cont.cons\u2082 v k)) }     s' \u22a2 update       (update (elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k))) stack         (Option.elim (some \u0393'.cons\u2097) id List.cons (trLList (contStack k))))       main (trList as ++ elim [] [] (trList v) (trContStack (Cont.cons\u2081 fs as k)) main) rev =     [] ** rfl ** ns : List \u2115 k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.cons\u2082 ns k) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.cons\u2082 ns k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 ns k)) }         b\u2082 ** obtain \u27e8c, h\u2081, h\u2082\u27e9 := IH (ns.headI :: v) none ** case intro.intro ns : List \u2115 k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' c : Cfg' h\u2081 : TrCfg (stepRet k (List.headI ns :: v)) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (\u039b'.ret (trCont k)), var := none, stk := elim (trList (List.headI ns :: v)) [] [] (trContStack k) } c \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.cons\u2082 ns k) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.cons\u2082 ns k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.cons\u2082 ns k)) }         b\u2082 ** exact \u27e8c, h\u2081, TransGen.head rfl <| head_stack_ok.trans h\u2082\u27e9 ** f : Code k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.comp f k) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.comp f k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.comp f k)) }         b\u2082 ** obtain \u27e8s', h\u2081, h\u2082\u27e9 := trNormal_respects f k v s ** case intro.intro f : Code k : Cont a_ih\u271d :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' s' : Cfg' h\u2081 : TrCfg (stepNormal f k v) s' h\u2082 :   Reaches\u2081 (TM2.step tr) { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }     s' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.comp f k) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.comp f k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.comp f k)) }         b\u2082 ** exact \u27e8_, h\u2081, TransGen.head rfl h\u2082\u27e9 ** f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' \u22a2 \u2203 b\u2082,     TrCfg (stepRet (Cont.fix f k) v) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }         b\u2082 ** rw [stepRet] ** f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' this :   if List.headI v = 0 then     natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)   else     natEnd (Option.iget (List.head? (trList v))) = false \u2227       List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) \u22a2 \u2203 b\u2082,     TrCfg (if List.headI v = 0 then stepRet k (List.tail v) else stepNormal f (Cont.fix f k) (List.tail v)) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }         b\u2082 ** by_cases v.headI = 0 <;> simp only [h, ite_true, ite_false] at this \u22a2 ** f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' \u22a2 if List.headI v = 0 then     natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v)   else     natEnd (Option.iget (List.head? (trList v))) = false \u2227       List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) ** cases' v with n ** case cons f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 s : Option \u0393' n : \u2115 tail\u271d : List \u2115 \u22a2 if List.headI (n :: tail\u271d) = 0 then     natEnd (Option.iget (List.head? (trList (n :: tail\u271d)))) = true \u2227       List.tail (trList (n :: tail\u271d)) = trList (List.tail (n :: tail\u271d))   else     natEnd (Option.iget (List.head? (trList (n :: tail\u271d)))) = false \u2227       List.tail (trList (n :: tail\u271d)) =         List.tail (trNat (List.headI (n :: tail\u271d))) ++ \u0393'.cons :: trList (List.tail (n :: tail\u271d)) ** cases' n with n ** case cons.succ f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 s : Option \u0393' tail\u271d : List \u2115 n : \u2115 \u22a2 if List.headI (Nat.succ n :: tail\u271d) = 0 then     natEnd (Option.iget (List.head? (trList (Nat.succ n :: tail\u271d)))) = true \u2227       List.tail (trList (Nat.succ n :: tail\u271d)) = trList (List.tail (Nat.succ n :: tail\u271d))   else     natEnd (Option.iget (List.head? (trList (Nat.succ n :: tail\u271d)))) = false \u2227       List.tail (trList (Nat.succ n :: tail\u271d)) =         List.tail (trNat (List.headI (Nat.succ n :: tail\u271d))) ++ \u0393'.cons :: trList (List.tail (Nat.succ n :: tail\u271d)) ** rw [trList, List.headI, trNat, Nat.cast_succ, Num.add_one, Num.succ, List.tail] ** case cons.succ f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 s : Option \u0393' tail\u271d : List \u2115 n : \u2115 \u22a2 if       (match Nat.succ n :: tail\u271d with         | [] => default         | a :: tail => a) =         0 then     natEnd (Option.iget (List.head? (trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d))) = true \u2227       (match trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d with         | [] => []         | head :: as => as) =         trList (List.tail (Nat.succ n :: tail\u271d))   else     natEnd (Option.iget (List.head? (trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d))) = false \u2227       (match trNum (Num.pos (Num.succ' \u2191n)) ++ \u0393'.cons :: trList tail\u271d with         | [] => []         | head :: as => as) =         List.tail (trNum (Num.pos (Num.succ' \u2191n))) ++ \u0393'.cons :: trList (List.tail (Nat.succ n :: tail\u271d)) ** cases (n : Num).succ' <;> exact \u27e8rfl, rfl\u27e9 ** case nil f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 s : Option \u0393' \u22a2 if List.headI [] = 0 then     natEnd (Option.iget (List.head? (trList []))) = true \u2227 List.tail (trList []) = trList (List.tail [])   else     natEnd (Option.iget (List.head? (trList []))) = false \u2227       List.tail (trList []) = List.tail (trNat (List.headI [])) ++ \u0393'.cons :: trList (List.tail []) ** exact \u27e8rfl, rfl\u27e9 ** case cons.zero f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 s : Option \u0393' tail\u271d : List \u2115 \u22a2 if List.headI (Nat.zero :: tail\u271d) = 0 then     natEnd (Option.iget (List.head? (trList (Nat.zero :: tail\u271d)))) = true \u2227       List.tail (trList (Nat.zero :: tail\u271d)) = trList (List.tail (Nat.zero :: tail\u271d))   else     natEnd (Option.iget (List.head? (trList (Nat.zero :: tail\u271d)))) = false \u2227       List.tail (trList (Nat.zero :: tail\u271d)) =         List.tail (trNat (List.headI (Nat.zero :: tail\u271d))) ++ \u0393'.cons :: trList (List.tail (Nat.zero :: tail\u271d)) ** simp ** case pos f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : List.headI v = 0 this : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v) \u22a2 \u2203 b\u2082,     TrCfg (stepRet k (List.tail v)) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }         b\u2082 ** obtain \u27e8c, h\u2081, h\u2082\u27e9 := IH v.tail (trList v).head? ** case pos.intro.intro f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : List.headI v = 0 this : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v) c : Cfg' h\u2081 : TrCfg (stepRet k (List.tail v)) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),       stk := elim (trList (List.tail v)) [] [] (trContStack k) }     c \u22a2 \u2203 b\u2082,     TrCfg (stepRet k (List.tail v)) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }         b\u2082 ** refine' \u27e8c, h\u2081, TransGen.head rfl _\u27e9 ** case pos.intro.intro f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : List.headI v = 0 this : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v) c : Cfg' h\u2081 : TrCfg (stepRet k (List.tail v)) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),       stk := elim (trList (List.tail v)) [] [] (trContStack k) }     c \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     (TM2.stepAux (tr (\u039b'.ret (trCont (Cont.fix f k)))) s (elim (trList v) [] [] (trContStack (Cont.fix f k)))) c ** simp only [Option.mem_def, TM2.stepAux, trContStack, contStack, elim_main, this, cond_true,\n  elim_update_main] ** case pos.intro.intro f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : List.headI v = 0 this : natEnd (Option.iget (List.head? (trList v))) = true \u2227 List.tail (trList v) = trList (List.tail v) c : Cfg' h\u2081 : TrCfg (stepRet k (List.tail v)) c h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),       stk := elim (trList (List.tail v)) [] [] (trContStack k) }     c \u22a2 TransGen (fun a b => TM2.step tr a = some b)     { l := some (\u039b'.ret (trCont k)), var := List.head? (trList v),       stk := elim (trList (List.tail v)) [] [] (trLList (contStack k)) }     c ** exact h\u2082 ** case neg f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : \u00acList.headI v = 0 this :   natEnd (Option.iget (List.head? (trList v))) = false \u2227     List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) \u22a2 \u2203 b\u2082,     TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }         b\u2082 ** obtain \u27e8s', h\u2081, h\u2082\u27e9 := trNormal_respects f (Cont.fix f k) v.tail (some \u0393'.cons) ** case neg.intro.intro f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : \u00acList.headI v = 0 this :   natEnd (Option.iget (List.head? (trList v))) = false \u2227     List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) s' : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,       stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }     s' \u22a2 \u2203 b\u2082,     TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) b\u2082 \u2227       Reaches\u2081 (TM2.step tr)         { l := some (\u039b'.ret (trCont (Cont.fix f k))), var := s,           stk := elim (trList v) [] [] (trContStack (Cont.fix f k)) }         b\u2082 ** refine' \u27e8_, h\u2081, TransGen.head rfl <| TransGen.trans _ h\u2082\u27e9 ** case neg.intro.intro f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : \u00acList.headI v = 0 this :   natEnd (Option.iget (List.head? (trList v))) = false \u2227     List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) s' : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,       stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }     s' \u22a2 TransGen (fun a b => b \u2208 TM2.step tr a)     (TM2.stepAux (tr (\u039b'.ret (trCont (Cont.fix f k)))) s (elim (trList v) [] [] (trContStack (Cont.fix f k))))     { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,       stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) } ** simp only [Option.mem_def, TM2.stepAux, elim_main, this.1, cond_false, elim_update_main,\n  trCont] ** case neg.intro.intro f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : \u00acList.headI v = 0 this :   natEnd (Option.iget (List.head? (trList v))) = false \u2227     List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) s' : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,       stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }     s' \u22a2 TransGen (fun a b => TM2.step tr a = some b)     { l := some (\u039b'.clear natEnd main (trNormal f (Cont'.fix f (trCont k)))), var := List.head? (trList v),       stk := elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k)) }     { l := some (trNormal f (Cont'.fix f (trCont k))), var := some \u0393'.cons,       stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) } ** convert clear_ok (splitAtPred_eq _ _ (trNat v.headI).tail (some \u0393'.cons) _ _ _) using 2 ** case h.e'_2.h.e'_7 f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : \u00acList.headI v = 0 this :   natEnd (Option.iget (List.head? (trList v))) = false \u2227     List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) s' : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,       stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }     s' \u22a2 elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) =     update (elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k))) main ?neg.intro.intro.convert_6\u271d ** simp ** case h.e'_2.h.e'_7 f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : \u00acList.headI v = 0 this :   natEnd (Option.iget (List.head? (trList v))) = false \u2227     List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) s' : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,       stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }     s' \u22a2 elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) =     elim ?neg.intro.intro.convert_6\u271d [] [] (trContStack (Cont.fix f k)) ** convert rfl ** case neg.intro.intro.convert_7 f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : \u00acList.headI v = 0 this :   natEnd (Option.iget (List.head? (trList v))) = false \u2227     List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) s' : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,       stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }     s' \u22a2 \u2200 (x : \u0393'), x \u2208 List.tail (trNat (List.headI v)) \u2192 natEnd x = false ** exact fun x h => trNat_natEnd _ _ (List.tail_subset _ h) ** case neg.intro.intro.convert_8 f : Code k : Cont IH :   \u2200 (v : List \u2115) (s : Option \u0393'),     \u2203 b\u2082,       TrCfg (stepRet k v) b\u2082 \u2227         Reaches\u2081 (TM2.step tr) { l := some (\u039b'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }           b\u2082 v : List \u2115 s : Option \u0393' h : \u00acList.headI v = 0 this :   natEnd (Option.iget (List.head? (trList v))) = false \u2227     List.tail (trList v) = List.tail (trNat (List.headI v)) ++ \u0393'.cons :: trList (List.tail v) s' : Cfg' h\u2081 : TrCfg (stepNormal f (Cont.fix f k) (List.tail v)) s' h\u2082 :   Reaches\u2081 (TM2.step tr)     { l := some (trNormal f (trCont (Cont.fix f k))), var := some \u0393'.cons,       stk := elim (trList (List.tail v)) [] [] (trContStack (Cont.fix f k)) }     s' \u22a2 Option.elim'     (elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k)) main = List.tail (trNat (List.headI v)) \u2227       trList (List.tail v) = [])     (fun a =>       natEnd a = true \u2227         elim (List.tail (trList v)) [] [] (trContStack (Cont.fix f k)) main =           List.tail (trNat (List.headI v)) ++ a :: trList (List.tail v))     (some \u0393'.cons) ** exact \u27e8rfl, this.2\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "WithTop.image_coe_Iio ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Iio a = Iio \u2191a ** rw [\u2190 preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Iio_subset_Iio le_top)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.negOfNat_mul_ofNat ** m n : Nat \u22a2 negOfNat m * \u2191n = negOfNat (m * n) ** rw [Int.mul_comm] ** m n : Nat \u22a2 \u2191n * negOfNat m = negOfNat (m * n) ** simp [ofNat_mul_negOfNat, Nat.mul_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.divInt_sub_divInt ** n\u2081 n\u2082 d\u2081 d\u2082 : Int z\u2081 : d\u2081 \u2260 0 z\u2082 : d\u2082 \u2260 0 \u22a2 n\u2081 /. d\u2081 - n\u2082 /. d\u2082 = (n\u2081 * d\u2082 - n\u2082 * d\u2081) /. (d\u2081 * d\u2082) ** simp only [Rat.sub_eq_add_neg, neg_divInt,\n  divInt_add_divInt _ _ z\u2081 z\u2082, Int.neg_mul, Int.sub_eq_add_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.ae_eq_of_forall_set_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' hm : m \u2264 m0 f g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc \u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191g ** suffices h_sub : \u21d1(f - g) =\u1d50[\u03bc] 0 ** case h_sub \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 g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc \u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0 ** have hfg' : \u2200 s : Set \u03b1, MeasurableSet[m] s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 := by\n  intro s hs h\u03bcs\n  rw [integral_congr_ae (ae_restrict_of_ae (Lp.coeFn_sub f g))]\n  rw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs)]\n  exact sub_eq_zero.mpr (hfg s hs h\u03bcs) ** case h_sub \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 g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc hfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0 \u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0 ** have hfg_int : \u2200 s, MeasurableSet[m] s \u2192 \u03bc s < \u221e \u2192 IntegrableOn (\u21d1(f - g)) s \u03bc := by\n  intro s hs h\u03bcs\n  rw [IntegrableOn, integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub f g))]\n  exact (hf_int_finite s hs h\u03bcs).sub (hg_int_finite s hs h\u03bcs) ** case h_sub \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 g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc hfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0 hfg_int : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(f - g)) s \u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0 ** have hfg_meas : AEStronglyMeasurable' m (\u21d1(f - g)) \u03bc :=\n  AEStronglyMeasurable'.congr (hf_meas.sub hg_meas) (Lp.coeFn_sub f g).symm ** case h_sub \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 g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc hfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0 hfg_int : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(f - g)) s hfg_meas : AEStronglyMeasurable' m (\u2191\u2191(f - g)) \u03bc \u22a2 \u2191\u2191(f - g) =\u1d50[\u03bc] 0 ** exact\n  Lp.ae_eq_zero_of_forall_set_integral_eq_zero' \ud835\udd5c hm (f - g) hp_ne_zero hp_ne_top hfg_int hfg'\n    hfg_meas ** \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 g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc h_sub : \u2191\u2191(f - g) =\u1d50[\u03bc] 0 \u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191g ** rw [\u2190 sub_ae_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 g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc h_sub : \u2191\u2191(f - g) =\u1d50[\u03bc] 0 \u22a2 \u2191\u2191f - \u2191\u2191g =\u1d50[\u03bc] 0 ** exact (Lp.coeFn_sub f g).symm.trans h_sub ** \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 g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0 ** intro s 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' hm : m \u2264 m0 f g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0 ** rw [integral_congr_ae (ae_restrict_of_ae (Lp.coeFn_sub f g))] ** \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 g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (a : \u03b1) in s, (\u2191\u2191f - \u2191\u2191g) a \u2202\u03bc = 0 ** rw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s 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' hm : m \u2264 m0 f g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (a : \u03b1) in s, \u2191\u2191f a \u2202\u03bc - \u222b (a : \u03b1) in s, \u2191\u2191g a \u2202\u03bc = 0 ** exact sub_eq_zero.mpr (hfg s 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' hm : m \u2264 m0 f g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc hfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0 \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(f - g)) s ** intro s 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' hm : m \u2264 m0 f g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc hfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 IntegrableOn (\u2191\u2191(f - g)) s ** rw [IntegrableOn, integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub f g))] ** \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 g : { 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 hg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191g) s hfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191g x \u2202\u03bc hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc hg_meas : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc hfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191(f - g) x \u2202\u03bc = 0 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 Integrable fun x => (\u2191\u2191f - \u2191\u2191g) x ** exact (hf_int_finite s hs h\u03bcs).sub (hg_int_finite s hs h\u03bcs) ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.pderiv_X ** R : Type u \u03c3 : Type v a a' a\u2081 a\u2082 : R s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : DecidableEq \u03c3 i j : \u03c3 \u22a2 \u2191(pderiv i) (X j) = Pi.single i 1 j ** rw [pderiv_def, mkDerivation_X] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Ordered.map ** \u03b1 : Type u_1 \u03b2 : Type u_2 cmp\u03b1 : \u03b1 \u2192 \u03b1 \u2192 Ordering cmp\u03b2 : \u03b2 \u2192 \u03b2 \u2192 Ordering f : \u03b1 \u2192 \u03b2 inst\u271d : IsMonotone cmp\u03b1 cmp\u03b2 f c\u271d : RBColor a : RBNode \u03b1 x : \u03b1 b : RBNode \u03b1 ax : All (fun x_1 => cmpLT cmp\u03b1 x_1 x) a xb : All (fun x_1 => cmpLT cmp\u03b1 x x_1) b ha : Ordered cmp\u03b1 a hb : Ordered cmp\u03b1 b \u22a2 Ordered cmp\u03b2 (map f (node c\u271d a x b)) ** refine \u27e8ax.map ?_, xb.map ?_, ha.map f, hb.map f\u27e9 <;> exact IsMonotone.lt_mono ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.biSup_measure_Iic ** \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\u271d s' t : Set \u03b1 inst\u271d : Preorder \u03b1 s : Set \u03b1 hsc : Set.Countable s hst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y hdir : DirectedOn (fun x x_1 => x \u2264 x_1) s \u22a2 \u2a06 x \u2208 s, \u2191\u2191\u03bc (Iic x) = \u2191\u2191\u03bc univ ** rw [\u2190 measure_biUnion_eq_iSup hsc] ** \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\u271d s' t : Set \u03b1 inst\u271d : Preorder \u03b1 s : Set \u03b1 hsc : Set.Countable s hst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y hdir : DirectedOn (fun x x_1 => x \u2264 x_1) s \u22a2 \u2191\u2191\u03bc (\u22c3 i \u2208 s, Iic i) = \u2191\u2191\u03bc univ ** congr ** case e_a \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\u271d s' t : Set \u03b1 inst\u271d : Preorder \u03b1 s : Set \u03b1 hsc : Set.Countable s hst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y hdir : DirectedOn (fun x x_1 => x \u2264 x_1) s \u22a2 \u22c3 i \u2208 s, Iic i = univ ** simp only [\u2190 bex_def] at hst ** case e_a \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\u271d s' t : Set \u03b1 inst\u271d : Preorder \u03b1 s : Set \u03b1 hsc : Set.Countable s hdir : DirectedOn (fun x x_1 => x \u2264 x_1) s hst : \u2200 (x : \u03b1), \u2203 x_1 x_2, x \u2264 x_1 \u22a2 \u22c3 i \u2208 s, Iic i = univ ** exact iUnion\u2082_eq_univ_iff.2 hst ** \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\u271d s' t : Set \u03b1 inst\u271d : Preorder \u03b1 s : Set \u03b1 hsc : Set.Countable s hst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y hdir : DirectedOn (fun x x_1 => x \u2264 x_1) s \u22a2 DirectedOn ((fun x x_1 => x \u2286 x_1) on fun x => Iic x) s ** exact directedOn_iff_directed.2 (hdir.directed_val.mono_comp _ fun x y => Iic_subset_Iic.2) ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.lcm_one_right ** i : \u2124 \u22a2 lcm i 1 = natAbs i ** rw [Int.lcm] ** i : \u2124 \u22a2 Nat.lcm (natAbs i) (natAbs 1) = natAbs i ** apply Nat.lcm_one_right ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_stronglyMeasurable_mul_of_bound ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m] ** let fs := hf.approxBounded c ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m] ** have hfs_tendsto : \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) :=\n  hf.tendsto_approxBounded_ae hf_bound ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m] ** by_cases h\u03bc : \u03bc = 0 ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m] ** have : \u03bc.ae.NeBot := by simp only [h\u03bc, ae_neBot, Ne.def, not_false_iff] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this : NeBot (Measure.ae \u03bc) \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m] ** have hc : 0 \u2264 c :=\n  haveI h_exists : \u2203 x, \u2016f x\u2016 \u2264 c := Eventually.exists hf_bound\n  (norm_nonneg _).trans h_exists.choose_spec ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m] ** have hfs_bound : \u2200 n x, \u2016fs n x\u2016 \u2264 c := hf.norm_approxBounded_le hc ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m] ** have : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] := by\n  refine' condexp_of_stronglyMeasurable hm (hf.mul stronglyMeasurable_condexp) _\n  exact integrable_condexp.bdd_mul' (hf.mono hm).aestronglyMeasurable hf_bound ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m] ** rw [\u2190 this] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] \u03bc[f * \u03bc[g|m]|m] ** refine' tendsto_condexp_unique (fun n x => fs n x * g x) (fun n x => fs n x * (\u03bc[g|m]) x) (f * g)\n  (f * \u03bc[g|m]) _ _ _ _ (fun x => c * \u2016g x\u2016) _ (fun x => c * \u2016(\u03bc[g|m]) x\u2016) _ _ _ _ ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u03bc = 0 \u22a2 \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m] ** simp only [h\u03bc, ae_zero] ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u03bc = 0 \u22a2 0[f * g|m] =\u1da0[\u22a5] f * 0[g|m] ** norm_cast ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 \u22a2 NeBot (Measure.ae \u03bc) ** simp only [h\u03bc, ae_neBot, Ne.def, not_false_iff] ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c \u22a2 \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] ** refine' condexp_of_stronglyMeasurable hm (hf.mul stronglyMeasurable_condexp) _ ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c \u22a2 Integrable (f * \u03bc[g|m]) ** exact integrable_condexp.bdd_mul' (hf.mono hm).aestronglyMeasurable hf_bound ** case neg.refine'_1 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 \u2200 (n : \u2115), Integrable ((fun n x => \u2191(fs n) x * g x) n) ** exact fun n => hg.bdd_mul' ((SimpleFunc.stronglyMeasurable (fs n)).mono hm).aestronglyMeasurable\n  (eventually_of_forall (hfs_bound n)) ** case neg.refine'_2 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 \u2200 (n : \u2115), Integrable ((fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n) ** exact fun n => integrable_condexp.bdd_mul'\n  ((SimpleFunc.stronglyMeasurable (fs n)).mono hm).aestronglyMeasurable\n  (eventually_of_forall (hfs_bound n)) ** case neg.refine'_3 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => (fun n x => \u2191(fs n) x * g x) n x) atTop (\ud835\udcdd ((f * g) x)) ** filter_upwards [hfs_tendsto] with x hx ** case h \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] x : \u03b1 hx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) \u22a2 Tendsto (fun n => \u2191(fs n) x * g x) atTop (\ud835\udcdd ((f * g) x)) ** rw [Pi.mul_apply] ** case h \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] x : \u03b1 hx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) \u22a2 Tendsto (fun n => \u2191(fs n) x * g x) atTop (\ud835\udcdd (f x * g x)) ** exact Tendsto.mul hx tendsto_const_nhds ** case neg.refine'_4 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => (fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n x) atTop (\ud835\udcdd ((f * \u03bc[g|m]) x)) ** filter_upwards [hfs_tendsto] with x hx ** case h \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] x : \u03b1 hx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) \u22a2 Tendsto (fun n => \u2191(fs n) x * (\u03bc[g|m]) x) atTop (\ud835\udcdd ((f * \u03bc[g|m]) x)) ** rw [Pi.mul_apply] ** case h \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] x : \u03b1 hx : Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) \u22a2 Tendsto (fun n => \u2191(fs n) x * (\u03bc[g|m]) x) atTop (\ud835\udcdd (f x * (\u03bc[g|m]) x)) ** exact Tendsto.mul hx tendsto_const_nhds ** case neg.refine'_5 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 Integrable fun x => c * \u2016g x\u2016 ** exact hg.norm.const_mul c ** case neg.refine'_6 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 Integrable fun x => c * \u2016(\u03bc[g|m]) x\u2016 ** exact integrable_condexp.norm.const_mul c ** case neg.refine'_7 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016(fun n x => \u2191(fs n) x * g x) n x\u2016 \u2264 (fun x => c * \u2016g x\u2016) x ** refine' fun n => eventually_of_forall fun x => _ ** case neg.refine'_7 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] n : \u2115 x : \u03b1 \u22a2 \u2016(fun n x => \u2191(fs n) x * g x) n x\u2016 \u2264 (fun x => c * \u2016g x\u2016) x ** exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hfs_bound n x) (norm_nonneg _)) ** case neg.refine'_8 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n x\u2016 \u2264 (fun x => c * \u2016(\u03bc[g|m]) x\u2016) x ** refine' fun n => eventually_of_forall fun x => _ ** case neg.refine'_8 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] n : \u2115 x : \u03b1 \u22a2 \u2016(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n x\u2016 \u2264 (fun x => c * \u2016(\u03bc[g|m]) x\u2016) x ** exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hfs_bound n x) (norm_nonneg _)) ** case neg.refine'_9 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] \u22a2 \u2200 (n : \u2115), \u03bc[(fun n x => \u2191(fs n) x * g x) n|m] =\u1d50[\u03bc] \u03bc[(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n|m] ** intro n ** case neg.refine'_9 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] n : \u2115 \u22a2 \u03bc[(fun n x => \u2191(fs n) x * g x) n|m] =\u1d50[\u03bc] \u03bc[(fun n x => \u2191(fs n) x * (\u03bc[g|m]) x) n|m] ** simp_rw [\u2190 Pi.mul_apply] ** case neg.refine'_9 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] n : \u2115 \u22a2 \u03bc[fun x => (\u2191(StronglyMeasurable.approxBounded hf c n) * g) x|m] =\u1d50[\u03bc]     \u03bc[fun x => (\u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m]) x|m] ** refine' (condexp_stronglyMeasurable_simpleFunc_mul hm _ hg).trans _ ** case neg.refine'_9 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] n : \u2115 \u22a2 \u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m] =\u1d50[\u03bc]     \u03bc[fun x => (\u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m]) x|m] ** rw [condexp_of_stronglyMeasurable hm\n  ((SimpleFunc.stronglyMeasurable _).mul stronglyMeasurable_condexp) _] ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : IsFiniteMeasure \u03bc f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hg : Integrable g c : \u211d hf_bound : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c fs : \u2115 \u2192 SimpleFunc \u03b1 \u211d := StronglyMeasurable.approxBounded hf c hfs_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) h\u03bc : \u00ac\u03bc = 0 this\u271d : NeBot (Measure.ae \u03bc) hc : 0 \u2264 c hfs_bound : \u2200 (n : \u2115) (x : \u03b1), \u2016\u2191(fs n) x\u2016 \u2264 c this : \u03bc[f * \u03bc[g|m]|m] = f * \u03bc[g|m] n : \u2115 \u22a2 Integrable (\u2191(StronglyMeasurable.approxBounded hf c n) * \u03bc[g|m]) ** exact integrable_condexp.bdd_mul'\n  ((SimpleFunc.stronglyMeasurable (fs n)).mono hm).aestronglyMeasurable\n  (eventually_of_forall (hfs_bound n)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) \u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc a b, \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =     \u2211 i : Fin 2,       ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),           Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -         \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),           Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x))) ** refine' integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv e _ _ ![f, g]\n  ![f', g'] s hs a b hle _ (fun x hx => _) _ _ Hi ** case refine'_1 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) \u22a2 \u2200 (x y : \u211d \u00d7 \u211d), \u2191e x \u2264 \u2191e y \u2194 x \u2264 y ** exact fun x y => (OrderIso.finTwoArrowIso \u211d).symm.le_iff_le ** case refine'_2 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) \u22a2 MeasurePreserving \u2191e ** exact (volume_preserving_finTwoArrow \u211d).symm _ ** case refine'_3 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) \u22a2 \u2200 (i : Fin (1 + 1)), ContinuousOn (Matrix.vecCons f ![g] i) (Set.Icc a b) ** exact Fin.forall_fin_two.2 \u27e8Hcf, Hcg\u27e9 ** case refine'_4 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) x : \u211d \u00d7 \u211d hx : x \u2208 interior (Set.Icc a b) \\ s \u22a2 \u2200 (i : Fin (1 + 1)), HasFDerivAt (Matrix.vecCons f ![g] i) (Matrix.vecCons f' ![g'] i x) x ** rw [Icc_prod_eq, interior_prod_eq, interior_Icc, interior_Icc] at hx ** case refine'_4 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) x : \u211d \u00d7 \u211d hx : x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u22a2 \u2200 (i : Fin (1 + 1)), HasFDerivAt (Matrix.vecCons f ![g] i) (Matrix.vecCons f' ![g'] i x) x ** exact Fin.forall_fin_two.2 \u27e8Hdf x hx, Hdg x hx\u27e9 ** case refine'_5 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) \u22a2 \u2200 (x : \u211d \u00d7 \u211d),     \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =       \u2211 i : Fin (1 + 1), \u2191(Matrix.vecCons f' ![g'] i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1)) ** intro x ** case refine'_5 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) x : \u211d \u00d7 \u211d \u22a2 \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =     \u2211 i : Fin (1 + 1), \u2191(Matrix.vecCons f' ![g'] i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1)) ** rw [Fin.sum_univ_two] ** case refine'_5 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) x : \u211d \u00d7 \u211d \u22a2 \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =     \u2191(Matrix.vecCons f' ![g'] 0 x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single 0 1)) +       \u2191(Matrix.vecCons f' ![g'] 1 x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single 1 1)) ** rfl ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) \u22a2 \u2211 i : Fin 2,       ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),           Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -         \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),           Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x))) =     ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +       ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) ** have : \u2200 (a b : \u211d\u00b9) (f : \u211d\u00b9 \u2192 E),\n    \u222b x in Icc a b, f x = \u222b x in Icc (a 0) (b 0), f fun _ => x := fun a b f \u21a6 by\n  convert (((volume_preserving_funUnique (Fin 1) \u211d).symm _).set_integral_preimage_emb\n    (MeasurableEquiv.measurableEmbedding _) f _).symm\n  exact ((OrderIso.funUnique (Fin 1) \u211d).symm.preimage_Icc a b).symm ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) this :   \u2200 (a b : Fin 1 \u2192 \u211d) (f : (Fin 1 \u2192 \u211d) \u2192 E),     \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x \u22a2 \u2211 i : Fin 2,       ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),           Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -         \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),           Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x))) =     ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +       ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) ** simp only [Fin.sum_univ_two, this] ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) this :   \u2200 (a b : Fin 1 \u2192 \u211d) (f : (Fin 1 \u2192 \u211d) \u2192 E),     \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x \u22a2 ((\u222b (x : \u211d) in           Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 0) 0)             ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 0) 0),           Matrix.vecCons f ![g] 0             (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))               (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b 0) fun x_1 =>                 x))) -         \u222b (x : \u211d) in           Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 0) 0)             ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 0) 0),           Matrix.vecCons f ![g] 0             (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))               (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a 0) fun x_1 =>                 x))) +       ((\u222b (x : \u211d) in           Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 1) 0)             ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 1) 0),           Matrix.vecCons f ![g] 1             (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))               (Fin.insertNth 1 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b 1) fun x_1 =>                 x))) -         \u222b (x : \u211d) in           Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 1) 0)             ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 1) 0),           Matrix.vecCons f ![g] 1             (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))               (Fin.insertNth 1 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a 1) fun x_1 =>                 x))) =     ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +       ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) ** rfl ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u271d b\u271d : \u211d \u00d7 \u211d hle : a\u271d \u2264 b\u271d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f\u271d (Set.Icc a\u271d b\u271d) Hcg : ContinuousOn g (Set.Icc a\u271d b\u271d) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt f\u271d (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u271d b\u271d) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) a b : Fin 1 \u2192 \u211d f : (Fin 1 \u2192 \u211d) \u2192 E \u22a2 \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x ** convert (((volume_preserving_funUnique (Fin 1) \u211d).symm _).set_integral_preimage_emb\n  (MeasurableEquiv.measurableEmbedding _) f _).symm ** case h.e'_3.h.e'_6.h.e'_4 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a\u271d b\u271d : \u211d \u00d7 \u211d hle : a\u271d \u2264 b\u271d s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f\u271d (Set.Icc a\u271d b\u271d) Hcg : ContinuousOn g (Set.Icc a\u271d b\u271d) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt f\u271d (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.1 b\u271d.1 \u00d7\u02e2 Set.Ioo a\u271d.2 b\u271d.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u271d b\u271d) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) a b : Fin 1 \u2192 \u211d f : (Fin 1 \u2192 \u211d) \u2192 E \u22a2 Set.Icc (a 0) (b 0) = \u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Set.Icc a b ** exact ((OrderIso.funUnique (Fin 1) \u211d).symm.preimage_Icc a b).symm ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) \u22a2 ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +       ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) =     (((\u222b (x : \u211d) in a.1 ..b.1, g (x, b.2)) - \u222b (x : \u211d) in a.1 ..b.1, g (x, a.2)) + \u222b (y : \u211d) in a.2 ..b.2, f (b.1, y)) -       \u222b (y : \u211d) in a.2 ..b.2, f (a.1, y) ** simp only [intervalIntegral.integral_of_le hle.1, intervalIntegral.integral_of_le hle.2,\n  set_integral_congr_set_ae (Ioc_ae_eq_Icc (\u03b1 := \u211d) (\u03bc := volume))] ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f g : \u211d \u00d7 \u211d \u2192 E f' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E a b : \u211d \u00d7 \u211d hle : a \u2264 b s : Set (\u211d \u00d7 \u211d) hs : Set.Countable s Hcf : ContinuousOn f (Set.Icc a b) Hcg : ContinuousOn g (Set.Icc a b) Hdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt f (f' x) x Hdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.1 b.1 \u00d7\u02e2 Set.Ioo a.2 b.2 \\ s \u2192 HasFDerivAt g (g' x) x Hi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b) e : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d) \u22a2 ((\u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) - \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y)) +       ((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) =     (((\u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, b.2)) - \u222b (x : \u211d) in Set.Icc a.1 b.1, g (x, a.2)) +         \u222b (y : \u211d) in Set.Icc a.2 b.2, f (b.1, y)) -       \u222b (y : \u211d) in Set.Icc a.2 b.2, f (a.1, y) ** abel ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.unifIntegrable_fin ** \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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 f : Fin n \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : Fin n), Mem\u2112p (f i) p \u22a2 UnifIntegrable f p \u03bc ** revert 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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 \u22a2 \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc ** induction' n with n h ** case succ \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 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc \u22a2 \u2200 {f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc ** intro f hfLp \u03b5 h\u03b5 ** case succ \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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 \u03b4 x,     \u2200 (i : Fin (Nat.succ n)) (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 ** let g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f k ** case succ \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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k \u22a2 \u2203 \u03b4 x,     \u2200 (i : Fin (Nat.succ n)) (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 ** have hgLp : \u2200 i, Mem\u2112p (g i) p \u03bc := fun i => hfLp i ** case succ \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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u22a2 \u2203 \u03b4 x,     \u2200 (i : Fin (Nat.succ n)) (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 \u27e8\u03b4\u2081, h\u03b4\u2081pos, h\u03b4\u2081\u27e9 := h hgLp h\u03b5 ** case succ.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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 \u03b4 x,     \u2200 (i : Fin (Nat.succ n)) (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 \u27e8\u03b4\u2082, h\u03b4\u2082pos, h\u03b4\u2082\u27e9 := (hfLp n).snorm_indicator_le \u03bc hp_one hp_top h\u03b5 ** case succ.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\u271d : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 \u03b4 x,     \u2200 (i : Fin (Nat.succ n)) (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' \u27e8min \u03b4\u2081 \u03b4\u2082, lt_min h\u03b4\u2081pos h\u03b4\u2082pos, fun i s hs h\u03bcs => _\u27e9 ** case succ.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\u271d : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** by_cases hi : i.val < n ** case zero \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 \u22a2 \u2200 {f : Fin Nat.zero \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin Nat.zero), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc ** intro f hf ** case zero \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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 f : Fin Nat.zero \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : Fin Nat.zero), Mem\u2112p (f i) p \u22a2 UnifIntegrable f p \u03bc ** have : Subsingleton (Fin Nat.zero) := subsingleton_fin_zero ** case zero \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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 f : Fin Nat.zero \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : Fin Nat.zero), Mem\u2112p (f i) p this : Subsingleton (Fin Nat.zero) \u22a2 UnifIntegrable f p \u03bc ** exact unifIntegrable_subsingleton \u03bc hp_one hp_top hf ** 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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : \u2191i < n \u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** rw [(_ : f i = g \u27e8i.val, hi\u27e9)] ** 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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : \u2191i < n \u22a2 snorm (indicator s (g { val := \u2191i, isLt := hi })) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** exact h\u03b4\u2081 _ s hs (le_trans h\u03bcs <| ENNReal.ofReal_le_ofReal <| min_le_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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : \u2191i < n \u22a2 f i = g { val := \u2191i, isLt := hi } ** simp ** 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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : \u00ac\u2191i < n \u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** rw [(_ : i = n)] ** 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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : \u00ac\u2191i < n \u22a2 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** exact h\u03b4\u2082 _ hs (le_trans h\u03bcs <| ENNReal.ofReal_le_ofReal <| min_le_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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : \u00ac\u2191i < n \u22a2 i = \u2191n ** have hi' := Fin.is_lt 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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : \u00ac\u2191i < n hi' : \u2191i < Nat.succ n \u22a2 i = \u2191n ** rw [Nat.lt_succ_iff] at hi' ** \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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : \u00ac\u2191i < n hi' : \u2191i \u2264 n \u22a2 i = \u2191n ** rw [not_lt] at hi ** \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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : n \u2264 \u2191i hi' : \u2191i \u2264 n \u22a2 i = \u2191n ** 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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : n \u2264 \u2191i hi' : \u2191i \u2264 n \u22a2 \u2191i = \u2191\u2191n ** symm ** 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 : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 n : \u2115 h : \u2200 {f : Fin n \u2192 \u03b1 \u2192 \u03b2}, (\u2200 (i : Fin n), Mem\u2112p (f i) p) \u2192 UnifIntegrable f p \u03bc f : Fin (Nat.succ n) \u2192 \u03b1 \u2192 \u03b2 hfLp : \u2200 (i : Fin (Nat.succ n)), Mem\u2112p (f i) p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 g : Fin n \u2192 \u03b1 \u2192 \u03b2 := fun k => f \u2191\u2191k hgLp : \u2200 (i : Fin n), Mem\u2112p (g i) p \u03b4\u2081 : \u211d h\u03b4\u2081pos : 0 < \u03b4\u2081 h\u03b4\u2081 :   \u2200 (i : Fin n) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s ((fun i => g i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u03b4\u2082 : \u211d h\u03b4\u2082pos : 0 < \u03b4\u2082 h\u03b4\u2082 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s (f \u2191n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : Fin (Nat.succ n) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) hi : n \u2264 \u2191i hi' : \u2191i \u2264 n \u22a2 \u2191\u2191n = \u2191i ** rw [Fin.coe_ofNat_eq_mod, le_antisymm hi' hi, Nat.mod_succ_eq_iff_lt, Nat.lt_succ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.disjUnion_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : Finset \u03b1 a : \u03b1 h : _root_.Disjoint s {a} \u22a2 disjUnion s {a} h = cons a s (_ : \u00aca \u2208 s) ** rw [disjUnion_comm, singleton_disjUnion] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.indep_limsup_atTop_self ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s \u22a2 Indep (limsup s atTop) (limsup s atTop) ** let ns : \u03b9 \u2192 Set \u03b9 := Set.Iic ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic \u22a2 Indep (limsup s atTop) (limsup s atTop) ** have hnsp : \u2200 i, BddAbove (ns i) := fun i => bddAbove_Iic ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) \u22a2 Indep (limsup s atTop) (limsup s atTop) ** refine' indep_limsup_self h_le h_indep _ _ hnsp _ ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) \u22a2 \u2200 (t : Set \u03b9), BddAbove t \u2192 t\u1d9c \u2208 atTop ** simp only [mem_atTop_sets, ge_iff_le, Set.mem_compl_iff, BddAbove, upperBounds, Set.Nonempty] ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) \u22a2 \u2200 (t : Set \u03b9), (\u2203 x, x \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x}) \u2192 \u2203 a, \u2200 (b : \u03b9), a \u2264 b \u2192 \u00acb \u2208 t ** rintro t \u27e8a, ha\u27e9 ** case refine'_1.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x} \u22a2 \u2203 a, \u2200 (b : \u03b9), a \u2264 b \u2192 \u00acb \u2208 t ** obtain \u27e8b, hb\u27e9 : \u2203 b, a < b := exists_gt a ** case refine'_1.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x} b : \u03b9 hb : a < b \u22a2 \u2203 a, \u2200 (b : \u03b9), a \u2264 b \u2192 \u00acb \u2208 t ** refine' \u27e8b, fun c hc hct => _\u27e9 ** case refine'_1.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x} b : \u03b9 hb : a < b c : \u03b9 hc : b \u2264 c hct : c \u2208 t \u22a2 False ** suffices : \u2200 i \u2208 t, i < c ** case refine'_1.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x} b : \u03b9 hb : a < b c : \u03b9 hc : b \u2264 c hct : c \u2208 t this : \u2200 (i : \u03b9), i \u2208 t \u2192 i < c \u22a2 False  case this \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x} b : \u03b9 hb : a < b c : \u03b9 hc : b \u2264 c hct : c \u2208 t \u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 i < c ** exact lt_irrefl c (this c hct) ** case this \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 a \u2264 x} b : \u03b9 hb : a < b c : \u03b9 hc : b \u2264 c hct : c \u2208 t \u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 i < c ** exact fun i hi => (ha hi).trans_lt (hb.trans_le hc) ** case refine'_2 \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) \u22a2 Directed (fun x x_1 => x \u2264 x_1) fun a => ns a ** exact Monotone.directed_le fun i j hij k hki => le_trans hki hij ** case refine'_3 \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : NoMaxOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Iic hnsp : \u2200 (i : \u03b9), BddAbove (ns i) \u22a2 \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a ** exact fun n => \u27e8n, le_rfl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** by_cases hp_zero : 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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** have hp : 0 < p.toReal := toReal_pos hp_zero hp_ne_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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** suffices\n    Tendsto (fun n => \u222b\u207b x, (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) atTop\n      (\ud835\udcdd 0) by\n  simp only [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_ne_top]\n  convert continuous_rpow_const.continuousAt.tendsto.comp this\n  simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)] ** 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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** have hF_meas :\n  \u2200 n, Measurable fun x => (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal := by\n  simpa only [\u2190 edist_eq_coe_nnnorm_sub] using fun n =>\n    (approxOn f hf s y\u2080 h\u2080 n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>\n      (measurable_edist_right.comp hf).pow_const p.toReal ** 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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** have h_bound :\n  \u2200 n, (fun x => (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) \u2264\u1d50[\u03bc] fun x =>\n      (\u2016f x - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal :=\n  fun n =>\n  eventually_of_forall fun x =>\n    rpow_le_rpow (coe_mono (nnnorm_approxOn_le hf h\u2080 x n)) toReal_nonneg ** 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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound :   \u2200 (n : \u2115),     (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** have h_fin : (\u222b\u207b a : \u03b2, (\u2016f a - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) \u2260 \u22a4 :=\n  (lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top hp_zero hp_ne_top hi).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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound :   \u2200 (n : \u2115),     (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** have h_lim :\n  \u2200\u1d50 a : \u03b2 \u2202\u03bc,\n    Tendsto (fun n => (\u2016approxOn f hf s y\u2080 h\u2080 n a - f a\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) atTop (\ud835\udcdd 0) := by\n  filter_upwards [h\u03bc] with a ha\n  have : Tendsto (fun n => (approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) :=\n    (tendsto_approxOn hf h\u2080 ha).sub tendsto_const_nhds\n  convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm)\n  simp [zero_rpow_of_pos hp] ** 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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound :   \u2200 (n : \u2115),     (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b2) \u2202\u03bc, Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** simpa using tendsto_lintegral_of_dominated_convergence _ hF_meas h_bound h_fin h_lim ** 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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : p = 0 \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** simpa only [hp_zero, snorm_exponent_zero] using tendsto_const_nhds ** \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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p this : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** simp only [snorm_eq_lintegral_rpow_nnnorm hp_zero 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\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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p this : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => (\u222b\u207b (x : \u03b2), \u2191\u2016(\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p))     atTop (\ud835\udcdd 0) ** convert continuous_rpow_const.continuousAt.tendsto.comp this ** case h.e'_5.h.e'_3 \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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p this : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) \u22a2 0 = 0 ^ (1 / ENNReal.toReal p) ** simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)] ** \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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p \u22a2 \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p ** simpa only [\u2190 edist_eq_coe_nnnorm_sub] using fun n =>\n  (approxOn f hf s y\u2080 h\u2080 n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>\n    (measurable_edist_right.comp hf).pow_const p.toReal ** \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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound :   \u2200 (n : \u2115),     (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 \u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u03bc, Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0) ** filter_upwards [h\u03bc] with a ha ** 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 \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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound :   \u2200 (n : \u2115),     (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 a : \u03b2 ha : f a \u2208 closure s \u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0) ** have : Tendsto (fun n => (approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) :=\n  (tendsto_approxOn hf h\u2080 ha).sub tendsto_const_nhds ** 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 \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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound :   \u2200 (n : \u2115),     (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 a : \u03b2 ha : f a \u2208 closure s this : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) \u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0) ** convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm) ** case h.e'_5.h.e'_3 \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 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound :   \u2200 (n : \u2115),     (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 a : \u03b2 ha : f a \u2208 closure s this : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) \u22a2 0 = \u2191\u2016f a - f a\u2016\u208a ^ ENNReal.toReal p ** simp [zero_rpow_of_pos hp] ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.Valid.isEmpty ** x\u271d : Substring h\u271d : Valid x\u271d l m r : List Char h : ValidFor l m r x\u271d \u22a2 Substring.isEmpty x\u271d = true \u2194 toString x\u271d = \"\" ** simp [h.isEmpty, h.toString, ext_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ProbabilityMeasure.eq_of_forall_toMeasure_apply_eq ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc \u03bd : ProbabilityMeasure \u03a9 h : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s \u22a2 \u03bc = \u03bd ** apply toMeasure_injective ** case a \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc \u03bd : ProbabilityMeasure \u03a9 h : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s \u22a2 \u2191\u03bc = \u2191\u03bd ** ext1 s s_mble ** case a.h \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc \u03bd : ProbabilityMeasure \u03a9 h : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s s : Set \u03a9 s_mble : MeasurableSet s \u22a2 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s ** exact h s s_mble ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.inv_mul_of_unit ** n : \u2115 a : ZMod n h : IsUnit a \u22a2 a\u207b\u00b9 * a = 1 ** rw [mul_comm, mul_inv_of_unit a h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.get_ofFn ** n\u271d : \u2115 \u03b1 : Type u_1 n : \u2115 f : Fin n \u2192 \u03b1 i : Fin n \u22a2 get (ofFn f) i = f i ** conv_rhs => erw [\u2190 List.get_ofFn f \u27e8i, by simp\u27e9] ** n\u271d : \u2115 \u03b1 : Type u_1 n : \u2115 f : Fin n \u2192 \u03b1 i : Fin n \u22a2 get (ofFn f) i = List.get (List.ofFn f) { val := \u2191i, isLt := (_ : \u2191i < List.length (List.ofFn f)) } ** simp only [get_eq_get] ** n\u271d : \u2115 \u03b1 : Type u_1 n : \u2115 f : Fin n \u2192 \u03b1 i : Fin n \u22a2 List.get (toList (ofFn f)) (Fin.cast (_ : n = List.length (toList (ofFn f))) i) =     List.get (List.ofFn f) { val := \u2191i, isLt := (_ : \u2191i < List.length (List.ofFn f)) } ** congr <;> simp [Fin.heq_ext_iff] ** n\u271d : \u2115 \u03b1 : Type u_1 n : \u2115 f : Fin n \u2192 \u03b1 i : Fin n \u22a2 \u2191i < List.length (List.ofFn f) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.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 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9 ** 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 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9 ** exact Integrable.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": "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 a \u2208 [[b, c]] \u2192 b \u2208 [[a, c]] \u2192 a = b ** 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 b \u2293 c \u2264 a \u2227 a \u2264 b \u2294 c \u2192 a \u2293 c \u2264 b \u2227 b \u2264 a \u2294 c \u2192 a = b ** exact Set.eq_of_mem_uIcc_of_mem_uIcc ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.uIcc_injective_left ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : DistribLattice \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d a\u2081 a\u2082 b b\u2081 b\u2082 c x a : \u03b1 \u22a2 Injective (uIcc a) ** simpa only [uIcc_comm] using uIcc_injective_right a ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Primrec'.sub ** \u22a2 Primrec' fun v => Vector.head v - Vector.head (Vector.tail v) ** suffices ** this : ?m.396872 \u22a2 Primrec' fun v => Vector.head v - Vector.head (Vector.tail v)  case this  \u22a2 ?m.396872 ** simpa using comp\u2082 (fun a b => b - a) this (tail head) head ** case this  \u22a2 Primrec' fun v => (fun a b => b - a) (Vector.head v) (Vector.head (Vector.tail v)) ** refine' (prec head (pred.comp\u2081 _ (tail head))).of_eq fun v => _ ** case this v : Vector \u2115 (Nat.succ 0 + 1) \u22a2 Nat.rec (Vector.head (Vector.tail v)) (fun y IH => Nat.pred (Vector.head (Vector.tail (y ::\u1d65 IH ::\u1d65 Vector.tail v))))       (Vector.head v) =     (fun a b => b - a) (Vector.head v) (Vector.head (Vector.tail v)) ** simp ** case this v : Vector \u2115 (Nat.succ 0 + 1) \u22a2 Nat.rec (Vector.head (Vector.tail v)) (fun y IH => Nat.pred IH) (Vector.head v) =     Vector.head (Vector.tail v) - Vector.head v ** induction v.head <;> simp [*, Nat.sub_succ] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.val_sub ** n : \u2115 inst\u271d : NeZero n a b : ZMod n h : val b \u2264 val a \u22a2 val (a - b) = val a - val b ** by_cases hb : b = 0 ** case pos n : \u2115 inst\u271d : NeZero n a b : ZMod n h : val b \u2264 val a hb : b = 0 \u22a2 val (a - b) = val a - val b ** cases hb ** case pos.refl n : \u2115 inst\u271d : NeZero n a : ZMod n h : val 0 \u2264 val a \u22a2 val (a - 0) = val a - val 0 ** simp ** case neg n : \u2115 inst\u271d : NeZero n a b : ZMod n h : val b \u2264 val a hb : \u00acb = 0 \u22a2 val (a - b) = val a - val b ** have : NeZero b := \u27e8hb\u27e9 ** case neg n : \u2115 inst\u271d : NeZero n a b : ZMod n h : val b \u2264 val a hb : \u00acb = 0 this : NeZero b \u22a2 val (a - b) = val a - val b ** rw [sub_eq_add_neg, val_add, val_neg_of_ne_zero, \u2190 Nat.add_sub_assoc (le_of_lt (val_lt _)),\n  add_comm, Nat.add_sub_assoc h, Nat.add_mod_left] ** case neg n : \u2115 inst\u271d : NeZero n a b : ZMod n h : val b \u2264 val a hb : \u00acb = 0 this : NeZero b \u22a2 (val a - val b) % n = val a - val b ** apply Nat.mod_eq_of_lt (tsub_lt_of_lt (val_lt _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.lintegral_eq_of_subset ** \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 : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s \u22a2 lintegral f \u03bc = \u2211 x in s, x * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {x}) ** refine' Finset.sum_bij_ne_zero (fun r _ _ => r) _ _ _ _ ** 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e s : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s \u22a2 \u2200 (a : \u211d\u22650\u221e) (h\u2081 : a \u2208 SimpleFunc.range f) (h\u2082 : a * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) \u2260 0), (fun r x x => r) a h\u2081 h\u2082 \u2208 s ** simpa only [forall_range_iff, mul_ne_zero_iff, and_imp] ** 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e s : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s \u22a2 \u2200 (a\u2081 a\u2082 : \u211d\u22650\u221e) (h\u2081\u2081 : a\u2081 \u2208 SimpleFunc.range f) (h\u2081\u2082 : a\u2081 * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2081}) \u2260 0) (h\u2082\u2081 : a\u2082 \u2208 SimpleFunc.range f)     (h\u2082\u2082 : a\u2082 * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2082}) \u2260 0), (fun r x x => r) a\u2081 h\u2081\u2081 h\u2081\u2082 = (fun r x x => r) a\u2082 h\u2082\u2081 h\u2082\u2082 \u2192 a\u2081 = a\u2082 ** intros ** 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e s : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s a\u2081\u271d a\u2082\u271d : \u211d\u22650\u221e h\u2081\u2081\u271d : a\u2081\u271d \u2208 SimpleFunc.range f h\u2081\u2082\u271d : a\u2081\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2081\u271d}) \u2260 0 h\u2082\u2081\u271d : a\u2082\u271d \u2208 SimpleFunc.range f h\u2082\u2082\u271d : a\u2082\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u2082\u271d}) \u2260 0 a\u271d : (fun r x x => r) a\u2081\u271d h\u2081\u2081\u271d h\u2081\u2082\u271d = (fun r x x => r) a\u2082\u271d h\u2082\u2081\u271d h\u2082\u2082\u271d \u22a2 a\u2081\u271d = a\u2082\u271d ** assumption ** 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e s : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s \u22a2 \u2200 (b : \u211d\u22650\u221e), b \u2208 s \u2192 b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0 \u2192 \u2203 a h\u2081 h\u2082, b = (fun r x x => r) a h\u2081 h\u2082 ** intro b _ hb ** 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e s : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s b : \u211d\u22650\u221e a\u271d : b \u2208 s hb : b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0 \u22a2 \u2203 a h\u2081 h\u2082, b = (fun r x x => r) a h\u2081 h\u2082 ** refine' \u27e8b, _, hb, rfl\u27e9 ** 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e s : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s b : \u211d\u22650\u221e a\u271d : b \u2208 s hb : b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0 \u22a2 b \u2208 SimpleFunc.range f ** rw [mem_range, \u2190 preimage_singleton_nonempty] ** 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 : \u03b1 \u2192\u209b \u211d\u22650\u221e s : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s b : \u211d\u22650\u221e a\u271d : b \u2208 s hb : b * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {b}) \u2260 0 \u22a2 Set.Nonempty (\u2191f \u207b\u00b9' {b}) ** exact nonempty_of_measure_ne_zero (mul_ne_zero_iff.1 hb).2 ** case refine'_4 \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 : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s \u22a2 \u2200 (a : \u211d\u22650\u221e) (h\u2081 : a \u2208 SimpleFunc.range f) (h\u2082 : a * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) \u2260 0),     a * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) = (fun r x x => r) a h\u2081 h\u2082 * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {(fun r x x => r) a h\u2081 h\u2082}) ** intros ** case refine'_4 \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 : Finset \u211d\u22650\u221e hs : \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {\u2191f x}) \u2260 0 \u2192 \u2191f x \u2208 s a\u271d : \u211d\u22650\u221e h\u2081\u271d : a\u271d \u2208 SimpleFunc.range f h\u2082\u271d : a\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u271d}) \u2260 0 \u22a2 a\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a\u271d}) = (fun r x x => r) a\u271d h\u2081\u271d h\u2082\u271d * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {(fun r x x => r) a\u271d h\u2081\u271d h\u2082\u271d}) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.ToPartrec.Code.id_eval ** v : List \u2115 \u22a2 eval id v = pure v ** simp [id] ** Qed",
        "informal": ""
    },
    {
        "formal": "PosNum.pred'_to_nat ** \u03b1 : Type u_1 n : PosNum \u22a2 Nat.succ \u2191(pred' n) = \u2191n ** rw [pred'_to_nat n, Nat.succ_pred_eq_of_pos (to_nat_pos n)] ** \u03b1 : Type u_1 n : PosNum this : Nat.succ \u2191(pred' n) = \u2191n h : \u21911 = \u2191n \u22a2 \u2191(Num.casesOn 0 1 bit1) = Nat.pred (_root_.bit0 \u2191n) ** rw [\u2190 to_nat_inj.1 h] ** \u03b1 : Type u_1 n : PosNum this : Nat.succ \u2191(pred' n) = \u2191n h : \u21911 = \u2191n \u22a2 \u2191(Num.casesOn 0 1 bit1) = Nat.pred (_root_.bit0 \u21911) ** rfl ** \u03b1 : Type u_1 n : PosNum this : Nat.succ \u2191(pred' n) = \u2191n p : PosNum h : Nat.succ \u2191p = \u2191n \u22a2 \u2191(Num.casesOn (pos p) 1 bit1) = Nat.pred (_root_.bit0 \u2191n) ** rw [\u2190 h] ** \u03b1 : Type u_1 n : PosNum this : Nat.succ \u2191(pred' n) = \u2191n p : PosNum h : Nat.succ \u2191p = \u2191n \u22a2 \u2191(Num.casesOn (pos p) 1 bit1) = Nat.pred (_root_.bit0 (Nat.succ \u2191p)) ** exact (Nat.succ_add p p).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "circleIntegral.integral_smul_const ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 \u2102 a : E c : \u2102 R : \u211d \u22a2 (\u222e (z : \u2102) in C(c, R), f z \u2022 a) = (\u222e (z : \u2102) in C(c, R), f z) \u2022 a ** simp only [circleIntegral, intervalIntegral.integral_smul_const, \u2190 smul_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpL1Clm_lpMeas ** \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 lpMeas F' \u211d m 1 \u03bc } \u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191f = \u2191f ** let g := lpMeasToLpTrimLie F' \u211d 1 \u03bc hm f ** \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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f \u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191f = \u2191f ** have hfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g := by\n  simp only [LinearIsometryEquiv.symm_apply_apply] ** \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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g \u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191f = \u2191f ** rw [hfg] ** \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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g \u22a2 \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) ** refine' @Lp.induction \u03b1 F' m _ 1 (\u03bc.trim hm) _ ENNReal.coe_ne_top (fun g : \u03b1 \u2192\u2081[\u03bc.trim hm] F' =>\n  condexpL1Clm F' hm \u03bc ((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g : \u03b1 \u2192\u2081[\u03bc] F') =\n  \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g)) _ _ _ g ** \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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f \u22a2 f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g ** simp only [LinearIsometryEquiv.symm_apply_apply] ** 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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g \u22a2 \u2200 (c : F') {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4),     (fun g =>         \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =           \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))       \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191(Measure.trim \u03bc hm) 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\u271d : \u03b1 \u2192 F' s\u271d : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g c : F' s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u22a2 \u2191(condexpL1Clm F' hm \u03bc)       \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm))           \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4) c)) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm))         \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4) c)) ** rw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator hs h\u03bcs.ne c,\n  condexpL1Clm_indicatorConstLp] ** 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\u271d : \u03b1 \u2192 F' s\u271d : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g c : F' s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4 \u22a2 \u2191(condexpInd F' hm \u03bc s) c = indicatorConstLp 1 (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c ** exact condexpInd_of_measurable hs ((le_trim hm).trans_lt 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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g \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 g =>             \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =               \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))           (Mem\u2112p.toLp f hf) \u2192         (fun g =>               \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =                 \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))             (Mem\u2112p.toLp g hg) \u2192           (fun g =>               \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =                 \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))             (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** intro f g hf hg _ hf_eq hg_eq ** 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\u00b9 : \u03b1 \u2192 F' s : Set \u03b1 f\u271d : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g\u271d : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d hfg : f\u271d = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\u271d f g : \u03b1 \u2192 F' hf : Mem\u2112p f 1 hg : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hf_eq :   \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) hg_eq :   \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) \u22a2 \u2191(condexpL1Clm F' hm \u03bc)       \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)) ** rw [LinearIsometryEquiv.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\u00b9 : \u03b1 \u2192 F' s : Set \u03b1 f\u271d : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g\u271d : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d hfg : f\u271d = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\u271d f g : \u03b1 \u2192 F' hf : Mem\u2112p f 1 hg : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hf_eq :   \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) hg_eq :   \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) \u22a2 \u2191(condexpL1Clm F' hm \u03bc)       \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf) +           \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf) +         \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) ** push_cast ** 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\u00b9 : \u03b1 \u2192 F' s : Set \u03b1 f\u271d : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g\u271d : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d hfg : f\u271d = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g\u271d f g : \u03b1 \u2192 F' hf : Mem\u2112p f 1 hg : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hf_eq :   \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) hg_eq :   \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) \u22a2 \u2191(condexpL1Clm F' hm \u03bc)       (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) +         \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg))) =     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp f hf)) +       \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) (Mem\u2112p.toLp g hg)) ** rw [map_add, hf_eq, hg_eq] ** 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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g \u22a2 IsClosed     {f |       (fun g =>           \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g) =             \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g))         f} ** refine' isClosed_eq _ _ ** case refine'_3.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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g \u22a2 Continuous fun f => \u2191(condexpL1Clm F' hm \u03bc) \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f) ** refine' (condexpL1Clm F' hm \u03bc).continuous.comp (continuous_induced_dom.comp _) ** case refine'_3.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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g \u22a2 Continuous fun f => \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f ** exact LinearIsometryEquiv.continuous _ ** case refine'_3.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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g \u22a2 Continuous fun f => \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f) ** refine' continuous_induced_dom.comp _ ** case refine'_3.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\u271d : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 lpMeas F' \u211d m 1 \u03bc } g : { x // x \u2208 Lp F' 1 } := \u2191(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f hfg : f = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) g \u22a2 Continuous fun f => \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm)) f ** exact LinearIsometryEquiv.continuous _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Isometry.hausdorffMeasure_image ** \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 f : X \u2192 Y d : \u211d hf : Isometry f hd : 0 \u2264 d \u2228 Surjective f s : Set X \u22a2 \u2191\u2191\u03bcH[d] (f '' s) = \u2191\u2191\u03bcH[d] s ** simp only [hausdorffMeasure, \u2190 OuterMeasure.coe_mkMetric, \u2190 OuterMeasure.comap_apply] ** \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 f : X \u2192 Y d : \u211d hf : Isometry f hd : 0 \u2264 d \u2228 Surjective f s : Set X \u22a2 \u2191(\u2191(OuterMeasure.comap f) \u2191(mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s ** simp only [mkMetric_toOuterMeasure] ** \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 f : X \u2192 Y d : \u211d hf : Isometry f hd : 0 \u2264 d \u2228 Surjective f s : Set X \u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s ** have : 0 \u2264 d \u2192 Monotone fun r \u21a6 @HPow.hPow \u211d\u22650\u221e \u211d \u211d\u22650\u221e instHPow r d := by\n  exact fun hd x y hxy => ENNReal.rpow_le_rpow hxy hd ** \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 f : X \u2192 Y d : \u211d hf : Isometry f hd : 0 \u2264 d \u2228 Surjective f s : Set X this : 0 \u2264 d \u2192 Monotone fun r => r ^ d \u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s ** have := OuterMeasure.isometry_comap_mkMetric (fun (r : \u211d\u22650\u221e) => r ^ d) hf (hd.imp_left this) ** \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 f : X \u2192 Y d : \u211d hf : Isometry f hd : 0 \u2264 d \u2228 Surjective f s : Set X this\u271d : 0 \u2264 d \u2192 Monotone fun r => r ^ d this : \u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d) = OuterMeasure.mkMetric fun r => r ^ d \u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s ** congr ** \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 f : X \u2192 Y d : \u211d hf : Isometry f hd : 0 \u2264 d \u2228 Surjective f s : Set X \u22a2 0 \u2264 d \u2192 Monotone fun r => r ^ d ** exact fun hd x y hxy => ENNReal.rpow_le_rpow hxy hd ** Qed",
        "informal": ""
    },
    {
        "formal": "QPF.Fix.rec_eq ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x : F (Fix F) \u22a2 rec g (mk x) = g (rec g <$> x) ** have : recF g \u2218 fixToW = Fix.rec g := by\n  apply funext\n  apply Quotient.ind\n  intro x\n  apply recF_eq_of_Wequiv\n  rw [fixToW]\n  apply Wrepr_equiv ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x : F (Fix F) this : recF g \u2218 fixToW = rec g \u22a2 rec g (mk x) = g (rec g <$> x) ** conv =>\n  lhs\n  rw [Fix.rec, Fix.mk]\n  dsimp ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x : F (Fix F) this : recF g \u2218 fixToW = rec g \u22a2 recF g (PFunctor.W.mk (fixToW <$> repr x)) = g (rec g <$> x) ** cases' h : repr x with a f ** case mk F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x : F (Fix F) this : recF g \u2218 fixToW = rec g a : (P F).A f : PFunctor.B (P F) a \u2192 Fix F h : repr x = { fst := a, snd := f } \u22a2 recF g (PFunctor.W.mk (fixToW <$> { fst := a, snd := f })) = g (rec g <$> x) ** rw [PFunctor.map_eq, recF_eq, \u2190 PFunctor.map_eq, PFunctor.W.dest_mk, \u2190 PFunctor.comp_map, abs_map,\n  \u2190 h, abs_repr, this] ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x : F (Fix F) \u22a2 recF g \u2218 fixToW = rec g ** apply funext ** case h F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x : F (Fix F) \u22a2 \u2200 (x : Fix F), (recF g \u2218 fixToW) x = rec g x ** apply Quotient.ind ** case h.a F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x : F (Fix F) \u22a2 \u2200 (a : PFunctor.W (P F)), (recF g \u2218 fixToW) (Quotient.mk Wsetoid a) = rec g (Quotient.mk Wsetoid a) ** intro x ** case h.a F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x\u271d : F (Fix F) x : PFunctor.W (P F) \u22a2 (recF g \u2218 fixToW) (Quotient.mk Wsetoid x) = rec g (Quotient.mk Wsetoid x) ** apply recF_eq_of_Wequiv ** case h.a.a F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x\u271d : F (Fix F) x : PFunctor.W (P F) \u22a2 Wequiv (fixToW (Quotient.mk Wsetoid x)) x ** rw [fixToW] ** case h.a.a F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 x\u271d : F (Fix F) x : PFunctor.W (P F) \u22a2 Wequiv     (Quotient.lift Wrepr       (_ :         \u2200 (x y : PFunctor.W (P F)),           Wequiv x y \u2192 recF (fun x => PFunctor.W.mk (repr x)) x = recF (fun x => PFunctor.W.mk (repr x)) y)       (Quotient.mk Wsetoid x))     x ** apply Wrepr_equiv ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.inv_smul_integral_comp_div_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 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 (x / c - d) = \u222b (x : \u211d) in a / c - d..b / c - d, f x ** by_cases hc : c = 0 <;> simp [hc, integral_comp_div_sub] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.measurable_kernel_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 inst\u271d : IsSFiniteKernel \u03ba t : Set (\u03b1 \u00d7 \u03b2) ht : MeasurableSet t \u22a2 Measurable fun a => \u2191\u2191(\u2191\u03ba a) (Prod.mk a \u207b\u00b9' t) ** rw [\u2190 kernel.kernel_sum_seq \u03ba] ** \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 : IsSFiniteKernel \u03ba t : Set (\u03b1 \u00d7 \u03b2) ht : MeasurableSet t \u22a2 Measurable fun a => \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) ** have : \u2200 a, kernel.sum (kernel.seq \u03ba) a (Prod.mk a \u207b\u00b9' t) =\n    \u2211' n, kernel.seq \u03ba n a (Prod.mk a \u207b\u00b9' t) := fun a =>\n  kernel.sum_apply' _ _ (measurable_prod_mk_left 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 inst\u271d : IsSFiniteKernel \u03ba t : Set (\u03b1 \u00d7 \u03b2) ht : MeasurableSet t this : \u2200 (a : \u03b1), \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) = \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t) \u22a2 Measurable fun a => \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) ** 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 : IsSFiniteKernel \u03ba t : Set (\u03b1 \u00d7 \u03b2) ht : MeasurableSet t this : \u2200 (a : \u03b1), \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) = \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t) \u22a2 Measurable fun a => \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t) ** refine' Measurable.ennreal_tsum 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 : IsSFiniteKernel \u03ba t : Set (\u03b1 \u00d7 \u03b2) ht : MeasurableSet t this : \u2200 (a : \u03b1), \u2191\u2191(\u2191(kernel.sum (seq \u03ba)) a) (Prod.mk a \u207b\u00b9' t) = \u2211' (n : \u2115), \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t) n : \u2115 \u22a2 Measurable fun a => \u2191\u2191(\u2191(seq \u03ba n) a) (Prod.mk a \u207b\u00b9' t) ** exact measurable_kernel_prod_mk_left_of_finite ht inferInstance ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.inv_smul_integral_comp_div_add ** \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 (x / c + d) = \u222b (x : \u211d) in a / c + d..b / c + d, f x ** by_cases hc : c = 0 <;> simp [hc, integral_comp_div_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.Heap.realSize_deleteMin ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool a : \u03b1 s' s : Heap \u03b1 eq : deleteMin le s = some (a, s') \u22a2 realSize s = realSize s' + 1 ** cases s with cases eq | cons r a c s => ?_ ** case cons.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 \u22a2 realSize (cons r a c s) =     realSize         (merge le (HeapNode.toHeap (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)           (FindMin.before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })             (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) +       1 ** have : (s.findMin le (cons r a c) \u27e8id, a, c, s\u27e9).HasSize (c.realSize + s.realSize + 1) :=\n  Heap.realSize_findMin (c.realSize + 1) (by simp) (Nat.add_right_comm ..) \u27e80, by simp\u27e9 ** case cons.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 this :   Std.BinomialHeap.Imp.FindMin.HasSize (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })     (HeapNode.realSize c + realSize s + 1) \u22a2 realSize (cons r a c s) =     realSize         (merge le (HeapNode.toHeap (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)           (FindMin.before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })             (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) +       1 ** revert this ** case cons.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 \u22a2 Std.BinomialHeap.Imp.FindMin.HasSize (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })       (HeapNode.realSize c + realSize s + 1) \u2192     realSize (cons r a c s) =       realSize           (merge le (HeapNode.toHeap (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)             (FindMin.before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s })               (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) +         1 ** match s.findMin le (cons r a c) \u27e8id, a, c, s\u27e9 with\n| { before, val, node, next } =>\n  intro \u27e8m, ih\u2081, ih\u2082\u27e9; dsimp only at ih\u2081 ih\u2082\n  rw [realSize, Nat.add_right_comm, ih\u2082]\n  simp only [realSize_merge, HeapNode.realSize_toHeap, ih\u2081, Nat.add_assoc, Nat.add_left_comm] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 \u22a2 \u2200 (s : Heap \u03b1), realSize (cons r a c s) = HeapNode.realSize c + 1 + realSize s ** simp ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 \u22a2 (\u2200 (s_1 : Heap \u03b1),       realSize (FindMin.before { before := id, val := a, node := c, next := s } s_1) = 0 + realSize s_1) \u2227     HeapNode.realSize c + realSize s + 1 =       0 + HeapNode.realSize { before := id, val := a, node := c, next := s }.node +           realSize { before := id, val := a, node := c, next := s }.next +         1 ** simp ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 before : Heap \u03b1 \u2192 Heap \u03b1 val : \u03b1 node : HeapNode \u03b1 next : Heap \u03b1 \u22a2 Std.BinomialHeap.Imp.FindMin.HasSize { before := before, val := val, node := node, next := next }       (HeapNode.realSize c + realSize s + 1) \u2192     realSize (cons r a c s) =       realSize           (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)) +         1 ** intro \u27e8m, ih\u2081, ih\u2082\u27e9 ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 before : Heap \u03b1 \u2192 Heap \u03b1 val : \u03b1 node : HeapNode \u03b1 next : Heap \u03b1 m : Nat ih\u2081 :   \u2200 (s : Heap \u03b1),     realSize (FindMin.before { before := before, val := val, node := node, next := next } s) = m + realSize s ih\u2082 :   HeapNode.realSize c + realSize s + 1 =     m + HeapNode.realSize { before := before, val := val, node := node, next := next }.node +         realSize { before := before, val := val, node := node, next := next }.next +       1 \u22a2 realSize (cons r a c s) =     realSize         (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)) +       1 ** dsimp only at ih\u2081 ih\u2082 ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 before : Heap \u03b1 \u2192 Heap \u03b1 val : \u03b1 node : HeapNode \u03b1 next : Heap \u03b1 m : Nat ih\u2081 : \u2200 (s : Heap \u03b1), realSize (before s) = m + realSize s ih\u2082 : HeapNode.realSize c + realSize s + 1 = m + HeapNode.realSize node + realSize next + 1 \u22a2 realSize (cons r a c s) =     realSize         (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)) +       1 ** rw [realSize, Nat.add_right_comm, ih\u2082] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 before : Heap \u03b1 \u2192 Heap \u03b1 val : \u03b1 node : HeapNode \u03b1 next : Heap \u03b1 m : Nat ih\u2081 : \u2200 (s : Heap \u03b1), realSize (before s) = m + realSize s ih\u2082 : HeapNode.realSize c + realSize s + 1 = m + HeapNode.realSize node + realSize next + 1 \u22a2 m + HeapNode.realSize node + realSize next + 1 =     realSize         (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)) +       1 ** simp only [realSize_merge, HeapNode.realSize_toHeap, ih\u2081, Nat.add_assoc, Nat.add_left_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.neg_add_lt_right_of_lt_add ** a b c : Int h : a < b + c \u22a2 -c + a < b ** rw [Int.add_comm] at h ** a b c : Int h : a < c + b \u22a2 -c + a < b ** exact Int.neg_add_lt_left_of_lt_add h ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condexp_ae_eq_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 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9 ** have hX : @Measurable \u03a9 \u03a9 m\u03a9 (m \u2293 m\u03a9) id := measurable_id.mono le_rfl (inf_le_right : m \u2293 m\u03a9 \u2264 m\u03a9) ** \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 hX : Measurable id \u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9 ** 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_int : Integrable f hX : Measurable id \u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id \u03c9) ** have h := condexp_ae_eq_integral_condDistrib_id hX hf_int ** \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 hX : Measurable id h : \u03bc[f|MeasurableSpace.comap id (m \u2293 m\u03a9)] =\u1d50[\u03bc] fun a => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id a) \u22a2 \u03bc[f|m \u2293 m\u03a9] =\u1d50[\u03bc] fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id \u03c9) ** simpa only [MeasurableSpace.comap_id, id_eq] using h ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.fiber_card_ne_zero_iff_mem_image ** \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t : Finset \u03b1 f\u271d : \u03b1 \u2192 \u03b2 n : \u2115 s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 inst\u271d : DecidableEq \u03b2 y : \u03b2 \u22a2 card (filter (fun x => f x = y) s) \u2260 0 \u2194 y \u2208 image f s ** rw [\u2190 pos_iff_ne_zero, card_pos, fiber_nonempty_iff_mem_image] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mem\u2112p_const_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 c : E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 \u22a2 Mem\u2112p (fun x => c) p \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4 ** rw [\u2190 snorm_const_lt_top_iff hp_ne_zero hp_ne_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 : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e c : E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 \u22a2 Mem\u2112p (fun x => c) p \u2194 snorm (fun x => c) p \u03bc < \u22a4 ** exact \u27e8fun h => h.2, fun h => \u27e8aestronglyMeasurable_const, h\u27e9\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.exists_tendsto_of_abs_bddAbove_aux ** \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 hf0 : f 0 = 0 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) \u2192 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** have ht :\n  \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 i : \u2115, \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f i n) \u03c9) atTop (\ud835\udcdd c) := by\n  rw [ae_all_iff]\n  exact fun i => Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedValue_leastGE i)\n    (hf.stoppedValue_leastGE_snorm_le' i.cast_nonneg hf0 hbdd) ** \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2192 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** filter_upwards [ht] with \u03c9 h\u03c9 h\u03c9b ** 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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : BddAbove (Set.range fun n => f n \u03c9) \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** rw [BddAbove] at h\u03c9b ** 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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** obtain \u27e8i, hi\u27e9 := exists_nat_gt h\u03c9b.some ** case h.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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** have hib : \u2200 n, f n \u03c9 < i := by\n  intro n\n  exact lt_of_le_of_lt ((mem_upperBounds.1 h\u03c9b.some_mem) _ \u27e8n, rfl\u27e9) hi ** case h.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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** have heq : \u2200 n, stoppedValue f (leastGE f i n) \u03c9 = f n \u03c9 := by\n  intro n\n  rw [leastGE]; unfold hitting; rw [stoppedValue]\n  rw [if_neg]\n  simp only [Set.mem_Icc, Set.mem_union, Set.mem_Ici]\n  push_neg\n  exact fun j _ => hib j ** case h.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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i heq : \u2200 (n : \u2115), stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9 \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** simp only [\u2190 heq, h\u03c9 i] ** \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 hf0 : f 0 = 0 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), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) ** rw [ae_all_iff] ** \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u22a2 \u2200 (i : \u2115), \u2200\u1d50 (a : \u03a9) \u2202\u03bc, \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) a) atTop (\ud835\udcdd c) ** exact fun i => Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedValue_leastGE i)\n  (hf.stoppedValue_leastGE_snorm_le' i.cast_nonneg hf0 hbdd) ** \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i \u22a2 \u2200 (n : \u2115), f n \u03c9 < \u2191i ** intro n ** \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i n : \u2115 \u22a2 f n \u03c9 < \u2191i ** exact lt_of_le_of_lt ((mem_upperBounds.1 h\u03c9b.some_mem) _ \u27e8n, rfl\u27e9) hi ** \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i \u22a2 \u2200 (n : \u2115), stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9 ** intro n ** \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i n : \u2115 \u22a2 stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9 ** rw [leastGE] ** \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i n : \u2115 \u22a2 stoppedValue f (hitting f (Set.Ici \u2191i) 0 n) \u03c9 = f n \u03c9 ** unfold hitting ** \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i n : \u2115 \u22a2 stoppedValue f       (fun x =>         if \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j x \u2208 Set.Ici \u2191i then sInf (Set.Icc 0 n \u2229 {i_1 | f i_1 x \u2208 Set.Ici \u2191i}) else n)       \u03c9 =     f n \u03c9 ** rw [stoppedValue] ** \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i n : \u2115 \u22a2 f (if \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j \u03c9 \u2208 Set.Ici \u2191i then sInf (Set.Icc 0 n \u2229 {i_1 | f i_1 \u03c9 \u2208 Set.Ici \u2191i}) else n) \u03c9 =     f n \u03c9 ** rw [if_neg] ** case hnc \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i n : \u2115 \u22a2 \u00ac\u2203 j, j \u2208 Set.Icc 0 n \u2227 f j \u03c9 \u2208 Set.Ici \u2191i ** simp only [Set.mem_Icc, Set.mem_union, Set.mem_Ici] ** case hnc \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i n : \u2115 \u22a2 \u00ac\u2203 j, (0 \u2264 j \u2227 j \u2264 n) \u2227 \u2191i \u2264 f j \u03c9 ** push_neg ** case hnc \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 hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R ht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c) h\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9)) i : \u2115 hi : Set.Nonempty.some h\u03c9b < \u2191i hib : \u2200 (n : \u2115), f n \u03c9 < \u2191i n : \u2115 \u22a2 \u2200 (j : \u2115), 0 \u2264 j \u2227 j \u2264 n \u2192 f j \u03c9 < \u2191i ** exact fun j _ => hib j ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero ** \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 : \u03b1 \u2192 E hf : Integrable f 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 \u22a2 f =\u1d50[\u03bc] 0 ** have hf_Lp : Mem\u2112p f 1 \u03bc := mem\u2112p_one_iff_integrable.mpr hf ** \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 : \u03b1 \u2192 E hf : Integrable f 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 hf_Lp : Mem\u2112p f 1 \u22a2 f =\u1d50[\u03bc] 0 ** let f_Lp := hf_Lp.toLp f ** \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 : \u03b1 \u2192 E hf : Integrable f 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 hf_Lp : Mem\u2112p f 1 f_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp \u22a2 f =\u1d50[\u03bc] 0 ** have hf_f_Lp : f =\u1d50[\u03bc] f_Lp := (Mem\u2112p.coeFn_toLp hf_Lp).symm ** \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 : \u03b1 \u2192 E hf : Integrable f 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 hf_Lp : Mem\u2112p f 1 f_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp hf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp \u22a2 f =\u1d50[\u03bc] 0 ** refine' hf_f_Lp.trans _ ** \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 : \u03b1 \u2192 E hf : Integrable f 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 hf_Lp : Mem\u2112p f 1 f_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp hf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp \u22a2 \u2191\u2191f_Lp =\u1d50[\u03bc] 0 ** refine' Lp.ae_eq_zero_of_forall_set_integral_eq_zero f_Lp one_ne_zero ENNReal.coe_ne_top _ _ ** case refine'_1 \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 : \u03b1 \u2192 E hf : Integrable f 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 hf_Lp : Mem\u2112p f 1 f_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp hf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f_Lp) s ** exact fun s _ _ => Integrable.integrableOn (L1.integrable_coeFn _) ** case refine'_2 \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 : \u03b1 \u2192 E hf : Integrable f 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 hf_Lp : Mem\u2112p f 1 f_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp hf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f_Lp x \u2202\u03bc = 0 ** intro s hs h\u03bcs ** case refine'_2 \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 : \u03b1 \u2192 E hf : Integrable f 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 hf_Lp : Mem\u2112p f 1 f_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp hf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191f_Lp x \u2202\u03bc = 0 ** rw [integral_congr_ae (ae_restrict_of_ae hf_f_Lp.symm)] ** case refine'_2 \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 : \u03b1 \u2192 E hf : Integrable f 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 hf_Lp : Mem\u2112p f 1 f_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp hf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = 0 ** exact hf_zero s hs h\u03bcs ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.inter_get_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Inter \u03b1 a b : Part \u03b1 hab : (a \u2229 b).Dom \u22a2 get (a \u2229 b) hab = get a (_ : a.Dom) \u2229 get b (_ : b.Dom) ** simp [inter_def] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Inter \u03b1 a b : Part \u03b1 hab : (a \u2229 b).Dom \u22a2 get (Part.bind a fun y => map (fun x => y \u2229 x) b) (_ : (Part.bind a fun y => map (fun x => y \u2229 x) b).Dom) =     get a (_ : a.Dom) \u2229 get b (_ : b.Dom) ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.Heap.WF.size_eq ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n a\u271d\u00b3 : Nat a\u271d\u00b2 : \u03b1 a\u271d\u00b9 : HeapNode \u03b1 a\u271d : Heap \u03b1 left\u271d : n \u2264 a\u271d\u00b3 h\u2081 : HeapNode.WF le a\u271d\u00b2 a\u271d\u00b9 a\u271d\u00b3 h\u2082 : WF le (a\u271d\u00b3 + 1) a\u271d \u22a2 size (cons a\u271d\u00b3 a\u271d\u00b2 a\u271d\u00b9 a\u271d) = realSize (cons a\u271d\u00b3 a\u271d\u00b2 a\u271d\u00b9 a\u271d) ** simp [size, Nat.shiftLeft, size_eq h\u2082, Nat.pow_succ, Nat.mul_succ] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n a\u271d\u00b3 : Nat a\u271d\u00b2 : \u03b1 a\u271d\u00b9 : HeapNode \u03b1 a\u271d : Heap \u03b1 left\u271d : n \u2264 a\u271d\u00b3 h\u2081 : HeapNode.WF le a\u271d\u00b2 a\u271d\u00b9 a\u271d\u00b3 h\u2082 : WF le (a\u271d\u00b3 + 1) a\u271d \u22a2 2 ^ a\u271d\u00b3 + realSize a\u271d = HeapNode.realSize a\u271d\u00b9 + 1 + realSize a\u271d ** simp [Nat.add_assoc, Nat.one_shiftLeft, h\u2081.realSize_eq, h\u2082.size_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.exists_disjoint_closedBall_covering_ae_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 : SigmaFinite \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 exists_absolutelyContinuous_isFiniteMeasure \u03bc with \u27e8\u03bd, h\u03bd, h\u03bc\u03bd\u27e9 ** case 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 : SigmaFinite \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) \u03bd : Measure \u03b1 h\u03bd : IsFiniteMeasure \u03bd h\u03bc\u03bd : \u03bc \u226a \u03bd \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 exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux \u03bd f s hf with\n  \u27e8t, t_count, ts, tr, t\u03bd, tdisj\u27e9 ** case intro.intro.intro.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 : SigmaFinite \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) \u03bd : Measure \u03b1 h\u03bd : IsFiniteMeasure \u03bd h\u03bc\u03bd : \u03bc \u226a \u03bd t : Set (\u03b1 \u00d7 \u211d) t_count : Set.Countable t ts : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s tr : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t\u03bd : \u2191\u2191\u03bd (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 tdisj : PairwiseDisjoint t fun p => 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 ** exact \u27e8t, t_count, ts, tr, h\u03bc\u03bd t\u03bd, tdisj\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.gcd_least_linear ** a b : \u2124 ha : a \u2260 0 \u22a2 IsLeast {n | 0 < n \u2227 \u2203 x y, \u2191n = a * x + b * y} (gcd a b) ** simp_rw [\u2190 gcd_dvd_iff] ** a b : \u2124 ha : a \u2260 0 \u22a2 IsLeast {n | 0 < n \u2227 gcd a b \u2223 n} (gcd a b) ** constructor ** case left a b : \u2124 ha : a \u2260 0 \u22a2 gcd a b \u2208 {n | 0 < n \u2227 gcd a b \u2223 n} ** simpa [and_true_iff, dvd_refl, Set.mem_setOf_eq] using gcd_pos_of_ne_zero_left b ha ** case right a b : \u2124 ha : a \u2260 0 \u22a2 gcd a b \u2208 lowerBounds {n | 0 < n \u2227 gcd a b \u2223 n} ** simp only [lowerBounds, and_imp, Set.mem_setOf_eq] ** case right a b : \u2124 ha : a \u2260 0 \u22a2 \u2200 \u2983a_1 : \u2115\u2984, 0 < a_1 \u2192 gcd a b \u2223 a_1 \u2192 gcd a b \u2264 a_1 ** exact fun n hn_pos hn => Nat.le_of_dvd hn_pos hn ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Martingale.stoppedValue_ae_eq_restrict_eq ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i\u271d n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) i : \u03b9 \u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x = i}] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4] ** refine' Filter.EventuallyEq.trans _\n  (condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const h h\u03c4 h\u03c4_le i).symm ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i\u271d n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) i : \u03b9 \u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x = i}] f i ** rw [Filter.EventuallyEq, ae_restrict_iff' (\u2131.le _ _ (h\u03c4.measurableSet_eq i))] ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i\u271d n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) i : \u03b9 \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 {\u03c9 | \u03c4 \u03c9 = i} \u2192 stoppedValue f \u03c4 x = f i x ** refine' Filter.eventually_of_forall fun x hx => _ ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i\u271d n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) i : \u03b9 x : \u03a9 hx : x \u2208 {\u03c9 | \u03c4 \u03c9 = i} \u22a2 stoppedValue f \u03c4 x = f i x ** rw [Set.mem_setOf_eq] at hx ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i\u271d n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) i : \u03b9 x : \u03a9 hx : \u03c4 x = i \u22a2 stoppedValue f \u03c4 x = f i x ** simp_rw [stoppedValue, hx] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.mkMetric_top ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X \u22a2 (mkMetric fun x => \u22a4) = \u22a4 ** apply toOuterMeasure_injective ** case a \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X \u22a2 \u2191(mkMetric fun x => \u22a4) = \u2191\u22a4 ** rw [mkMetric_toOuterMeasure, OuterMeasure.mkMetric_top, toOuterMeasure_top] ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.decode\u2082 ** \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 \u22a2 Primrec\u2082 fun p a => encode a ** exact encode_iff.mpr snd ** \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 \u22a2 Primrec\u2082 fun p a => p.1 ** exact fst.comp fst ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.funext ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 p q : MvPolynomial \u03c3 R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q \u22a2 p = q ** suffices \u2200 p, (\u2200 x : \u03c3 \u2192 R, eval x p = 0) \u2192 p = 0 by\n  rw [\u2190 sub_eq_zero, this (p - q)]\n  simp only [h, RingHom.map_sub, forall_const, sub_self] ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 p q : MvPolynomial \u03c3 R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q \u22a2 \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 ** clear h p q ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 \u22a2 \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 ** intro p h ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 p : MvPolynomial \u03c3 R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0 \u22a2 p = 0 ** obtain \u27e8n, f, hf, p, rfl\u27e9 := exists_fin_rename p ** case intro.intro.intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 n : \u2115 f : Fin n \u2192 \u03c3 hf : Function.Injective f p : MvPolynomial (Fin n) R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0 \u22a2 \u2191(rename f) p = 0 ** suffices p = 0 by rw [this, AlgHom.map_zero] ** case intro.intro.intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 n : \u2115 f : Fin n \u2192 \u03c3 hf : Function.Injective f p : MvPolynomial (Fin n) R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0 \u22a2 p = 0 ** apply funext_fin ** case intro.intro.intro.intro.h R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 n : \u2115 f : Fin n \u2192 \u03c3 hf : Function.Injective f p : MvPolynomial (Fin n) R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0 \u22a2 \u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0 ** intro x ** case intro.intro.intro.intro.h R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 n : \u2115 f : Fin n \u2192 \u03c3 hf : Function.Injective f p : MvPolynomial (Fin n) R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0 x : Fin n \u2192 R \u22a2 \u2191(eval x) p = 0 ** classical\n  convert h (Function.extend f x 0)\n  simp only [eval, eval\u2082Hom_rename, Function.extend_comp hf] ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 p q : MvPolynomial \u03c3 R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q this : \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 \u22a2 p = q ** rw [\u2190 sub_eq_zero, this (p - q)] ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 p q : MvPolynomial \u03c3 R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = \u2191(eval x) q this : \u2200 (p : MvPolynomial \u03c3 R), (\u2200 (x : \u03c3 \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 \u22a2 \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (p - q) = 0 ** simp only [h, RingHom.map_sub, forall_const, sub_self] ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 n : \u2115 f : Fin n \u2192 \u03c3 hf : Function.Injective f p : MvPolynomial (Fin n) R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0 this : p = 0 \u22a2 \u2191(rename f) p = 0 ** rw [this, AlgHom.map_zero] ** case intro.intro.intro.intro.h R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 n : \u2115 f : Fin n \u2192 \u03c3 hf : Function.Injective f p : MvPolynomial (Fin n) R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0 x : Fin n \u2192 R \u22a2 \u2191(eval x) p = 0 ** convert h (Function.extend f x 0) ** case h.e'_2 R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R \u03c3 : Type u_2 n : \u2115 f : Fin n \u2192 \u03c3 hf : Function.Injective f p : MvPolynomial (Fin n) R h : \u2200 (x : \u03c3 \u2192 R), \u2191(eval x) (\u2191(rename f) p) = 0 x : Fin n \u2192 R \u22a2 \u2191(eval x) p = \u2191(eval (Function.extend f x 0)) (\u2191(rename f) p) ** simp only [eval, eval\u2082Hom_rename, Function.extend_comp hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 \u22a2 StronglyMeasurable f ** have hs_m\u2082 : MeasurableSet[m\u2082] s := by\n  rw [\u2190 Set.inter_univ s]\n  refine' hs Set.univ _\n  rwa [Set.inter_univ] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s \u22a2 StronglyMeasurable f ** obtain \u27e8g_seq_s, hg_seq_tendsto, hg_seq_zero\u27e9 := stronglyMeasurable_in_set hs_m hf hf_zero ** case intro.intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 g_seq_s\u2082 : \u2115 \u2192 \u03b1 \u2192\u209b E :=   fun n =>     { toFun := \u2191(g_seq_s n), measurableSet_fiber' := (_ : \u2200 (x : E), MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x})),       finite_range' := (_ : Set.Finite (range \u2191(g_seq_s n))) } \u22a2 StronglyMeasurable f ** exact \u27e8g_seq_s\u2082, hg_seq_tendsto\u27e9 ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 \u22a2 MeasurableSet s ** rw [\u2190 Set.inter_univ s] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 \u22a2 MeasurableSet (s \u2229 univ) ** rwa [Set.inter_univ] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E \u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x}) ** rw [\u2190 Set.inter_univ (g_seq_s n \u207b\u00b9' {x}), \u2190 Set.union_compl_self s,\n  Set.inter_union_distrib_left, Set.inter_comm (g_seq_s n \u207b\u00b9' {x})] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E \u22a2 MeasurableSet (s \u2229 \u2191(g_seq_s n) \u207b\u00b9' {x} \u222a \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c) ** refine' MeasurableSet.union (hs _ (hs_m.inter _)) _ ** case refine'_2 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E \u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c) ** by_cases hx : x = 0 ** case refine'_1 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E \u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x}) ** exact @SimpleFunc.measurableSet_fiber _ _ m _ _ ** case pos \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : x = 0 \u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c) ** suffices g_seq_s n \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c by\n  rw [this]\n  exact hs_m\u2082.compl ** case pos \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : x = 0 \u22a2 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c ** ext1 y ** case pos.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : x = 0 y : \u03b1 \u22a2 y \u2208 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c \u2194 y \u2208 s\u1d9c ** rw [hx, Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff] ** case pos.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : x = 0 y : \u03b1 \u22a2 \u2191(g_seq_s n) y = 0 \u2227 y \u2208 s\u1d9c \u2194 y \u2208 s\u1d9c ** exact \u27e8fun h => h.2, fun h => \u27e8hg_seq_zero y h n, h\u27e9\u27e9 ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : x = 0 this : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c \u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c) ** rw [this] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : x = 0 this : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c \u22a2 MeasurableSet s\u1d9c ** exact hs_m\u2082.compl ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : \u00acx = 0 \u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c) ** suffices g_seq_s n \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205 by\n  rw [this]\n  exact MeasurableSet.empty ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : \u00acx = 0 \u22a2 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205 ** ext1 y ** case neg.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : \u00acx = 0 y : \u03b1 \u22a2 y \u2208 \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c \u2194 y \u2208 \u2205 ** simp only [mem_inter_iff, mem_preimage, mem_singleton_iff, mem_compl_iff,\n  mem_empty_iff_false, iff_false_iff, not_and, not_not_mem] ** case neg.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : \u00acx = 0 y : \u03b1 \u22a2 \u2191(g_seq_s n) y = x \u2192 y \u2208 s ** refine' Function.mtr fun hys => _ ** case neg.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : \u00acx = 0 y : \u03b1 hys : \u00acy \u2208 s \u22a2 \u00ac\u2191(g_seq_s n) y = x ** rw [hg_seq_zero y hys n] ** case neg.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : \u00acx = 0 y : \u03b1 hys : \u00acy \u2208 s \u22a2 \u00ac0 = x ** exact Ne.symm hx ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : \u00acx = 0 this : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205 \u22a2 MeasurableSet (\u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c) ** rw [this] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : Countable \u03b9 f\u271d g : \u03b1\u271d \u2192 \u03b2 \u03b1 : Type u_5 E : Type u_6 m m\u2082 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace E inst\u271d : Zero E s : Set \u03b1 f : \u03b1 \u2192 E hs_m : MeasurableSet s hs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t) hf : StronglyMeasurable f hf_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 f x = 0 hs_m\u2082 : MeasurableSet s g_seq_s : \u2115 \u2192 \u03b1 \u2192\u209b E hg_seq_tendsto : \u2200 (x : \u03b1), Tendsto (fun n => \u2191(g_seq_s n) x) atTop (\ud835\udcdd (f x)) hg_seq_zero : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2200 (n : \u2115), \u2191(g_seq_s n) x = 0 n : \u2115 x : E hx : \u00acx = 0 this : \u2191(g_seq_s n) \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205 \u22a2 MeasurableSet \u2205 ** exact MeasurableSet.empty ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.eq_singleton_or_nontrivial ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : Finset \u03b1 a b : \u03b1 ha : a \u2208 s \u22a2 s = {a} \u2228 Finset.Nontrivial s ** rw [\u2190 coe_eq_singleton] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : Finset \u03b1 a b : \u03b1 ha : a \u2208 s \u22a2 \u2191s = {a} \u2228 Finset.Nontrivial s ** exact Set.eq_singleton_or_nontrivial ha ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.valMinAbs_natAbs_eq_min ** n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n \u22a2 Int.natAbs (valMinAbs a) = min (val a) (n - val a) ** rw [valMinAbs_def_pos] ** n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n \u22a2 Int.natAbs (if val a \u2264 n / 2 then \u2191(val a) else \u2191(val a) - \u2191n) = min (val a) (n - val a) ** split_ifs with h ** case pos n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n h : val a \u2264 n / 2 \u22a2 Int.natAbs \u2191(val a) = min (val a) (n - val a) ** rw [Int.natAbs_ofNat] ** case pos n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n h : val a \u2264 n / 2 \u22a2 val a = min (val a) (n - val a) ** symm ** case pos n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n h : val a \u2264 n / 2 \u22a2 min (val a) (n - val a) = val a ** apply\n  min_eq_left (le_trans h (le_trans (Nat.half_le_of_sub_le_half _) (Nat.sub_le_sub_left n h))) ** n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n h : val a \u2264 n / 2 \u22a2 n - (n - n / 2) \u2264 n / 2 ** rw [Nat.sub_sub_self (Nat.div_le_self _ _)] ** case neg n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n h : \u00acval a \u2264 n / 2 \u22a2 Int.natAbs (\u2191(val a) - \u2191n) = min (val a) (n - val a) ** rw [\u2190 Int.natAbs_neg, neg_sub, \u2190 Nat.cast_sub a.val_le] ** case neg n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n h : \u00acval a \u2264 n / 2 \u22a2 Int.natAbs \u2191(n - val a) = min (val a) (n - val a) ** symm ** case neg n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n h : \u00acval a \u2264 n / 2 \u22a2 min (val a) (n - val a) = Int.natAbs \u2191(n - val a) ** apply\n  min_eq_right\n    (le_trans (le_trans (Nat.sub_le_sub_left n (lt_of_not_ge h)) (Nat.le_half_of_half_lt_sub _))\n      (le_of_not_ge h)) ** n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n h : \u00acval a \u2264 n / 2 \u22a2 n / 2 < n - (n - Nat.succ (n / 2)) ** rw [Nat.sub_sub_self (Nat.div_lt_self (lt_of_le_of_ne' (Nat.zero_le _) hpos.1) one_lt_two)] ** n\u271d a\u271d n : \u2115 hpos : NeZero n a : ZMod n h : \u00acval a \u2264 n / 2 \u22a2 n / 2 < Nat.succ (n / 2) ** apply Nat.lt_succ_self ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.count_ne_zero' ** \u03b1 : Type u_1 \u03b2 : Type ?u.20372 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 hs' : Set.Nonempty s s_mble : MeasurableSet s \u22a2 \u2191\u2191count s \u2260 0 ** rw [Ne.def, count_eq_zero_iff' s_mble] ** \u03b1 : Type u_1 \u03b2 : Type ?u.20372 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 hs' : Set.Nonempty s s_mble : MeasurableSet s \u22a2 \u00acs = \u2205 ** exact hs'.ne_empty ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.set_lintegral_condKernelReal_prod ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) ** apply MeasurableSpace.induction_on_inter (borel_eq_generateFrom_Iic \u211d) isPiSystem_Iic _ _ _ _ ht ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) \u2205 \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) ** simp only [measure_empty, lintegral_const, zero_mul, prod_empty] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t \u22a2 \u2200 (t : Set \u211d), t \u2208 range Iic \u2192 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) ** rintro t \u27e8q, rfl\u27e9 ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t q : \u211d \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (Iic q) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic q) ** exact set_lintegral_condKernelReal_Iic \u03c1 q hs ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t \u22a2 \u2200 (t : Set \u211d),     MeasurableSet t \u2192       \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u2192         \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t\u1d9c) ** intro t ht ht_lintegral ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 ** congr with a ** case e_f.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) a : \u03b1 \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c = \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) t ** rw [measure_compl 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 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -       \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 ** rw [lintegral_sub (kernel.measurable_coe (condKernelReal \u03c1) ht)] ** case hg_fin \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 \u2260 \u22a4 ** rw [ht_lintegral] ** case hg_fin \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u2260 \u22a4 ** exact measure_ne_top \u03c1 _ ** case h_le \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) t) \u2264\u1d50[Measure.restrict (Measure.fst \u03c1) s] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ ** exact 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 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -       \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 =     \u2191\u2191\u03c1 (s \u00d7\u02e2 univ) - \u2191\u2191\u03c1 (s \u00d7\u02e2 t) ** rw [set_lintegral_condKernelReal_univ \u03c1 hs, ht_lintegral] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 univ) - \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191\u03c1 (s \u00d7\u02e2 t\u1d9c) ** rw [\u2190 measure_diff _ (hs.prod ht) (measure_ne_top \u03c1 _)] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 univ \\ s \u00d7\u02e2 t) = \u2191\u2191\u03c1 (s \u00d7\u02e2 t\u1d9c) ** rw [prod_diff_prod, compl_eq_univ_diff] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 (univ \\ t) \u222a (s \\ s) \u00d7\u02e2 univ) = \u2191\u2191\u03c1 (s \u00d7\u02e2 (univ \\ t)) ** simp only [diff_self, empty_prod, union_empty] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 s \u00d7\u02e2 t \u2286 s \u00d7\u02e2 univ ** rw [prod_subset_prod_iff] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t\u271d : Set \u211d ht\u271d : MeasurableSet t\u271d t : Set \u211d ht : MeasurableSet t ht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u22a2 s \u2286 s \u2227 t \u2286 univ \u2228 s = \u2205 \u2228 t = \u2205 ** exact Or.inl \u27e8subset_rfl, subset_univ t\u27e9 ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t \u22a2 \u2200 (f : \u2115 \u2192 Set \u211d),     Pairwise (Disjoint on f) \u2192       (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192         (\u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)) \u2192           \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \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 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t f : \u2115 \u2192 Set \u211d hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i) \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c3 i, f i) ** simp_rw [measure_iUnion hf_disj hf_meas] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t f : \u2115 \u2192 Set \u211d hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i) \u22a2 \u222b\u207b (a : \u03b1) in s, \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c3 i, f i) ** rw [lintegral_tsum fun i => (kernel.measurable_coe _ (hf_meas i)).aemeasurable.restrict,\n  prod_iUnion, measure_iUnion] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t f : \u2115 \u2192 Set \u211d hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i) \u22a2 \u2211' (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2211' (i : \u2115), \u2191\u2191\u03c1 (s \u00d7\u02e2 f i) ** simp_rw [hf_eq] ** case hn \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t f : \u2115 \u2192 Set \u211d hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i) \u22a2 Pairwise (Disjoint on fun i => s \u00d7\u02e2 f i) ** intro i j hij ** case hn \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t f : \u2115 \u2192 Set \u211d hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i) i j : \u2115 hij : i \u2260 j \u22a2 (Disjoint on fun i => s \u00d7\u02e2 f i) i j ** rw [Function.onFun, disjoint_prod] ** case hn \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t f : \u2115 \u2192 Set \u211d hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i) i j : \u2115 hij : i \u2260 j \u22a2 Disjoint s s \u2228 Disjoint (f i) (f j) ** exact Or.inr (hf_disj hij) ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s t : Set \u211d ht : MeasurableSet t f : \u2115 \u2192 Set \u211d hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i) \u22a2 \u2200 (i : \u2115), MeasurableSet (s \u00d7\u02e2 f i) ** exact fun i => MeasurableSet.prod hs (hf_meas i) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.setToL1_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\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 0 C f : { x // x \u2208 Lp E 1 } \u22a2 \u2191(setToL1 hT) f = 0 ** suffices setToL1 hT = 0 by rw [this]; simp ** \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 0 C f : { x // x \u2208 Lp E 1 } \u22a2 setToL1 hT = 0 ** 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 0 C f : { x // x \u2208 Lp E 1 } \u22a2 ContinuousLinearMap.comp 0 (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 0 C f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(ContinuousLinearMap.comp 0 (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** rw [setToL1SCLM_zero_left hT f, ContinuousLinearMap.zero_comp, ContinuousLinearMap.zero_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 0 C f : { x // x \u2208 Lp E 1 } this : setToL1 hT = 0 \u22a2 \u2191(setToL1 hT) f = 0 ** 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 T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc 0 C f : { x // x \u2208 Lp E 1 } this : setToL1 hT = 0 \u22a2 \u21910 f = 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.eq_zero_iff_ae_eq_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 f : { x // x \u2208 Lp E p } \u22a2 f = 0 \u2194 \u2191\u2191f =\u1d50[\u03bc] 0 ** rw [\u2190 (Lp.mem\u2112p f).toLp_eq_toLp_iff zero_mem\u2112p, Mem\u2112p.toLp_zero, toLp_coeFn] ** Qed",
        "informal": ""
    },
    {
        "formal": "QPF.Cofix.bisim ** 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 Liftr r (dest x) (dest y) \u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y ** apply Cofix.bisim_rel ** 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 Liftr r (dest x) (dest y) \u22a2 \u2200 (x y : Cofix F), r x y \u2192 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y ** intro x y 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 Liftr r (dest x) (dest y) x y : Cofix F rxy : r x y \u22a2 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y ** rcases (liftr_iff r _ _).mp (h x y rxy) with \u27e8a, f\u2080, f\u2081, dxeq, dyeq, h'\u27e9 ** case h.intro.intro.intro.intro.intro 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 Liftr r (dest x) (dest y) x y : Cofix F rxy : r x y a : (P F).A f\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F dxeq : dest x = abs { fst := a, snd := f\u2080 } dyeq : dest y = abs { fst := a, snd := f\u2081 } h' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i) \u22a2 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y ** rw [dxeq, dyeq, \u2190 abs_map, \u2190 abs_map, PFunctor.map_eq, PFunctor.map_eq] ** case h.intro.intro.intro.intro.intro 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 Liftr r (dest x) (dest y) x y : Cofix F rxy : r x y a : (P F).A f\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F dxeq : dest x = abs { fst := a, snd := f\u2080 } dyeq : dest y = abs { fst := a, snd := f\u2081 } h' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i) \u22a2 abs { fst := a, snd := (Quot.mk fun x y => r x y) \u2218 f\u2080 } = abs { fst := a, snd := (Quot.mk fun x y => r x y) \u2218 f\u2081 } ** congr 2 with i ** case h.intro.intro.intro.intro.intro.e_a.e_snd.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 Liftr r (dest x) (dest y) x y : Cofix F rxy : r x y a : (P F).A f\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F dxeq : dest x = abs { fst := a, snd := f\u2080 } dyeq : dest y = abs { fst := a, snd := f\u2081 } h' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i) i : PFunctor.B (P F) a \u22a2 ((Quot.mk fun x y => r x y) \u2218 f\u2080) i = ((Quot.mk fun x y => r x y) \u2218 f\u2081) i ** apply Quot.sound ** case h.intro.intro.intro.intro.intro.e_a.e_snd.h.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 Liftr r (dest x) (dest y) x y : Cofix F rxy : r x y a : (P F).A f\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F dxeq : dest x = abs { fst := a, snd := f\u2080 } dyeq : dest y = abs { fst := a, snd := f\u2081 } h' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i) i : PFunctor.B (P F) a \u22a2 r (f\u2080 i) (f\u2081 i) ** apply h' ** Qed",
        "informal": ""
    },
    {
        "formal": "ContinuousMap.toLp_denseRange ** \u03b1 : Type u_1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b1 inst\u271d\u2079 : T4Space \u03b1 inst\u271d\u2078 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u2076 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 \ud835\udd5c : Type u_3 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedAlgebra \u211d \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : CompactSpace \u03b1 inst\u271d\u00b9 : Measure.WeaklyRegular \u03bc inst\u271d : IsFiniteMeasure \u03bc \u22a2 DenseRange \u2191(toLp p \u03bc \ud835\udd5c) ** refine (BoundedContinuousFunction.toLp_denseRange _ _ hp \ud835\udd5c).mono ?_ ** \u03b1 : Type u_1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b1 inst\u271d\u2079 : T4Space \u03b1 inst\u271d\u2078 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u2076 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 \ud835\udd5c : Type u_3 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedAlgebra \u211d \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : CompactSpace \u03b1 inst\u271d\u00b9 : Measure.WeaklyRegular \u03bc inst\u271d : IsFiniteMeasure \u03bc \u22a2 range \u2191(BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c) \u2286 range \u2191(toLp p \u03bc \ud835\udd5c) ** refine range_subset_iff.2 fun f \u21a6 ?_ ** \u03b1 : Type u_1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b1 inst\u271d\u2079 : T4Space \u03b1 inst\u271d\u2078 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u2076 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 \ud835\udd5c : Type u_3 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedAlgebra \u211d \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : CompactSpace \u03b1 inst\u271d\u00b9 : Measure.WeaklyRegular \u03bc inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192\u1d47 E \u22a2 \u2191(BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c) f \u2208 range \u2191(toLp p \u03bc \ud835\udd5c) ** exact \u27e8f.toContinuousMap, rfl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Iterator.ValidFor.extract ** l m r : List Char it\u2081 it\u2082 : Iterator h\u2081 : ValidFor l (m ++ r) it\u2081 h\u2082 : ValidFor (List.reverse m ++ l) r it\u2082 \u22a2 Iterator.extract it\u2081 it\u2082 = { data := m } ** cases h\u2081.out ** case refl l m r : List Char it\u2082 : Iterator h\u2082 : ValidFor (List.reverse m ++ l) r it\u2082 h\u2081 : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } } \u22a2 Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } } it\u2082 = { data := m } ** cases h\u2082.out ** case refl.refl l m r : List Char h\u2081 : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } } h\u2082 :   ValidFor (List.reverse m ++ l) r     { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } } \u22a2 Iterator.extract { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } }       { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } } =     { data := m } ** simp [Iterator.extract, List.reverseAux_eq, Nat.not_lt.2 (Nat.le_add_left ..)] ** case refl.refl l m r : List Char h\u2081 : ValidFor l (m ++ r) { s := { data := List.reverseAux l (m ++ r) }, i := { byteIdx := utf8Len l } } h\u2082 :   ValidFor (List.reverse m ++ l) r     { s := { data := List.reverseAux (List.reverse m ++ l) r }, i := { byteIdx := utf8Len (List.reverse m ++ l) } } \u22a2 String.extract { data := List.reverse l ++ (m ++ r) } { byteIdx := utf8Len l } { byteIdx := utf8Len m + utf8Len l } =     { data := m } ** simpa [Nat.add_comm] using extract_of_valid l.reverse m r ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.prod_smul ** \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 inst\u271d\u00b2 : SigmaFinite \u03bd \ud835\udd5c : Type u_7 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hf : Integrable f hg : Integrable g \u22a2 Integrable fun z => f z.1 \u2022 g z.2 ** refine' (integrable_prod_iff _).2 \u27e8_, _\u27e9 ** case refine'_1 \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 inst\u271d\u00b2 : SigmaFinite \u03bd \ud835\udd5c : Type u_7 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hf : Integrable f hg : Integrable g \u22a2 AEStronglyMeasurable (fun z => f z.1 \u2022 g z.2) (Measure.prod \u03bc \u03bd) ** exact hf.1.fst.smul hg.1.snd ** case refine'_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\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 inst\u271d\u00b2 : SigmaFinite \u03bd \ud835\udd5c : Type u_7 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hf : Integrable f hg : Integrable g \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Integrable fun y => f (x, y).1 \u2022 g (x, y).2 ** exact eventually_of_forall fun x => hg.smul (f x) ** case refine'_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\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 inst\u271d\u00b2 : SigmaFinite \u03bd \ud835\udd5c : Type u_7 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c g : \u03b2 \u2192 E hf : Integrable f hg : Integrable g \u22a2 Integrable fun x => \u222b (y : \u03b2), \u2016f (x, y).1 \u2022 g (x, y).2\u2016 \u2202\u03bd ** simpa only [norm_smul, integral_mul_left] using hf.norm.mul_const _ ** Qed",
        "informal": ""
    },
    {
        "formal": "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 Ioi (-c), f x ** have A : MeasurableEmbedding fun x : \u211d => -x :=\n  (Homeomorph.neg \u211d).closedEmbedding.measurableEmbedding ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E c : \u211d f : \u211d \u2192 E A : MeasurableEmbedding fun x => -x \u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x ** have := MeasurableEmbedding.set_integral_map (\u03bc := volume) A f (Ici (-c)) ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E c : \u211d f : \u211d \u2192 E A : MeasurableEmbedding fun x => -x this : \u222b (y : \u211d) in Ici (-c), f y \u2202Measure.map (fun x => -x) volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x) \u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x ** rw [Measure.map_neg_eq_self (volume : Measure \u211d)] at this ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E c : \u211d f : \u211d \u2192 E A : MeasurableEmbedding fun x => -x this : \u222b (y : \u211d) in Ici (-c), f y = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x) \u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x ** simp_rw [\u2190 integral_Ici_eq_integral_Ioi, this, neg_preimage, preimage_neg_Ici, neg_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.jordanDecomposition_add_withDensity_mutuallySingular ** \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 ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc \u22a2 ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098     (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x) ** rw [mutuallySingular_ennreal_iff, totalVariation_mutuallySingular_iff] at ht\u03bc ** \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 ht\u03bc :   (toJordanDecomposition t).posPart \u27c2\u2098 VectorMeasure.ennrealToMeasure (toENNRealVectorMeasure \u03bc) \u2227     (toJordanDecomposition t).negPart \u27c2\u2098 VectorMeasure.ennrealToMeasure (toENNRealVectorMeasure \u03bc) \u22a2 ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098     (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x) ** change\n  _ \u27c2\u2098 VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun \u03bc) \u2227\n    _ \u27c2\u2098 VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun \u03bc) at ht\u03bc ** \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 ht\u03bc :   (toJordanDecomposition t).posPart \u27c2\u2098       Equiv.toFun VectorMeasure.equivMeasure (Equiv.invFun VectorMeasure.equivMeasure \u03bc) \u2227     (toJordanDecomposition t).negPart \u27c2\u2098       Equiv.toFun VectorMeasure.equivMeasure (Equiv.invFun VectorMeasure.equivMeasure \u03bc) \u22a2 ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098     (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x) ** rw [VectorMeasure.equivMeasure.right_inv] at ht\u03bc ** \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 ht\u03bc : (toJordanDecomposition t).posPart \u27c2\u2098 \u03bc \u2227 (toJordanDecomposition t).negPart \u27c2\u2098 \u03bc \u22a2 ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098     (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x) ** exact\n  ((JordanDecomposition.mutuallySingular _).add_right\n        (ht\u03bc.1.mono_ac (refl _) (withDensity_absolutelyContinuous _ _))).add_left\n    ((ht\u03bc.2.symm.mono_ac (withDensity_absolutelyContinuous _ _) (refl _)).add_right\n      (withDensity_ofReal_mutuallySingular hf)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.mem_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\u271d : \u03b1 m n\u271d : \u2115 a : \u03b1 n : \u2115 \u22a2 a \u2208 s ^ n \u2194 \u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = a ** simp [\u2190 mem_coe, coe_pow, Set.mem_pow] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.continuousWithinAt_primitive ** \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) \u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** by_cases h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 ** 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 \u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** have h\u2081\u2082 : b\u2081 \u2264 b\u2082 := h\u2080.1.trans h\u2080.2 ** 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 \u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** have min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 := min_eq_left h\u2081\u2082 ** 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this : \u2200 (b : \u211d), b \u2208 Icc b\u2081 b\u2082 \u2192 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc \u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** apply ContinuousWithinAt.congr _ this (this _ h\u2080) ** \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this : \u2200 (b : \u211d), b \u2208 Icc b\u2081 b\u2082 \u2192 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc \u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** clear 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x \u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** refine' continuousWithinAt_const.add _ ** \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x \u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** have :\n  (fun b => \u222b x in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b =>\n    \u222b x in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc := by\n  apply eventuallyEq_of_mem self_mem_nhdsWithin\n  exact fun b b_in => (integral_indicator b_in).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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc \u22a2 ContinuousWithinAt (fun y => \u222b (x : \u211d) in b\u2081..y, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** apply ContinuousWithinAt.congr_of_eventuallyEq _ this (integral_indicator h\u2080).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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc \u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** have : IntervalIntegrable (fun x => \u2016f x\u2016) \u03bc b\u2081 b\u2082 :=\n  IntervalIntegrable.norm (h_int' <| right_mem_Icc.mpr h\u2081\u2082) ** \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : IntervalIntegrable (fun x => \u2016f x\u2016) \u03bc b\u2081 b\u2082 \u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** refine' continuousWithinAt_of_dominated_interval _ _ this _ <;> clear 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 \u22a2 \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x ** rintro x \u27e8h\u2081, h\u2082\u27e9 ** case 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 x : \u211d h\u2081 : b\u2081 \u2264 x h\u2082 : x \u2264 b\u2082 \u22a2 IntervalIntegrable f \u03bc b\u2081 x ** apply h_int.mono_set ** case 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 x : \u211d h\u2081 : b\u2081 \u2264 x h\u2082 : x \u2264 b\u2082 \u22a2 [[b\u2081, x]] \u2286 [[min a b\u2081, max a b\u2082]] ** apply uIcc_subset_uIcc ** case intro.h\u2081 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 x : \u211d h\u2081 : b\u2081 \u2264 x h\u2082 : x \u2264 b\u2082 \u22a2 b\u2081 \u2208 [[min a b\u2081, max a b\u2082]] ** exact \u27e8min_le_of_left_le (min_le_right a b\u2081),\n  h\u2081.trans (h\u2082.trans <| le_max_of_le_right <| le_max_right _ _)\u27e9 ** case intro.h\u2082 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 x : \u211d h\u2081 : b\u2081 \u2264 x h\u2082 : x \u2264 b\u2082 \u22a2 x \u2208 [[min a b\u2081, max a b\u2082]] ** exact \u27e8min_le_of_left_le <| (min_le_right _ _).trans h\u2081,\n  le_max_of_le_right <| h\u2082.trans <| le_max_right _ _\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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x \u22a2 \u2200 (b : \u211d), b \u2208 Icc b\u2081 b\u2082 \u2192 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc ** rintro b \u27e8h\u2081, h\u2082\u27e9 ** case 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 a b\u271d b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x b : \u211d h\u2081 : b\u2081 \u2264 b h\u2082 : b \u2264 b\u2082 \u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b\u2081, f x \u2202\u03bc + \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc ** rw [\u2190 integral_add_adjacent_intervals _ (h_int' \u27e8h\u2081, h\u2082\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 a b\u271d b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x b : \u211d h\u2081 : b\u2081 \u2264 b h\u2082 : b \u2264 b\u2082 \u22a2 IntervalIntegrable f \u03bc a b\u2081 ** apply h_int.mono_set ** \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\u271d b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x b : \u211d h\u2081 : b\u2081 \u2264 b h\u2082 : b \u2264 b\u2082 \u22a2 [[a, b\u2081]] \u2286 [[min a b\u2081, max a b\u2082]] ** apply uIcc_subset_uIcc ** case h\u2081 \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\u271d b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x b : \u211d h\u2081 : b\u2081 \u2264 b h\u2082 : b \u2264 b\u2082 \u22a2 a \u2208 [[min a b\u2081, max a b\u2082]] ** exact \u27e8min_le_of_left_le (min_le_left a b\u2081), le_max_of_le_right (le_max_left _ _)\u27e9 ** case h\u2082 \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\u271d b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x b : \u211d h\u2081 : b\u2081 \u2264 b h\u2082 : b \u2264 b\u2082 \u22a2 b\u2081 \u2208 [[min a b\u2081, max a b\u2082]] ** exact \u27e8min_le_of_left_le (min_le_right _ _),\n  le_max_of_le_right (h\u2081.trans <| h\u2082.trans (le_max_right a b\u2082))\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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x \u22a2 (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc ** apply eventuallyEq_of_mem self_mem_nhdsWithin ** \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x \u22a2 EqOn (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) (fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc) (Icc b\u2081 b\u2082) ** exact fun b b_in => (integral_indicator b_in).symm ** case refine'_1 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc \u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080,     AEStronglyMeasurable (fun x_1 => indicator {x_2 | x_2 \u2264 x} f x_1) (Measure.restrict \u03bc (\u0399 b\u2081 b\u2082)) ** apply Eventually.mono self_mem_nhdsWithin ** case refine'_1 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc \u22a2 \u2200 (x : \u211d),     b\u2081 \u2264 x \u2227 x \u2264 b\u2082 \u2192 AEStronglyMeasurable (fun x_1 => indicator {x_2 | x_2 \u2264 x} f x_1) (Measure.restrict \u03bc (\u0399 b\u2081 b\u2082)) ** intro x hx ** case refine'_1 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc x : \u211d hx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082 \u22a2 AEStronglyMeasurable (fun x_1 => indicator {x_2 | x_2 \u2264 x} f x_1) (Measure.restrict \u03bc (\u0399 b\u2081 b\u2082)) ** erw [aestronglyMeasurable_indicator_iff, Measure.restrict_restrict, Iic_inter_Ioc_of_le] ** case refine'_1 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc x : \u211d hx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082 \u22a2 MeasurableSet {x_1 | x_1 \u2264 x}  case refine'_1 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc x : \u211d hx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082 \u22a2 MeasurableSet {x_1 | x_1 \u2264 x} ** exacts [measurableSet_Iic, measurableSet_Iic] ** case refine'_1 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc x : \u211d hx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082 \u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc (min b\u2081 b\u2082) x)) ** rw [min\u2081\u2082] ** case refine'_1 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc x : \u211d hx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082 \u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc b\u2081 x)) ** exact (h_int' hx).1.aestronglyMeasurable ** case refine'_1 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc x : \u211d hx : b\u2081 \u2264 x \u2227 x \u2264 b\u2082 \u22a2 x \u2264 max b\u2081 b\u2082 ** exact le_max_of_le_right hx.2 ** case refine'_2 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc \u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 b\u2081 b\u2082 \u2192 \u2016indicator {x_1 | x_1 \u2264 x} f t\u2016 \u2264 \u2016f t\u2016 ** refine' eventually_of_forall fun x => eventually_of_forall fun t => _ ** case refine'_2 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc x t : \u211d \u22a2 t \u2208 \u0399 b\u2081 b\u2082 \u2192 \u2016indicator {x_1 | x_1 \u2264 x} f t\u2016 \u2264 \u2016f t\u2016 ** dsimp [indicator] ** case refine'_2 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc x t : \u211d \u22a2 t \u2208 \u0399 b\u2081 b\u2082 \u2192 \u2016if t \u2264 x then f t else 0\u2016 \u2264 \u2016f t\u2016 ** split_ifs <;> simp ** case refine'_3 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc \u22a2 \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 b\u2081 b\u2082 \u2192 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f t) (Icc b\u2081 b\u2082) b\u2080 ** have : \u2200\u1d50 t \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t := by\n  apply Eventually.mono (compl_mem_ae_iff.mpr hb\u2080)\n  intro x hx\n  exact Ne.lt_or_lt hx ** case refine'_3 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t \u22a2 \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 b\u2081 b\u2082 \u2192 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f t) (Icc b\u2081 b\u2082) b\u2080 ** apply this.mono ** case refine'_3 \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t \u22a2 \u2200 (x : \u211d), x < b\u2080 \u2228 b\u2080 < x \u2192 x \u2208 \u0399 b\u2081 b\u2082 \u2192 ContinuousWithinAt (fun x_1 => indicator {x | x \u2264 x_1} f x) (Icc b\u2081 b\u2082) b\u2080 ** rintro x\u2080 (hx\u2080 | hx\u2080) - ** \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc \u22a2 \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t ** apply Eventually.mono (compl_mem_ae_iff.mpr hb\u2080) ** \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc \u22a2 \u2200 (x : \u211d), \u00acx \u2208 {b\u2080} \u2192 x < b\u2080 \u2228 b\u2080 < x ** intro 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc x : \u211d hx : \u00acx \u2208 {b\u2080} \u22a2 x < b\u2080 \u2228 b\u2080 < x ** exact Ne.lt_or_lt hx ** case refine'_3.inl \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : x\u2080 < b\u2080 \u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080 ** have : \u2200\u1da0 x in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, {t : \u211d | t \u2264 x}.indicator f x\u2080 = f x\u2080 := by\n  apply mem_nhdsWithin_of_mem_nhds\n  apply Eventually.mono (Ioi_mem_nhds hx\u2080)\n  intro x hx\n  simp [hx.le] ** case refine'_3.inl \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d\u00b9 :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : x\u2080 < b\u2080 this : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = f x\u2080 \u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080 ** apply continuousWithinAt_const.congr_of_eventuallyEq this ** case refine'_3.inl \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d\u00b9 :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : x\u2080 < b\u2080 this : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = f x\u2080 \u22a2 indicator {t | t \u2264 b\u2080} f x\u2080 = f x\u2080 ** simp [hx\u2080.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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : x\u2080 < b\u2080 \u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = f x\u2080 ** apply mem_nhdsWithin_of_mem_nhds ** 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : x\u2080 < b\u2080 \u22a2 {x | (fun x => indicator {t | t \u2264 x} f x\u2080 = f x\u2080) x} \u2208 \ud835\udcdd b\u2080 ** apply Eventually.mono (Ioi_mem_nhds hx\u2080) ** 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : x\u2080 < b\u2080 \u22a2 \u2200 (x : \u211d), x\u2080 < x \u2192 indicator {t | t \u2264 x} f x\u2080 = f x\u2080 ** intro 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : x\u2080 < b\u2080 x : \u211d hx : x\u2080 < x \u22a2 indicator {t | t \u2264 x} f x\u2080 = f x\u2080 ** simp [hx.le] ** case refine'_3.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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : b\u2080 < x\u2080 \u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080 ** have : \u2200\u1da0 x in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, {t : \u211d | t \u2264 x}.indicator f x\u2080 = 0 := by\n  apply mem_nhdsWithin_of_mem_nhds\n  apply Eventually.mono (Iio_mem_nhds hx\u2080)\n  intro x hx\n  simp [hx] ** case refine'_3.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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d\u00b9 :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : b\u2080 < x\u2080 this : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = 0 \u22a2 ContinuousWithinAt (fun x => indicator {x_1 | x_1 \u2264 x} f x\u2080) (Icc b\u2081 b\u2082) b\u2080 ** apply continuousWithinAt_const.congr_of_eventuallyEq this ** case refine'_3.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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d\u00b9 :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this\u271d : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : b\u2080 < x\u2080 this : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = 0 \u22a2 indicator {t | t \u2264 b\u2080} f x\u2080 = 0 ** simp [hx\u2080] ** \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : b\u2080 < x\u2080 \u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc b\u2081 b\u2082] b\u2080, indicator {t | t \u2264 x} f x\u2080 = 0 ** apply mem_nhdsWithin_of_mem_nhds ** 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : b\u2080 < x\u2080 \u22a2 {x | (fun x => indicator {t | t \u2264 x} f x\u2080 = 0) x} \u2208 \ud835\udcdd b\u2080 ** apply Eventually.mono (Iio_mem_nhds hx\u2080) ** 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : b\u2080 < x\u2080 \u22a2 \u2200 (x : \u211d), x < x\u2080 \u2192 indicator {t | t \u2264 x} f x\u2080 = 0 ** intro 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : b\u2080 \u2208 Icc b\u2081 b\u2082 h\u2081\u2082 : b\u2081 \u2264 b\u2082 min\u2081\u2082 : min b\u2081 b\u2082 = b\u2081 h_int' : \u2200 {x : \u211d}, x \u2208 Icc b\u2081 b\u2082 \u2192 IntervalIntegrable f \u03bc b\u2081 x this\u271d :   (fun b => \u222b (x : \u211d) in b\u2081..b, f x \u2202\u03bc) =\u1da0[\ud835\udcdd[Icc b\u2081 b\u2082] b\u2080] fun b => \u222b (x : \u211d) in b\u2081..b\u2082, indicator {x | x \u2264 b} f x \u2202\u03bc this : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t < b\u2080 \u2228 b\u2080 < t x\u2080 : \u211d hx\u2080 : b\u2080 < x\u2080 x : \u211d hx : x < x\u2080 \u22a2 indicator {t | t \u2264 x} f x\u2080 = 0 ** simp [hx] ** 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 a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : \u00acb\u2080 \u2208 Icc b\u2081 b\u2082 \u22a2 ContinuousWithinAt (fun b => \u222b (x : \u211d) in a..b, f x \u2202\u03bc) (Icc b\u2081 b\u2082) b\u2080 ** apply continuousWithinAt_of_not_mem_closure ** case neg.a \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 b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E hb\u2080 : \u2191\u2191\u03bc {b\u2080} = 0 h_int : IntervalIntegrable f \u03bc (min a b\u2081) (max a b\u2082) h\u2080 : \u00acb\u2080 \u2208 Icc b\u2081 b\u2082 \u22a2 \u00acb\u2080 \u2208 closure (Icc b\u2081 b\u2082) ** rwa [closure_Icc] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.exists_ne_map_eq_of_ncard_lt_of_maps_to ** \u03b1 : Type u_2 s t\u271d : Set \u03b1 \u03b2 : Type u_1 t : Set \u03b2 hc : ncard t < ncard s f : \u03b1 \u2192 \u03b2 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t ht : autoParam (Set.Finite t) _auto\u271d \u22a2 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y ** by_contra h' ** \u03b1 : Type u_2 s t\u271d : Set \u03b1 \u03b2 : Type u_1 t : Set \u03b2 hc : ncard t < ncard s f : \u03b1 \u2192 \u03b2 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t ht : autoParam (Set.Finite t) _auto\u271d h' : \u00ac\u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y \u22a2 False ** simp only [Ne.def, exists_prop, not_exists, not_and, not_imp_not] at h' ** \u03b1 : Type u_2 s t\u271d : Set \u03b1 \u03b2 : Type u_1 t : Set \u03b2 hc : ncard t < ncard s f : \u03b1 \u2192 \u03b2 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t ht : autoParam (Set.Finite t) _auto\u271d h' : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 f x = f x_1 \u2192 x = x_1 \u22a2 False ** exact (ncard_le_ncard_of_injOn f hf h' ht).not_lt hc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_le_integral ** \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 \u22a2 \u2203 x, f x \u2264 \u222b (a : \u03b1), f a \u2202\u03bc ** simpa only [average_eq_integral] using exists_le_average (IsProbabilityMeasure.ne_zero \u03bc) hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.exists_intermediate_Set ** \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2081 : i + ncard s \u2264 ncard t h\u2082 : s \u2286 t \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s ** cases' t.finite_or_infinite with ht ht ** case inr \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2081 : i + ncard s \u2264 ncard t h\u2082 : s \u2286 t ht : Set.Infinite t \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s ** rw [ht.ncard] at h\u2081 ** case inr \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2081 : i + ncard s \u2264 0 h\u2082 : s \u2286 t ht : Set.Infinite t \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s ** have h\u2081' := Nat.eq_zero_of_le_zero h\u2081 ** case inr \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2081 : i + ncard s \u2264 0 h\u2082 : s \u2286 t ht : Set.Infinite t h\u2081' : i + ncard s = 0 \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s ** rw [add_eq_zero_iff] at h\u2081' ** case inr \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2081 : i + ncard s \u2264 0 h\u2082 : s \u2286 t ht : Set.Infinite t h\u2081' : i = 0 \u2227 ncard s = 0 \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s ** refine' \u27e8t, h\u2082, rfl.subset, _\u27e9 ** case inr \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2081 : i + ncard s \u2264 0 h\u2082 : s \u2286 t ht : Set.Infinite t h\u2081' : i = 0 \u2227 ncard s = 0 \u22a2 ncard t = i + ncard s ** rw [h\u2081'.2, h\u2081'.1, ht.ncard, add_zero] ** case inl \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2081 : i + ncard s \u2264 ncard t h\u2082 : s \u2286 t ht : Set.Finite t \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + ncard s ** rw [ncard_eq_toFinset_card _ (ht.subset h\u2082)] at h\u2081 \u22a2 ** case inl \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2082 : s \u2286 t ht : Set.Finite t h\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 ncard t \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ** rw [ncard_eq_toFinset_card t ht] at h\u2081 ** case inl \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2082 : s \u2286 t ht : Set.Finite t h\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht) \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ** obtain \u27e8r', hsr', hr't, hr'\u27e9 := Finset.exists_intermediate_set _ h\u2081 (by simpa) ** case inl.intro.intro.intro \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2082 : s \u2286 t ht : Set.Finite t h\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht) r' : Finset \u03b1 hsr' : Finite.toFinset (_ : Set.Finite s) \u2286 r' hr't : r' \u2286 Finite.toFinset ht hr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ** exact \u27e8r', by simpa using hsr', by simpa using hr't, by rw [\u2190 hr', ncard_coe_Finset]\u27e9 ** \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2082 : s \u2286 t ht : Set.Finite t h\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht) \u22a2 Finite.toFinset (_ : Set.Finite s) \u2286 Finite.toFinset ht ** simpa ** \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2082 : s \u2286 t ht : Set.Finite t h\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht) r' : Finset \u03b1 hsr' : Finite.toFinset (_ : Set.Finite s) \u2286 r' hr't : r' \u2286 Finite.toFinset ht hr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u22a2 s \u2286 \u2191r' ** simpa using hsr' ** \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2082 : s \u2286 t ht : Set.Finite t h\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht) r' : Finset \u03b1 hsr' : Finite.toFinset (_ : Set.Finite s) \u2286 r' hr't : r' \u2286 Finite.toFinset ht hr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u22a2 \u2191r' \u2286 t ** simpa using hr't ** \u03b1 : Type u_1 s t : Set \u03b1 i : \u2115 h\u2082 : s \u2286 t ht : Set.Finite t h\u2081 : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u2264 Finset.card (Finite.toFinset ht) r' : Finset \u03b1 hsr' : Finite.toFinset (_ : Set.Finite s) \u2286 r' hr't : r' \u2286 Finite.toFinset ht hr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s)) \u22a2 ncard \u2191r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ** rw [\u2190 hr', ncard_coe_Finset] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpIndL1Fin_disjoint_union ** \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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G \u22a2 condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) (_ : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4) x =     condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G \u22a2 \u2191\u2191(condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) (_ : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4) x) =\u1d50[\u03bc]     \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct x) ** have h\u03bcst :=\n  ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr \u27e8h\u03bcs, h\u03bct\u27e9))).ne ** 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 \u22a2 \u2191\u2191(condexpIndL1Fin hm (_ : MeasurableSet (s \u222a t)) (_ : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4) x) =\u1d50[\u03bc]     \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct x) ** refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) h\u03bcst x).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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 \u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]     \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm ht h\u03bct x) ** 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 \u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]     \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) ** have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h\u03bcs 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) \u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]     \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) ** have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht h\u03bct 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x) \u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc]     \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) ** refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x) \u22a2 \u2191\u2191(condexpIndSMul hm (_ : MeasurableSet (s \u222a t)) h\u03bcst x) =\u1d50[\u03bc] fun x_1 =>     \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1 ** rw [condexpIndSMul] ** 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x) \u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))           \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 (_ : MeasurableSet (s \u222a t)) h\u03bcst 1))) =\u1d50[\u03bc]     fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1 ** rw [indicatorConstLp_disjoint_union hs ht h\u03bcs h\u03bct hst (1 : \u211d)] ** 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x) \u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))           \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1 + indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]     fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1 ** rw [(condexpL2 \u211d \u211d hm).map_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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x) \u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))           \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1) +               \u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]     fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1 ** push_cast ** 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x) \u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x))           (\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +             \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1)))) =\u1d50[\u03bc]     fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1 ** rw [((toSpanSingleton \u211d x).compLpL 2 \u03bc).map_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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x) \u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +           \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]     fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1 ** refine' (Lp.coeFn_add _ _).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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x) \u22a2 \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) +       \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))) =\u1d50[\u03bc]     fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1 ** refine' eventually_of_forall fun y => _ ** 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 ht : MeasurableSet t h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 hst : s \u2229 t = \u2205 x : G h\u03bcst : \u2191\u2191\u03bc (s \u222a t) \u2260 \u22a4 hs_eq : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ht_eq : \u2191\u2191(condexpIndL1Fin hm ht h\u03bct x) =\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm ht h\u03bct x) y : \u03b1 \u22a2 (\u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) +         \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1))))       y =     (fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm ht h\u03bct x) x_1) y ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableEquiv.measurable_comp_iff ** \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 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f : \u03b2 \u2192 \u03b3 e : \u03b1 \u2243\u1d50 \u03b2 hfe : Measurable (f \u2218 \u2191e) \u22a2 Measurable f ** have : Measurable (f \u2218 (e.symm.trans e).toEquiv) := hfe.comp e.symm.measurable ** \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 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f : \u03b2 \u2192 \u03b3 e : \u03b1 \u2243\u1d50 \u03b2 hfe : Measurable (f \u2218 \u2191e) this : Measurable (f \u2218 \u2191(trans (symm e) e).toEquiv) \u22a2 Measurable f ** rwa [coe_toEquiv, symm_trans_self] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_liftCover ** \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 htm : \u2200 (i : \u03b9), MeasurableSet (t i) f : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 hfm : \u2200 (i : \u03b9), Measurable (f i) hf :   \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 } htU : \u22c3 i, t i = univ s : Set \u03b2 hs : MeasurableSet s \u22a2 MeasurableSet (liftCover t f hf htU \u207b\u00b9' s) ** rw [preimage_liftCover] ** \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 htm : \u2200 (i : \u03b9), MeasurableSet (t i) f : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 hfm : \u2200 (i : \u03b9), Measurable (f i) hf :   \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 } htU : \u22c3 i, t i = univ s : Set \u03b2 hs : MeasurableSet s \u22a2 MeasurableSet (\u22c3 i, Subtype.val '' (f i \u207b\u00b9' s)) ** exact .iUnion fun i => .subtype_image (htm i) <| hfm i hs ** Qed",
        "informal": ""
    },
    {
        "formal": "lipschitzWith_circleMap ** E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R \u03b8 : \u211d \u22a2 \u2191\u2016deriv (circleMap c R) \u03b8\u2016\u208a \u2264 \u2191(\u2191Real.nnabs R) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_ball_center ** E\u271d : Type u_1 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d inst\u271d\u00b9\u2070 : MeasurableSpace E\u271d inst\u271d\u2079 : BorelSpace E\u271d inst\u271d\u2078 : FiniteDimensional \u211d E\u271d \u03bc\u271d : Measure E\u271d inst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d F : Type u_2 inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F s : Set E\u271d E : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc x : E r : \u211d this : ball 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' ball x r \u22a2 \u2191\u2191\u03bc (ball x r) = \u2191\u2191\u03bc (ball 0 r) ** rw [this, measure_preimage_add] ** E\u271d : Type u_1 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d inst\u271d\u00b9\u2070 : MeasurableSpace E\u271d inst\u271d\u2079 : BorelSpace E\u271d inst\u271d\u2078 : FiniteDimensional \u211d E\u271d \u03bc\u271d : Measure E\u271d inst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d F : Type u_2 inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F s : Set E\u271d E : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc x : E r : \u211d \u22a2 ball 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' ball x r ** simp [preimage_add_ball] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_eq_zero_and_zero_of_ae_le_mul_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 f : \u03b1 \u2192 F g : \u03b1 \u2192 G c : \u211d h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c * \u2016g x\u2016 hc : c < 0 p : \u211d\u22650\u221e \u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 0 ** simp_rw [le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg (norm_nonneg _) hc (norm_nonneg _),\n  norm_eq_zero, eventually_and] at 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 f : \u03b1 \u2192 F g : \u03b1 \u2192 G c : \u211d hc : c < 0 p : \u211d\u22650\u221e h : (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x = 0) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x = 0 \u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 0 ** change f =\u1d50[\u03bc] 0 \u2227 g =\u1d50[\u03bc] 0 at 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 f : \u03b1 \u2192 F g : \u03b1 \u2192 G c : \u211d hc : c < 0 p : \u211d\u22650\u221e h : f =\u1d50[\u03bc] 0 \u2227 g =\u1d50[\u03bc] 0 \u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 0 ** simp [snorm_congr_ae h.1, snorm_congr_ae h.2] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.continuous_integral_integral ** \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' \u22a2 Continuous fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc ** rw [continuous_iff_continuousAt] ** \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' \u22a2 \u2200 (x : { x // x \u2208 Lp E 1 }), ContinuousAt (fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc) x ** intro g ** \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' g : { x // x \u2208 Lp E 1 } \u22a2 ContinuousAt (fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc) g ** refine'\n  tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_prod_left\n    (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_prod_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\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' g : { x // x \u2208 Lp E 1 } \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) \u2202\u03bd - \u222b (y : \u03b2), \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0) ** simp_rw [\u2190\n  lintegral_fn_integral_sub (fun x => (\u2016x\u2016\u208a : \u211d\u22650\u221e)) (L1.integrable_coeFn _)\n    (L1.integrable_coeFn g)] ** \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' g : { x // x \u2208 Lp E 1 } \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) (\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 \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' g : { x // x \u2208 Lp E 1 } \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)  case refine'_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' g : { x // x \u2208 Lp E 1 } \u22a2 (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) \u2264 fun i =>     \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc ** swap ** case refine'_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' g : { x // x \u2208 Lp E 1 } \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0) ** have : \u2200 i : \u03b1 \u00d7 \u03b2 \u2192\u2081[\u03bc.prod \u03bd] E, 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 \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' 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 : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0) ** conv =>\n  congr\n  ext\n  rw [\u2190 lintegral_prod_of_measurable _ (this _), \u2190 L1.ofReal_norm_sub_eq_lintegral] ** case refine'_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' 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 x => ENNReal.ofReal \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0) ** rw [\u2190 ofReal_zero] ** case refine'_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' 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 x => ENNReal.ofReal \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd (ENNReal.ofReal 0)) ** refine' (continuous_ofReal.tendsto 0).comp _ ** case refine'_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' 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 x => \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0) ** rw [\u2190 tendsto_iff_norm_sub_tendsto_zero] ** case refine'_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' 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 x => x) (\ud835\udcdd g) (\ud835\udcdd g) ** exact tendsto_id ** case refine'_1 \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' 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\u03bd \u2202\u03bc ** case refine'_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' g : { x // x \u2208 Lp E 1 } \u22a2 (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) \u2264 fun i =>     \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc ** exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.zoom_toList ** \u03b1 : Type u_1 cut : \u03b1 \u2192 Ordering t' : RBNode \u03b1 p' : Path \u03b1 t : RBNode \u03b1 eq : zoom cut t root = (t', p') \u22a2 withList p' (toList t') = toList t ** rw [\u2190 fill_toList, \u2190 zoom_fill eq] ** \u03b1 : Type u_1 cut : \u03b1 \u2192 Ordering t' : RBNode \u03b1 p' : Path \u03b1 t : RBNode \u03b1 eq : zoom cut t root = (t', p') \u22a2 toList (fill root t) = toList t ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.mapAccumr_eq_map ** \u03b1 : Type u_2 n : \u2115 \u03b2 : Type u_1 xs : Vector \u03b1 n ys : Vector \u03b2 n \u03c3 : Type f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2 s\u2080 : \u03c3 S : Set \u03c3 h\u2080 : s\u2080 \u2208 S closure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S out : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2 \u22a2 (mapAccumr f xs s\u2080).2 = map (fun x => (f x s\u2080).2) xs ** rw[Vector.map_eq_mapAccumr] ** \u03b1 : Type u_2 n : \u2115 \u03b2 : Type u_1 xs : Vector \u03b1 n ys : Vector \u03b2 n \u03c3 : Type f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2 s\u2080 : \u03c3 S : Set \u03c3 h\u2080 : s\u2080 \u2208 S closure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S out : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2 \u22a2 (mapAccumr f xs s\u2080).2 = (mapAccumr (fun x x_1 => ((), (f x s\u2080).2)) xs ()).2 ** apply mapAccumr_bisim_tail ** case h \u03b1 : Type u_2 n : \u2115 \u03b2 : Type u_1 xs : Vector \u03b1 n ys : Vector \u03b2 n \u03c3 : Type f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2 s\u2080 : \u03c3 S : Set \u03c3 h\u2080 : s\u2080 \u2208 S closure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S out : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2 \u22a2 \u2203 R, R s\u2080 () \u2227 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1), R s q \u2192 R (f a s).1 ((), (f a s\u2080).2).1 \u2227 (f a s).2 = ((), (f a s\u2080).2).2 ** use fun s _ => s \u2208 S, h\u2080 ** case right \u03b1 : Type u_2 n : \u2115 \u03b2 : Type u_1 xs : Vector \u03b1 n ys : Vector \u03b2 n \u03c3 : Type f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2 s\u2080 : \u03c3 S : Set \u03c3 h\u2080 : s\u2080 \u2208 S closure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).1 \u2208 S out : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).2 = (f a s').2 \u22a2 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1), s \u2208 S \u2192 (f a s).1 \u2208 S \u2227 (f a s).2 = ((), (f a s\u2080).2).2 ** exact @fun s _q a h => \u27e8closure a s h, out a s s\u2080 h h\u2080\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.mem_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 : LE \u03b1 x : \u03b1 z : \u03b2 \u22a2 z \u2208 Accumulate s x \u2194 \u2203 y, y \u2264 x \u2227 z \u2208 s y ** simp_rw [accumulate_def, mem_iUnion\u2082, exists_prop] ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.foldrM_eq_reverse_foldlM_data ** 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 init : \u03b2 arr : Array \u03b1 \u22a2 foldrM f init arr (size arr) = List.foldlM (fun x y => f y x) init (List.reverse arr.data) ** have : arr = #[] \u2228 0 < arr.size :=\n  match arr with | \u27e8[]\u27e9 => .inl rfl | \u27e8a::l\u27e9 => .inr (Nat.zero_lt_succ _) ** 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 init : \u03b2 arr : Array \u03b1 this : arr = #[] \u2228 0 < size arr \u22a2 foldrM f init arr (size arr) = List.foldlM (fun x y => f y x) init (List.reverse arr.data) ** match arr, this with | _, .inl rfl => rfl | arr, .inr 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 init : \u03b2 arr\u271d : Array \u03b1 this : arr\u271d = #[] \u2228 0 < size arr\u271d arr : Array \u03b1 h : 0 < size arr \u22a2 foldrM f init arr (size arr) = List.foldlM (fun x y => f y x) init (List.reverse arr.data) ** simp [foldrM, h, \u2190 foldrM_eq_reverse_foldlM_data.aux, List.take_length] ** 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 init : \u03b2 arr : Array \u03b1 this : arr = #[] \u2228 0 < size arr \u22a2 foldrM f init #[] (size #[]) = List.foldlM (fun x y => f y x) init (List.reverse #[].data) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.bind\u2081_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 \u03c5 : Type u_6 f : \u03c3 \u2192 MvPolynomial \u03c4 R g : \u03c4 \u2192 MvPolynomial \u03c5 R \u03c6 : MvPolynomial \u03c3 R \u22a2 \u2191(bind\u2081 g) (\u2191(bind\u2081 f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(bind\u2081 g) (f i)) \u03c6 ** simp [bind\u2081, \u2190 comp_aeval] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haarMeasure_apply ** G : Type u_1 inst\u271d\u2075 : Group G inst\u271d\u2074 : TopologicalSpace G inst\u271d\u00b3 : T2Space G inst\u271d\u00b2 : TopologicalGroup G inst\u271d\u00b9 : MeasurableSpace G inst\u271d : BorelSpace G K\u2080 : PositiveCompacts G s : Set G hs : MeasurableSet s \u22a2 \u2191\u2191(haarMeasure K\u2080) s = \u2191(Content.outerMeasure (haarContent K\u2080)) s / \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 ** change ((haarContent K\u2080).outerMeasure K\u2080)\u207b\u00b9 * (haarContent K\u2080).measure s = _ ** G : Type u_1 inst\u271d\u2075 : Group G inst\u271d\u2074 : TopologicalSpace G inst\u271d\u00b3 : T2Space G inst\u271d\u00b2 : TopologicalGroup G inst\u271d\u00b9 : MeasurableSpace G inst\u271d : BorelSpace G K\u2080 : PositiveCompacts G s : Set G hs : MeasurableSet s \u22a2 (\u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080)\u207b\u00b9 * \u2191\u2191(Content.measure (haarContent K\u2080)) s =     \u2191(Content.outerMeasure (haarContent K\u2080)) s / \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 ** simp only [hs, div_eq_mul_inv, mul_comm, Content.measure_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.dlookup_list_toFinmap ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 s : List (Sigma \u03b2) \u22a2 lookup a (List.toFinmap s) = dlookup a s ** rw [List.toFinmap, lookup_toFinmap, lookup_to_alist] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1to0.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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 l\u2081 : Option \u039b v : \u03c3 T : Tape \u0393 \u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := l\u2081, var := v, Tape := T })     (TM1.step M { l := l\u2081, var := v, Tape := T }) ** cases' l\u2081 with l\u2081 ** case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v : \u03c3 T : Tape \u0393 l\u2081 : \u039b \u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := some l\u2081, var := v, Tape := T })     (TM1.step M { l := some l\u2081, var := v, Tape := T }) ** simp only [trCfg, TM1.step, FRespects, Option.map] ** case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v : \u03c3 T : Tape \u0393 l\u2081 : \u039b \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (M l\u2081), v), Tape := T }     {       q :=         (match (TM1.stepAux (M l\u2081) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (M l\u2081) v T).var),       Tape := (TM1.stepAux (M l\u2081) v T).Tape } ** induction' M l\u2081 with _ q IH _ q IH _ q IH generalizing v T ** case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : Dir q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.move a\u271d q), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write a\u271d q), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }       {         q :=           (match (TM1.stepAux a\u271d\u00b9 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux a\u271d\u00b9 v T).var),         Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape } a_ih\u271d :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }       {         q :=           (match (TM1.stepAux a\u271d v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux a\u271d v T).var),         Tape := (TM1.stepAux a\u271d v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }     {       q :=         (match (TM1.stepAux TM1.Stmt.halt v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux TM1.Stmt.halt v T).var),       Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape } ** case move d q IH => exact TransGen.head rfl (IH _ _) ** case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write a\u271d q), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }       {         q :=           (match (TM1.stepAux a\u271d\u00b9 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux a\u271d\u00b9 v T).var),         Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape } a_ih\u271d :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }       {         q :=           (match (TM1.stepAux a\u271d v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux a\u271d v T).var),         Tape := (TM1.stepAux a\u271d v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }     {       q :=         (match (TM1.stepAux TM1.Stmt.halt v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux TM1.Stmt.halt v T).var),       Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape } ** case write a q IH => exact TransGen.head rfl (IH _ _) ** case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }       {         q :=           (match (TM1.stepAux a\u271d\u00b9 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux a\u271d\u00b9 v T).var),         Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape } a_ih\u271d :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }       {         q :=           (match (TM1.stepAux a\u271d v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux a\u271d v T).var),         Tape := (TM1.stepAux a\u271d v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }     {       q :=         (match (TM1.stepAux TM1.Stmt.halt v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux TM1.Stmt.halt v T).var),       Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape } ** case load a q IH => exact (reaches\u2081_eq (by rfl)).2 (IH _ _) ** case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }  case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }     {       q :=         (match (TM1.stepAux TM1.Stmt.halt v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux TM1.Stmt.halt v T).var),       Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape } ** iterate 2\n  exact TransGen.single (congr_arg some (congr (congr_arg TM0.Cfg.mk rfl) (Tape.write_self T))) ** case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v : \u03c3 T : Tape \u0393 \u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := none, var := v, Tape := T })     (TM1.step M { l := none, var := v, Tape := T }) ** 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 d : \u03c3 q\u271d : Tape \u0393 l\u2081 : \u039b IH\u271d : Dir q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.move IH\u271d q), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).Tape } ** exact TransGen.head rfl (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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 a : \u03c3 q\u271d : Tape \u0393 l\u2081 : \u039b IH\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write IH\u271d q), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).Tape } ** exact TransGen.head rfl (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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 a : \u03c3 q\u271d : Tape \u0393 l\u2081 : \u039b IH\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load IH\u271d q), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).Tape } ** exact (reaches\u2081_eq (by rfl)).2 (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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 a : \u03c3 q\u271d : Tape \u0393 l\u2081 : \u039b IH\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }       {         q :=           (match (TM1.stepAux q v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q v T).var),         Tape := (TM1.stepAux q v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 TM0.step (tr M) { q := (some (TM1.Stmt.load IH\u271d q), v), Tape := T } =     TM0.step (tr M) { q := (some q, IH\u271d T.head v), Tape := T } ** 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2081 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2081 v T).var),         Tape := (TM1.stepAux q\u2081 v T).Tape } IH\u2082 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2082 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2082 v T).var),         Tape := (TM1.stepAux q\u2082 v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }     {       q :=         (match (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).var),       Tape := (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).Tape } ** unfold TM1.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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2081 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2081 v T).var),         Tape := (TM1.stepAux q\u2081 v T).Tape } IH\u2082 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2082 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2082 v T).var),         Tape := (TM1.stepAux q\u2082 v T).Tape } v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }     {       q :=         (match (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with           | some x => some (M x)           | none => none,           (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),       Tape := (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape } ** cases e : p T.1 v ** case false \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 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2081 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2081 v T).var),         Tape := (TM1.stepAux q\u2081 v T).Tape } IH\u2082 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2082 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2082 v T).var),         Tape := (TM1.stepAux q\u2082 v T).Tape } v : \u03c3 T : Tape \u0393 e : p T.head v = false \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }     {       q :=         (match (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with           | some x => some (M x)           | none => none,           (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),       Tape := (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape } ** exact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (IH\u2082 _ _) ** \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 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2081 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2081 v T).var),         Tape := (TM1.stepAux q\u2081 v T).Tape } IH\u2082 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2082 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2082 v T).var),         Tape := (TM1.stepAux q\u2082 v T).Tape } v : \u03c3 T : Tape \u0393 e : p T.head v = false \u22a2 TM0.step (tr M) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T } =     TM0.step (tr M) { q := (some q\u2082, v), Tape := T } ** simp only [TM0.step, tr, trAux, e] ** \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 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2081 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2081 v T).var),         Tape := (TM1.stepAux q\u2081 v T).Tape } IH\u2082 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2082 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2082 v T).var),         Tape := (TM1.stepAux q\u2082 v T).Tape } v : \u03c3 T : Tape \u0393 e : p T.head v = false \u22a2 Option.map       (fun x =>         { q := x.1,           Tape :=             match x.2 with             | move d => Tape.move d T             | write a => Tape.write a T })       (some (bif false then trAux M T.head q\u2081 v else trAux M T.head q\u2082 v)) =     Option.map       (fun x =>         { q := x.1,           Tape :=             match x.2 with             | move d => Tape.move d T             | write a => Tape.write a T })       (some (trAux M T.head q\u2082 v)) ** rfl ** case true \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 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2081 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2081 v T).var),         Tape := (TM1.stepAux q\u2081 v T).Tape } IH\u2082 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2082 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2082 v T).var),         Tape := (TM1.stepAux q\u2082 v T).Tape } v : \u03c3 T : Tape \u0393 e : p T.head v = true \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }     {       q :=         (match (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with           | some x => some (M x)           | none => none,           (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),       Tape := (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape } ** exact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2081 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2081 v T).var),         Tape := (TM1.stepAux q\u2081 v T).Tape } IH\u2082 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2082 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2082 v T).var),         Tape := (TM1.stepAux q\u2082 v T).Tape } v : \u03c3 T : Tape \u0393 e : p T.head v = true \u22a2 TM0.step (tr M) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T } =     TM0.step (tr M) { q := (some q\u2081, v), Tape := T } ** simp only [TM0.step, tr, trAux, e] ** \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 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2081 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2081 v T).var),         Tape := (TM1.stepAux q\u2081 v T).Tape } IH\u2082 :   \u2200 (v : \u03c3) (T : Tape \u0393),     Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }       {         q :=           (match (TM1.stepAux q\u2082 v T).l with             | some x => some (M x)             | none => none,             (TM1.stepAux q\u2082 v T).var),         Tape := (TM1.stepAux q\u2082 v T).Tape } v : \u03c3 T : Tape \u0393 e : p T.head v = true \u22a2 Option.map       (fun x =>         { q := x.1,           Tape :=             match x.2 with             | move d => Tape.move d T             | write a => Tape.write a T })       (some (bif true then trAux M T.head q\u2081 v else trAux M T.head q\u2082 v)) =     Option.map       (fun x =>         { q := x.1,           Tape :=             match x.2 with             | move d => Tape.move d T             | write a => Tape.write a T })       (some (trAux M T.head q\u2081 v)) ** rfl ** case 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 M : \u039b \u2192 Stmt\u2081 x\u271d : Cfg\u2081 v\u271d : \u03c3 T\u271d : Tape \u0393 l\u2081 : \u039b v : \u03c3 T : Tape \u0393 \u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }     {       q :=         (match (TM1.stepAux TM1.Stmt.halt v T).l with           | some x => some (M x)           | none => none,           (TM1.stepAux TM1.Stmt.halt v T).var),       Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape } ** exact TransGen.single (congr_arg some (congr (congr_arg TM0.Cfg.mk rfl) (Tape.write_self T))) ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.bsize ** l m r : List Char \u22a2 Substring.bsize       { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },         stopPos := { byteIdx := utf8Len l + utf8Len m } } =     utf8Len m ** simp [Substring.bsize, Nat.add_sub_cancel_left] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.pdf.hasPDF_iff ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 E \u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc ** constructor ** case mp \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 E \u22a2 HasPDF X \u2119 \u2192 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc ** intro hX' ** case mp \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 E hX' : HasPDF X \u2119 \u22a2 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc ** exact \u27e8hX'.pdf'.1, haveLebesgueDecomposition_of_hasPDF, map_absolutelyContinuous\u27e9 ** case mpr \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 E \u22a2 Measurable X \u2227 HaveLebesgueDecomposition (map X \u2119) \u03bc \u2227 map X \u2119 \u226a \u03bc \u2192 HasPDF X \u2119 ** rintro \u27e8hX, h_decomp, h\u27e9 ** case mpr.intro.intro \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 E hX : Measurable X h_decomp : HaveLebesgueDecomposition (map X \u2119) \u03bc h : map X \u2119 \u226a \u03bc \u22a2 HasPDF X \u2119 ** haveI := h_decomp ** case mpr.intro.intro \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 E hX : Measurable X h_decomp : HaveLebesgueDecomposition (map X \u2119) \u03bc h : map X \u2119 \u226a \u03bc this : HaveLebesgueDecomposition (map X \u2119) \u03bc \u22a2 HasPDF X \u2119 ** refine' \u27e8\u27e8hX, (Measure.map X \u2119).rnDeriv \u03bc, measurable_rnDeriv _ _, _\u27e9\u27e9 ** case mpr.intro.intro \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 E hX : Measurable X h_decomp : HaveLebesgueDecomposition (map X \u2119) \u03bc h : map X \u2119 \u226a \u03bc this : HaveLebesgueDecomposition (map X \u2119) \u03bc \u22a2 map X \u2119 = withDensity \u03bc (rnDeriv (map X \u2119) \u03bc) ** rwa [withDensity_rnDeriv_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_closedBall' ** 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 x : E r : \u211d hr : 0 \u2264 r \u22a2 \u2191\u2191\u03bc (closedBall x r) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1) ** rw [\u2190 addHaar_closedBall_mul \u03bc x hr zero_le_one, mul_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.dropSlice_eq_dropSliceTR ** \u22a2 @dropSlice = @dropSliceTR ** funext \u03b1 n m l ** case h.h.h.h \u03b1 : Type u_1 n m : Nat l : List \u03b1 \u22a2 dropSlice n m l = dropSliceTR n m l ** simp [dropSliceTR] ** case h.h.h.h \u03b1 : Type u_1 n m : Nat l : List \u03b1 \u22a2 dropSlice n m l =     match m with     | 0 => l     | succ m => dropSliceTR.go l m l n #[] ** split ** case h.h.h.h.h_1 \u03b1 : Type u_1 n : Nat l : List \u03b1 m\u271d : Nat \u22a2 dropSlice n 0 l = l  case h.h.h.h.h_2 \u03b1 : Type u_1 n : Nat l : List \u03b1 m\u271d\u00b9 m\u271d : Nat \u22a2 dropSlice n (succ m\u271d) l = dropSliceTR.go l m\u271d l n #[] ** { rw [dropSlice_zero\u2082] } ** case h.h.h.h.h_2 \u03b1 : Type u_1 n : Nat l : List \u03b1 m\u271d\u00b9 m\u271d : Nat \u22a2 dropSlice n (succ m\u271d) l = dropSliceTR.go l m\u271d l n #[] ** rename_i m ** case h.h.h.h.h_2 \u03b1 : Type u_1 n : Nat l : List \u03b1 m\u271d m : Nat \u22a2 dropSlice n (succ m) l = dropSliceTR.go l m l n #[] ** exact (go #[] _ _ rfl).symm ** \u03b1 : Type u_1 n : Nat l : List \u03b1 m\u271d m : Nat acc : Array \u03b1 head\u271d : \u03b1 xs : List \u03b1 h : l = acc.data ++ head\u271d :: xs \u22a2 dropSliceTR.go l m (head\u271d :: xs) 0 acc = acc.data ++ dropSlice 0 (m + 1) (head\u271d :: xs) ** simp [dropSliceTR.go, dropSlice, h] ** \u03b1 : Type u_1 n\u271d : Nat l : List \u03b1 m\u271d m : Nat acc : Array \u03b1 x : \u03b1 xs : List \u03b1 n : Nat \u22a2 l = acc.data ++ x :: xs \u2192 dropSliceTR.go l m (x :: xs) (n + 1) acc = acc.data ++ dropSlice (n + 1) (m + 1) (x :: xs) ** simp [dropSliceTR.go, dropSlice] ** \u03b1 : Type u_1 n\u271d : Nat l : List \u03b1 m\u271d m : Nat acc : Array \u03b1 x : \u03b1 xs : List \u03b1 n : Nat \u22a2 l = acc.data ++ x :: xs \u2192 dropSliceTR.go l m xs n (Array.push acc x) = acc.data ++ x :: dropSlice n (m + 1) xs ** intro h ** \u03b1 : Type u_1 n\u271d : Nat l : List \u03b1 m\u271d m : Nat acc : Array \u03b1 x : \u03b1 xs : List \u03b1 n : Nat h : l = acc.data ++ x :: xs \u22a2 dropSliceTR.go l m xs n (Array.push acc x) = acc.data ++ x :: dropSlice n (m + 1) xs ** rw [go _ xs] ** \u03b1 : Type u_1 n\u271d : Nat l : List \u03b1 m\u271d m : Nat acc : Array \u03b1 x : \u03b1 xs : List \u03b1 n : Nat h : l = acc.data ++ x :: xs \u22a2 (Array.push acc x).data ++ dropSlice n (m + 1) xs = acc.data ++ x :: dropSlice n (m + 1) xs  case a \u03b1 : Type u_1 n\u271d : Nat l : List \u03b1 m\u271d m : Nat acc : Array \u03b1 x : \u03b1 xs : List \u03b1 n : Nat h : l = acc.data ++ x :: xs \u22a2 l = (Array.push acc x).data ++ xs ** {simp} ** case a \u03b1 : Type u_1 n\u271d : Nat l : List \u03b1 m\u271d m : Nat acc : Array \u03b1 x : \u03b1 xs : List \u03b1 n : Nat h : l = acc.data ++ x :: xs \u22a2 l = (Array.push acc x).data ++ xs ** simp [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.toIterator ** l m r : List Char x\u271d : Substring h : ValidFor l m r x\u271d \u22a2 Iterator.ValidFor (List.reverse l) (m ++ r) (Substring.toIterator x\u271d) ** simp [Substring.toIterator] ** l m r : List Char x\u271d : Substring h : ValidFor l m r x\u271d \u22a2 Iterator.ValidFor (List.reverse l) (m ++ r) { s := x\u271d.str, i := x\u271d.startPos } ** exact .of_eq _ (by simp [h.str, List.reverseAux_eq]) (by simp [h.startPos]) ** l m r : List Char x\u271d : Substring h : ValidFor l m r x\u271d \u22a2 { s := x\u271d.str, i := x\u271d.startPos }.s.data = List.reverseAux (List.reverse l) (m ++ r) ** simp [h.str, List.reverseAux_eq] ** l m r : List Char x\u271d : Substring h : ValidFor l m r x\u271d \u22a2 { s := x\u271d.str, i := x\u271d.startPos }.i.byteIdx = utf8Len (List.reverse l) ** simp [h.startPos] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.smul_testAgainstNN_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 : TopologicalSpace \u03a9 c : \u211d\u22650 \u03bc : FiniteMeasure \u03a9 f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 testAgainstNN (c \u2022 \u03bc) f = c \u2022 testAgainstNN \u03bc f ** simp only [testAgainstNN, toMeasure_smul, smul_eq_mul, \u2190 ENNReal.smul_toNNReal, ENNReal.smul_def,\n  lintegral_smul_measure] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec'.of_part ** this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v) n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : _root_.Partrec f \u22a2 Partrec' f ** let g := fun n\u2081 =>\n  (Part.ofOption (decode (\u03b1 := Vector \u2115 n) n\u2081)).bind (fun a => Part.map encode (f a)) ** this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v) n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : _root_.Partrec f g : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a) \u22a2 Partrec' f ** exact\n  (comp\u2081 g (this g hf) (prim Nat.Primrec'.encode)).of_eq fun i => by\n    dsimp only; simp [encodek, Part.map_id'] ** this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v) n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : _root_.Partrec f g : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a) i : Vector \u2115 n \u22a2 g (encode i) = f i ** dsimp only ** this : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v) n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : _root_.Partrec f g : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a) i : Vector \u2115 n \u22a2 (Part.bind \u2191(decode (encode i)) fun a => Part.map encode (f a)) = f i ** simp [encodek, Part.map_id'] ** f : \u2115 \u2192. \u2115 hf : Partrec f \u22a2 Partrec' fun v => f (Vector.head v) ** obtain \u27e8c, rfl\u27e9 := exists_code.1 hf ** case intro c : Code hf : Partrec (eval c) \u22a2 Partrec' fun v => eval c (Vector.head v) ** simpa [eval_eq_rfindOpt] using\n  rfindOpt <|\n    of_prim <|\n      Primrec.encode_iff.2 <|\n        evaln_prim.comp <|\n          (Primrec.vector_head.pair (_root_.Primrec.const c)).pair <|\n            Primrec.vector_head.comp Primrec.vector_tail ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_add_uIcc ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 (fun x => a + x) \u207b\u00b9' [[b, c]] = [[b - a, c - a]] ** simp only [\u2190 Icc_min_max, preimage_const_add_Icc, min_sub_sub_right, max_sub_sub_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.mem_degrees ** 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 p : MvPolynomial \u03c3 R i : \u03c3 \u22a2 i \u2208 degrees p \u2194 \u2203 d, coeff d p \u2260 0 \u2227 i \u2208 d.support ** classical\nsimp only [degrees_def, Multiset.mem_sup, \u2190 mem_support_iff, Finsupp.mem_toMultiset, exists_prop] ** 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 p : MvPolynomial \u03c3 R i : \u03c3 \u22a2 i \u2208 degrees p \u2194 \u2203 d, coeff d p \u2260 0 \u2227 i \u2208 d.support ** simp only [degrees_def, Multiset.mem_sup, \u2190 mem_support_iff, Finsupp.mem_toMultiset, exists_prop] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable ** \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 ** have : \u03b5 / 2 \u2260 0 := (ENNReal.half_pos \u03b50).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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 ** rcases exists_lt_lowerSemicontinuous_lintegral_ge \u03bc (fmeas.mk f) fmeas.measurable_mk this with\n  \u27e8g0, f_lt_g0, g0_cont, g0_int\u27e9 ** case 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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 ** rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with \u27e8s, hs, smeas, \u03bcs\u27e9 ** case 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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 ** rcases exists_le_lowerSemicontinuous_lintegral_ge \u03bc (s.indicator fun _x => \u221e)\n    (measurable_const.indicator smeas) this with\n  \u27e8g1, le_g1, g1_cont, g1_int\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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 \u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 ** refine' \u27e8fun x => g0 x + g1 x, fun x => _, g0_cont.add g1_cont, _\u27e9 ** case intro.intro.intro.intro.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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 x : \u03b1 \u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x ** by_cases h : x \u2208 s ** case pos \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 x : \u03b1 h : x \u2208 s \u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x ** have := le_g1 x ** case pos \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this\u271d : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 x : \u03b1 h : x \u2208 s this : Set.indicator s (fun _x => \u22a4) x \u2264 g1 x \u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x ** simp only [h, Set.indicator_of_mem, top_le_iff] at this ** case pos \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this\u271d : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 x : \u03b1 h : x \u2208 s this : g1 x = \u22a4 \u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x ** simp [this] ** case neg \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 x : \u03b1 h : \u00acx \u2208 s \u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x ** have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs; exact hs h ** case neg \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this\u271d : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 x : \u03b1 h : \u00acx \u2208 s this : f x = AEMeasurable.mk f fmeas x \u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x ** rw [this] ** case neg \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this\u271d : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 x : \u03b1 h : \u00acx \u2208 s this : f x = AEMeasurable.mk f fmeas x \u22a2 \u2191(AEMeasurable.mk f fmeas x) < (fun x => g0 x + g1 x) x ** exact (f_lt_g0 x).trans_le le_self_add ** \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 x : \u03b1 h : \u00acx \u2208 s \u22a2 f x = AEMeasurable.mk f fmeas x ** rw [Set.compl_subset_comm] at hs ** \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : s\u1d9c \u2286 {x | (fun x => f x = AEMeasurable.mk f fmeas x) x} smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 x : \u03b1 h : \u00acx \u2208 s \u22a2 f x = AEMeasurable.mk f fmeas x ** exact hs 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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc + \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2 + (0 + \u03b5 / 2) ** refine' add_le_add _ _ ** 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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2 ** convert g0_int using 2 ** case h.e'_4.h.e'_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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc ** exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _) ** 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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 0 + \u03b5 / 2 ** convert g1_int ** case h.e'_4.h.e'_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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 \u22a2 0 = \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc ** simp only [smeas, \u03bcs, lintegral_const, Set.univ_inter, MeasurableSet.univ,\n  lintegral_indicator, mul_zero, restrict_apply] ** \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\u22650 fmeas : AEMeasurable f \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 this : \u03b5 / 2 \u2260 0 g0 : \u03b1 \u2192 \u211d\u22650\u221e f_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x g0_cont : LowerSemicontinuous g0 g0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2 s : Set \u03b1 hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s smeas : MeasurableSet s \u03bcs : \u2191\u2191\u03bc s = 0 g1 : \u03b1 \u2192 \u211d\u22650\u221e le_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x g1_cont : LowerSemicontinuous g1 g1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2 + (0 + \u03b5 / 2) = \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 ** simp only [add_assoc, ENNReal.add_halves, zero_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.smul_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 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 f x : \u03c3 \u2192 R p : MvPolynomial \u03c3 R s : R \u22a2 \u2191(eval x) (s \u2022 p) = s * \u2191(eval x) p ** rw [smul_eq_C_mul, (eval x).map_mul, eval_C] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.evariance_eq_lintegral_ofReal ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 evariance X \u03bc = \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) \u2202\u03bc ** rw [evariance] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 \u2202\u03bc = \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) \u2202\u03bc ** congr ** case e_f \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) = fun \u03c9 => ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) ** ext1 \u03c9 ** case e_f.h \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \u22a2 \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 = ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) ** rw [pow_two, \u2190 ENNReal.coe_mul, \u2190 nnnorm_mul, \u2190 pow_two] ** case e_f.h \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \u22a2 \u2191\u2016(X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2\u2016\u208a = ENNReal.ofReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) ** congr ** case e_f.h.e_a \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u03c9 : \u03a9 \u22a2 \u2016(X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2\u2016\u208a = Real.toNNReal ((X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) ** exact (Real.toNNReal_eq_nnnorm_of_nonneg <| sq_nonneg _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "aemeasurable_of_re_im ** \u03b1 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \ud835\udd5c \u03bc : MeasureTheory.Measure \u03b1 hre : AEMeasurable fun x => \u2191IsROrC.re (f x) him : AEMeasurable fun x => \u2191IsROrC.im (f x) \u22a2 AEMeasurable f ** convert AEMeasurable.add (M := \ud835\udd5c) (IsROrC.measurable_ofReal.comp_aemeasurable hre)\n    ((IsROrC.measurable_ofReal.comp_aemeasurable him).mul_const IsROrC.I) ** case h.e'_5.h \u03b1 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b9 : IsROrC \ud835\udd5c inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \ud835\udd5c \u03bc : MeasureTheory.Measure \u03b1 hre : AEMeasurable fun x => \u2191IsROrC.re (f x) him : AEMeasurable fun x => \u2191IsROrC.im (f x) x\u271d : \u03b1 \u22a2 f x\u271d = (IsROrC.ofReal \u2218 fun x => \u2191IsROrC.re (f x)) x\u271d + (IsROrC.ofReal \u2218 fun x => \u2191IsROrC.im (f x)) x\u271d * IsROrC.I ** exact (IsROrC.re_add_im _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "ContinuousMap.range_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\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 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E \u22a2 Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c)) = Lp.boundedContinuousFunction E p \u03bc ** refine' SetLike.ext' _ ** \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 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E \u22a2 \u2191(Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u2191(Lp.boundedContinuousFunction E p \u03bc) ** have := (linearIsometryBoundedOfCompact \u03b1 E \ud835\udd5c).surjective ** \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 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E this : Function.Surjective \u2191(linearIsometryBoundedOfCompact \u03b1 E \ud835\udd5c) \u22a2 \u2191(Submodule.toAddSubgroup (LinearMap.range (toLp p \u03bc \ud835\udd5c))) = \u2191(Lp.boundedContinuousFunction E p \u03bc) ** convert Function.Surjective.range_comp this (BoundedContinuousFunction.toLp (E := E) p \u03bc \ud835\udd5c) ** case h.e'_3 \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 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E this : Function.Surjective \u2191(linearIsometryBoundedOfCompact \u03b1 E \ud835\udd5c) \u22a2 \u2191(Lp.boundedContinuousFunction E p \u03bc) = Set.range \u2191(BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c) ** rw [\u2190 BoundedContinuousFunction.range_toLp p \u03bc (\ud835\udd5c := \ud835\udd5c), Submodule.coe_toAddSubgroup,\n  LinearMap.range_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.map_add_left_Ioc ** \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) (Ioc a b) = Ioc (c + a) (c + b) ** rw [\u2190 coe_inj, coe_map, coe_Ioc, coe_Ioc] ** \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.Ioc a b = Set.Ioc (c + a) (c + b) ** exact Set.image_const_add_Ioc _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.map_add_left_Ioo ** \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) (Ioo a b) = Ioo (c + a) (c + b) ** rw [\u2190 coe_inj, coe_map, coe_Ioo, coe_Ioo] ** \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.Ioo a b = Set.Ioo (c + a) (c + b) ** exact Set.image_const_add_Ioo _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "integrable_of_isBigO_exp_neg ** f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) \u22a2 IntegrableOn f (Ioi a) ** cases' h2.isBigOWith with c h3 ** case intro f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c : \u211d h3 : Asymptotics.IsBigOWith c atTop f fun x => rexp (-b * x) \u22a2 IntegrableOn f (Ioi a) ** rw [Asymptotics.isBigOWith_iff, eventually_atTop] at h3 ** case intro f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c : \u211d h3 : \u2203 a, \u2200 (b_1 : \u211d), b_1 \u2265 a \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 \u22a2 IntegrableOn f (Ioi a) ** cases' h3 with r bdr ** case intro.intro f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 \u22a2 IntegrableOn f (Ioi a) ** let v := max a r ** case intro.intro f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r \u22a2 IntegrableOn f (Ioi a) ** have int_left : IntegrableOn f (Ioc a v) := by\n  rw [\u2190 intervalIntegrable_iff_integrable_Ioc_of_le (le_max_left a r)]\n  have u : Icc a v \u2286 Ici a := Icc_subset_Ici_self\n  exact (h1.mono u).intervalIntegrable_of_Icc (le_max_left a r) ** case intro.intro f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) \u22a2 IntegrableOn f (Ioi a) ** suffices IntegrableOn f (Ioi v) by\n  have t := integrableOn_union.mpr \u27e8int_left, this\u27e9\n  simpa only [Ioc_union_Ioi_eq_Ioi, le_max_iff, le_refl, true_or_iff] using t ** case intro.intro f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) \u22a2 IntegrableOn f (Ioi v) ** constructor ** case intro.intro.right f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) \u22a2 HasFiniteIntegral f ** have : HasFiniteIntegral (fun x : \u211d => c * exp (-b * x)) (volume.restrict (Ioi v)) :=\n  (exp_neg_integrableOn_Ioi v h0).hasFiniteIntegral.const_mul c ** case intro.intro.right f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) this : HasFiniteIntegral fun x => c * rexp (-b * x) \u22a2 HasFiniteIntegral f ** apply this.mono ** case intro.intro.right f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) this : HasFiniteIntegral fun x => c * rexp (-b * x) \u22a2 \u2200\u1d50 (a : \u211d) \u2202Measure.restrict volume (Ioi v), \u2016f a\u2016 \u2264 \u2016c * rexp (-b * a)\u2016 ** refine' (ae_restrict_iff' measurableSet_Ioi).mpr _ ** case intro.intro.right f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) this : HasFiniteIntegral fun x => c * rexp (-b * x) \u22a2 \u2200\u1d50 (x : \u211d), x \u2208 Ioi v \u2192 \u2016f x\u2016 \u2264 \u2016c * rexp (-b * x)\u2016 ** refine' ae_of_all _ fun x h1x => _ ** case intro.intro.right f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) this : HasFiniteIntegral fun x => c * rexp (-b * x) x : \u211d h1x : x \u2208 Ioi v \u22a2 \u2016f x\u2016 \u2264 \u2016c * rexp (-b * x)\u2016 ** rw [norm_mul, norm_eq_abs] ** case intro.intro.right f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) this : HasFiniteIntegral fun x => c * rexp (-b * x) x : \u211d h1x : x \u2208 Ioi v \u22a2 |f x| \u2264 \u2016c\u2016 * \u2016rexp (-b * x)\u2016 ** rw [mem_Ioi] at h1x ** case intro.intro.right f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) this : HasFiniteIntegral fun x => c * rexp (-b * x) x : \u211d h1x : v < x \u22a2 |f x| \u2264 \u2016c\u2016 * \u2016rexp (-b * x)\u2016 ** specialize bdr x ((le_max_right a r).trans h1x.le) ** case intro.intro.right f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) this : HasFiniteIntegral fun x => c * rexp (-b * x) x : \u211d h1x : v < x bdr : \u2016f x\u2016 \u2264 c * \u2016rexp (-b * x)\u2016 \u22a2 |f x| \u2264 \u2016c\u2016 * \u2016rexp (-b * x)\u2016 ** exact bdr.trans (mul_le_mul_of_nonneg_right (le_abs_self c) (norm_nonneg _)) ** f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r \u22a2 IntegrableOn f (Ioc a v) ** rw [\u2190 intervalIntegrable_iff_integrable_Ioc_of_le (le_max_left a r)] ** f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r \u22a2 IntervalIntegrable f volume a (max a r) ** have u : Icc a v \u2286 Ici a := Icc_subset_Ici_self ** f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r u : Icc a v \u2286 Ici a \u22a2 IntervalIntegrable f volume a (max a r) ** exact (h1.mono u).intervalIntegrable_of_Icc (le_max_left a r) ** f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) this : IntegrableOn f (Ioi v) \u22a2 IntegrableOn f (Ioi a) ** have t := integrableOn_union.mpr \u27e8int_left, this\u27e9 ** f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) this : IntegrableOn f (Ioi v) t : IntegrableOn f (Ioc a v \u222a Ioi v) \u22a2 IntegrableOn f (Ioi a) ** simpa only [Ioc_union_Ioi_eq_Ioi, le_max_iff, le_refl, true_or_iff] using t ** case intro.intro.left f : \u211d \u2192 \u211d a b : \u211d h0 : 0 < b h1 : ContinuousOn f (Ici a) h2 : f =O[atTop] fun x => rexp (-b * x) c r : \u211d bdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016rexp (-b * b_1)\u2016 v : \u211d := max a r int_left : IntegrableOn f (Ioc a v) \u22a2 AEStronglyMeasurable f (Measure.restrict volume (Ioi v)) ** exact (h1.mono <| Ioi_subset_Ici <| le_max_left a r).aestronglyMeasurable measurableSet_Ioi ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_not_mem_null_lintegral_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\u22650\u221e inst\u271d : IsProbabilityMeasure \u03bc hint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4 hN : \u2191\u2191\u03bc N = 0 \u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x ** simpa only [laverage_eq_lintegral] using\n  exists_not_mem_null_laverage_le (IsProbabilityMeasure.ne_zero \u03bc) hint hN ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.degrees_rename ** 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 f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R \u22a2 degrees (\u2191(rename f) \u03c6) \u2286 Multiset.map f (degrees \u03c6) ** classical\nintro i\nrw [mem_degrees, Multiset.mem_map]\nrintro \u27e8d, hd, hi\u27e9\nobtain \u27e8x, rfl, hx\u27e9 := coeff_rename_ne_zero _ _ _ hd\nsimp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi\nrw [sum_apply, Finsupp.sum] at hi\ncontrapose! hi\nrw [Finset.sum_eq_zero]\nintro j hj\nsimp only [exists_prop, mem_degrees] at hi\nspecialize hi j \u27e8x, hx, hj\u27e9\nrw [Finsupp.single_apply, if_neg hi] ** 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 f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R \u22a2 degrees (\u2191(rename f) \u03c6) \u2286 Multiset.map f (degrees \u03c6) ** intro 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 f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 \u22a2 i \u2208 degrees (\u2191(rename f) \u03c6) \u2192 i \u2208 Multiset.map f (degrees \u03c6) ** rw [mem_degrees, Multiset.mem_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 f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 \u22a2 (\u2203 d, coeff d (\u2191(rename f) \u03c6) \u2260 0 \u2227 i \u2208 d.support) \u2192 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i ** rintro \u27e8d, hd, hi\u27e9 ** case intro.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 q : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 d : \u03c4 \u2192\u2080 \u2115 hd : coeff d (\u2191(rename f) \u03c6) \u2260 0 hi : i \u2208 d.support \u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i ** obtain \u27e8x, rfl, hx\u27e9 := coeff_rename_ne_zero _ _ _ hd ** case intro.intro.intro.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 q : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 x : \u03c3 \u2192\u2080 \u2115 hx : coeff x \u03c6 \u2260 0 hd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0 hi : i \u2208 (Finsupp.mapDomain f x).support \u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i ** simp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi ** case intro.intro.intro.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 q : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 x : \u03c3 \u2192\u2080 \u2115 hx : coeff x \u03c6 \u2260 0 hd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0 hi : \u2191(sum x fun a => Finsupp.single (f a)) i \u2260 0 \u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i ** rw [sum_apply, Finsupp.sum] at hi ** case intro.intro.intro.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 q : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 x : \u03c3 \u2192\u2080 \u2115 hx : coeff x \u03c6 \u2260 0 hd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0 hi : \u2211 a in x.support, (\u2191fun\u2080 | f a => \u2191x a) i \u2260 0 \u22a2 \u2203 a, a \u2208 degrees \u03c6 \u2227 f a = i ** contrapose! hi ** case intro.intro.intro.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 q : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 x : \u03c3 \u2192\u2080 \u2115 hx : coeff x \u03c6 \u2260 0 hd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0 hi : \u2200 (a : \u03c3), a \u2208 degrees \u03c6 \u2192 f a \u2260 i \u22a2 \u2211 x_1 in x.support, (\u2191fun\u2080 | f x_1 => \u2191x x_1) i = 0 ** rw [Finset.sum_eq_zero] ** case intro.intro.intro.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 q : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 x : \u03c3 \u2192\u2080 \u2115 hx : coeff x \u03c6 \u2260 0 hd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0 hi : \u2200 (a : \u03c3), a \u2208 degrees \u03c6 \u2192 f a \u2260 i \u22a2 \u2200 (x_1 : \u03c3), x_1 \u2208 x.support \u2192 (\u2191fun\u2080 | f x_1 => \u2191x x_1) i = 0 ** intro j hj ** case intro.intro.intro.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 q : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 x : \u03c3 \u2192\u2080 \u2115 hx : coeff x \u03c6 \u2260 0 hd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0 hi : \u2200 (a : \u03c3), a \u2208 degrees \u03c6 \u2192 f a \u2260 i j : \u03c3 hj : j \u2208 x.support \u22a2 (\u2191fun\u2080 | f j => \u2191x j) i = 0 ** simp only [exists_prop, mem_degrees] at hi ** case intro.intro.intro.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 q : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 x : \u03c3 \u2192\u2080 \u2115 hx : coeff x \u03c6 \u2260 0 hd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0 j : \u03c3 hj : j \u2208 x.support hi : \u2200 (a : \u03c3), (\u2203 d, coeff d \u03c6 \u2260 0 \u2227 a \u2208 d.support) \u2192 f a \u2260 i \u22a2 (\u2191fun\u2080 | f j => \u2191x j) i = 0 ** specialize hi j \u27e8x, hx, hj\u27e9 ** case intro.intro.intro.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 q : MvPolynomial \u03c3 R f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R i : \u03c4 x : \u03c3 \u2192\u2080 \u2115 hx : coeff x \u03c6 \u2260 0 hd : coeff (Finsupp.mapDomain f x) (\u2191(rename f) \u03c6) \u2260 0 j : \u03c3 hj : j \u2208 x.support hi : f j \u2260 i \u22a2 (\u2191fun\u2080 | f j => \u2191x j) i = 0 ** rw [Finsupp.single_apply, if_neg hi] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrable_withDensity_iff ** \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 hf : Measurable f hflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 \u211d \u22a2 Integrable g \u2194 Integrable fun x => g x * ENNReal.toReal (f x) ** have : (fun x => g x * (f x).toReal) = fun x => (f x).toReal \u2022 g x := by simp [mul_comm] ** \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 hf : Measurable f hflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 \u211d this : (fun x => g x * ENNReal.toReal (f x)) = fun x => ENNReal.toReal (f x) \u2022 g x \u22a2 Integrable g \u2194 Integrable fun x => g x * ENNReal.toReal (f x) ** rw [this] ** \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 hf : Measurable f hflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 \u211d this : (fun x => g x * ENNReal.toReal (f x)) = fun x => ENNReal.toReal (f x) \u2022 g x \u22a2 Integrable g \u2194 Integrable fun x => ENNReal.toReal (f x) \u2022 g x ** exact integrable_withDensity_iff_integrable_smul' hf hflt ** \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 hf : Measurable f hflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 \u211d \u22a2 (fun x => g x * ENNReal.toReal (f x)) = fun x => ENNReal.toReal (f x) \u2022 g x ** simp [mul_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_image_le ** \u03b1 : Type u_2 s t : Set \u03b1 \u03b1\u271d : Type u_1 f : \u03b1 \u2192 \u03b1\u271d hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard (f '' s) \u2264 ncard s ** to_encard_tac ** \u03b1 : Type u_2 s t : Set \u03b1 \u03b1\u271d : Type u_1 f : \u03b1 \u2192 \u03b1\u271d hs : autoParam (Set.Finite s) _auto\u271d \u22a2 \u2191(ncard (f '' s)) \u2264 \u2191(ncard s) ** rw [hs.cast_ncard_eq, (hs.image _).cast_ncard_eq] ** \u03b1 : Type u_2 s t : Set \u03b1 \u03b1\u271d : Type u_1 f : \u03b1 \u2192 \u03b1\u271d hs : autoParam (Set.Finite s) _auto\u271d \u22a2 encard (f '' s) \u2264 encard s ** apply encard_image_le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.smul_boundedBy ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e c : \u211d\u22650\u221e hc : c \u2260 \u22a4 \u22a2 c \u2022 boundedBy m = boundedBy (c \u2022 m) ** simp only [boundedBy , smul_ofFunction hc] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e c : \u211d\u22650\u221e hc : c \u2260 \u22a4 \u22a2 OuterMeasure.ofFunction (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) (_ : c \u2022 \u2a06 (_ : Set.Nonempty \u2205), m \u2205 = 0) =     OuterMeasure.ofFunction (fun s => \u2a06 (_ : Set.Nonempty s), (c \u2022 m) s) (_ : \u2a06 (_ : Set.Nonempty \u2205), (c \u2022 m) \u2205 = 0) ** congr 1 with s : 1 ** case e_m.h \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e c : \u211d\u22650\u221e hc : c \u2260 \u22a4 s : Set \u03b1 \u22a2 (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) s = \u2a06 (_ : Set.Nonempty s), (c \u2022 m) s ** rcases s.eq_empty_or_nonempty with (rfl | hs) <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.fixInduction_spec ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 C : \u03b1 \u2192 Sort u_7 f : \u03b1 \u2192. \u03b2 \u2295 \u03b1 b : \u03b2 a : \u03b1 h : b \u2208 fix f a H : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a' \u22a2 fixInduction h H = H a h fun a' h' => fixInduction (_ : b \u2208 fix f a') H ** unfold fixInduction ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 C : \u03b1 \u2192 Sort u_7 f : \u03b1 \u2192. \u03b2 \u2295 \u03b1 b : \u03b2 a : \u03b1 h : b \u2208 fix f a H : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a' \u22a2 (let_fun h\u2082 :=       (_ :         b \u2208           WellFounded.fixF             (fun a IH =>               Part.assert (f a).Dom fun hf =>                 match e : Part.get (f a) hf with                 | Sum.inl b => Part.some b                 | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))             a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a));     Acc.rec (motive := fun {a} h\u2081 =>       b \u2208           WellFounded.fixF             (fun a IH =>               Part.assert (f a).Dom fun hf =>                 match e : Part.get (f a) hf with                 | Sum.inl b => Part.some b                 | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))             a h\u2081 \u2192         C a)       (fun a ha IH h\u2082 =>         let_fun h :=           (_ :             b \u2208               Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>                 WellFounded.fixF                   (fun a IH =>                     Part.assert (f a).Dom fun hf =>                       match e : Part.get (f a) hf with                       | Sum.inl b => Part.some b                       | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                   a h);         H a h fun a' fa' =>           IH a' fa'             (_ :               b \u2208                 WellFounded.fixF                   (fun a IH =>                     Part.assert (f a).Dom fun hf =>                       match e : Part.get (f a) hf with                       | Sum.inl b => Part.some b                       | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                   a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))       (_ : Acc (fun x y => Sum.inr x \u2208 f y) a) h\u2082) =     H a h fun a' h' =>       let_fun h\u2082 :=         (_ :           b \u2208             WellFounded.fixF               (fun a IH =>                 Part.assert (f a).Dom fun hf =>                   match e : Part.get (f a) hf with                   | Sum.inl b => Part.some b                   | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))               a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));       Acc.rec (motive := fun {a} h\u2081 =>         b \u2208             WellFounded.fixF               (fun a IH =>                 Part.assert (f a).Dom fun hf =>                   match e : Part.get (f a) hf with                   | Sum.inl b => Part.some b                   | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))               a h\u2081 \u2192           (fun x => C x) a)         (fun a ha IH h\u2082 =>           let_fun h :=             (_ :               b \u2208                 Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>                   WellFounded.fixF                     (fun a IH =>                       Part.assert (f a).Dom fun hf =>                         match e : Part.get (f a) hf with                         | Sum.inl b => Part.some b                         | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                     a h);           H a h fun a' fa' =>             IH a' fa'               (_ :                 b \u2208                   WellFounded.fixF                     (fun a IH =>                       Part.assert (f a).Dom fun hf =>                         match e : Part.get (f a) hf with                         | Sum.inl b => Part.some b                         | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                     a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))         (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082 ** generalize (Part.mem_assert_iff.1 h).fst = ha ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 C : \u03b1 \u2192 Sort u_7 f : \u03b1 \u2192. \u03b2 \u2295 \u03b1 b : \u03b2 a : \u03b1 h : b \u2208 fix f a H : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a' ha : Acc (fun x y => Sum.inr x \u2208 f y) a \u22a2 (let_fun h\u2082 :=       (_ :         b \u2208           WellFounded.fixF             (fun a IH =>               Part.assert (f a).Dom fun hf =>                 match e : Part.get (f a) hf with                 | Sum.inl b => Part.some b                 | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))             a (_ : Acc (fun x y => Sum.inr x \u2208 f y) a));     Acc.rec (motive := fun {a} h\u2081 =>       b \u2208           WellFounded.fixF             (fun a IH =>               Part.assert (f a).Dom fun hf =>                 match e : Part.get (f a) hf with                 | Sum.inl b => Part.some b                 | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))             a h\u2081 \u2192         C a)       (fun a ha IH h\u2082 =>         let_fun h :=           (_ :             b \u2208               Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>                 WellFounded.fixF                   (fun a IH =>                     Part.assert (f a).Dom fun hf =>                       match e : Part.get (f a) hf with                       | Sum.inl b => Part.some b                       | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                   a h);         H a h fun a' fa' =>           IH a' fa'             (_ :               b \u2208                 WellFounded.fixF                   (fun a IH =>                     Part.assert (f a).Dom fun hf =>                       match e : Part.get (f a) hf with                       | Sum.inl b => Part.some b                       | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                   a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))       ha h\u2082) =     H a h fun a' h' =>       let_fun h\u2082 :=         (_ :           b \u2208             WellFounded.fixF               (fun a IH =>                 Part.assert (f a).Dom fun hf =>                   match e : Part.get (f a) hf with                   | Sum.inl b => Part.some b                   | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))               a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));       Acc.rec (motive := fun {a} h\u2081 =>         b \u2208             WellFounded.fixF               (fun a IH =>                 Part.assert (f a).Dom fun hf =>                   match e : Part.get (f a) hf with                   | Sum.inl b => Part.some b                   | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))               a h\u2081 \u2192           (fun x => C x) a)         (fun a ha IH h\u2082 =>           let_fun h :=             (_ :               b \u2208                 Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>                   WellFounded.fixF                     (fun a IH =>                       Part.assert (f a).Dom fun hf =>                         match e : Part.get (f a) hf with                         | Sum.inl b => Part.some b                         | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                     a h);           H a h fun a' fa' =>             IH a' fa'               (_ :                 b \u2208                   WellFounded.fixF                     (fun a IH =>                       Part.assert (f a).Dom fun hf =>                         match e : Part.get (f a) hf with                         | Sum.inl b => Part.some b                         | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                     a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))         (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082 ** induction ha ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 C : \u03b1 \u2192 Sort u_7 f : \u03b1 \u2192. \u03b2 \u2295 \u03b1 b : \u03b2 a : \u03b1 H : (a' : \u03b1) \u2192 b \u2208 fix f a' \u2192 ((a'' : \u03b1) \u2192 Sum.inr a'' \u2208 f a' \u2192 C a'') \u2192 C a' x\u271d : \u03b1 h\u271d : \u2200 (y : \u03b1), Sum.inr y \u2208 f x\u271d \u2192 Acc (fun x y => Sum.inr x \u2208 f y) y h_ih\u271d :   \u2200 (y : \u03b1) (a : Sum.inr y \u2208 f x\u271d) (h : b \u2208 fix f y),     (let_fun h\u2082 :=         (_ :           b \u2208             WellFounded.fixF               (fun a IH =>                 Part.assert (f a).Dom fun hf =>                   match e : Part.get (f a) hf with                   | Sum.inl b => Part.some b                   | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))               y (_ : Acc (fun x y => Sum.inr x \u2208 f y) y));       Acc.rec (motive := fun {a} h\u2081 =>         b \u2208             WellFounded.fixF               (fun a IH =>                 Part.assert (f a).Dom fun hf =>                   match e : Part.get (f a) hf with                   | Sum.inl b => Part.some b                   | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))               a h\u2081 \u2192           C a)         (fun a ha IH h\u2082 =>           let_fun h :=             (_ :               b \u2208                 Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>                   WellFounded.fixF                     (fun a IH =>                       Part.assert (f a).Dom fun hf =>                         match e : Part.get (f a) hf with                         | Sum.inl b => Part.some b                         | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                     a h);           H a h fun a' fa' =>             IH a' fa'               (_ :                 b \u2208                   WellFounded.fixF                     (fun a IH =>                       Part.assert (f a).Dom fun hf =>                         match e : Part.get (f a) hf with                         | Sum.inl b => Part.some b                         | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                     a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))         (_ : Acc ?m.25674 y) h\u2082) =       H y h fun a' h' =>         let_fun h\u2082 :=           (_ :             b \u2208               WellFounded.fixF                 (fun a IH =>                   Part.assert (f a).Dom fun hf =>                     match e : Part.get (f a) hf with                     | Sum.inl b => Part.some b                     | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                 a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));         Acc.rec (motive := fun {a} h\u2081 =>           b \u2208               WellFounded.fixF                 (fun a IH =>                   Part.assert (f a).Dom fun hf =>                     match e : Part.get (f a) hf with                     | Sum.inl b => Part.some b                     | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                 a h\u2081 \u2192             (fun x => C x) a)           (fun a ha IH h\u2082 =>             let_fun h :=               (_ :                 b \u2208                   Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>                     WellFounded.fixF                       (fun a IH =>                         Part.assert (f a).Dom fun hf =>                           match e : Part.get (f a) hf with                           | Sum.inl b => Part.some b                           | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                       a h);             H a h fun a' fa' =>               IH a' fa'                 (_ :                   b \u2208                     WellFounded.fixF                       (fun a IH =>                         Part.assert (f a).Dom fun hf =>                           match e : Part.get (f a) hf with                           | Sum.inl b => Part.some b                           | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                       a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))           (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082 h : b \u2208 fix f x\u271d \u22a2 (let_fun h\u2082 :=       (_ :         b \u2208           WellFounded.fixF             (fun a IH =>               Part.assert (f a).Dom fun hf =>                 match e : Part.get (f a) hf with                 | Sum.inl b => Part.some b                 | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))             x\u271d (_ : Acc (fun x y => Sum.inr x \u2208 f y) x\u271d));     Acc.rec (motive := fun {a} h\u2081 =>       b \u2208           WellFounded.fixF             (fun a IH =>               Part.assert (f a).Dom fun hf =>                 match e : Part.get (f a) hf with                 | Sum.inl b => Part.some b                 | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))             a h\u2081 \u2192         C a)       (fun a ha IH h\u2082 =>         let_fun h :=           (_ :             b \u2208               Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>                 WellFounded.fixF                   (fun a IH =>                     Part.assert (f a).Dom fun hf =>                       match e : Part.get (f a) hf with                       | Sum.inl b => Part.some b                       | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                   a h);         H a h fun a' fa' =>           IH a' fa'             (_ :               b \u2208                 WellFounded.fixF                   (fun a IH =>                     Part.assert (f a).Dom fun hf =>                       match e : Part.get (f a) hf with                       | Sum.inl b => Part.some b                       | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                   a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))       (_ : Acc (fun x y => Sum.inr x \u2208 f y) x\u271d) h\u2082) =     H x\u271d h fun a' h' =>       let_fun h\u2082 :=         (_ :           b \u2208             WellFounded.fixF               (fun a IH =>                 Part.assert (f a).Dom fun hf =>                   match e : Part.get (f a) hf with                   | Sum.inl b => Part.some b                   | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))               a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a'));       Acc.rec (motive := fun {a} h\u2081 =>         b \u2208             WellFounded.fixF               (fun a IH =>                 Part.assert (f a).Dom fun hf =>                   match e : Part.get (f a) hf with                   | Sum.inl b => Part.some b                   | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))               a h\u2081 \u2192           (fun x => C x) a)         (fun a ha IH h\u2082 =>           let_fun h :=             (_ :               b \u2208                 Part.assert (Acc (fun x y => Sum.inr x \u2208 f y) a) fun h =>                   WellFounded.fixF                     (fun a IH =>                       Part.assert (f a).Dom fun hf =>                         match e : Part.get (f a) hf with                         | Sum.inl b => Part.some b                         | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                     a h);           H a h fun a' fa' =>             IH a' fa'               (_ :                 b \u2208                   WellFounded.fixF                     (fun a IH =>                       Part.assert (f a).Dom fun hf =>                         match e : Part.get (f a) hf with                         | Sum.inl b => Part.some b                         | Sum.inr a' => IH a' (_ : \u2203 h, Part.get (f a) h = Sum.inr a'))                     a' (_ : Acc (fun x y => Sum.inr x \u2208 f y) a')))         (_ : Acc (fun x y => Sum.inr x \u2208 f y) a') h\u2082 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.div_eq_iff_eq_mul_left ** a b c : Int H : b \u2260 0 H' : b \u2223 a \u22a2 div a b = c \u2194 a = c * b ** rw [Int.mul_comm] ** a b c : Int H : b \u2260 0 H' : b \u2223 a \u22a2 div a b = c \u2194 a = b * c ** exact Int.div_eq_iff_eq_mul_right H H' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.toJordanDecomposition_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 \u22a2 toJordanDecomposition 0 = 0 ** apply toSignedMeasure_injective ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 \u22a2 toSignedMeasure (toJordanDecomposition 0) = toSignedMeasure 0 ** simp [toSignedMeasure_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.sum_mul_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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 \u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc ** have h\u03bebdd : \u2200 i, \u2203 C, \u2200 \u03c9, |\u03be i \u03c9| \u2264 C := fun i =>\n  \u27e8R, fun \u03c9 => (abs_of_nonneg (hnonneg i \u03c9)).trans_le (hbdd i \u03c9)\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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 h\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C \u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc ** have hint : \u2200 m, Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) \u03bc := fun m =>\n  integrable_finset_sum' _ fun i _ => Integrable.bdd_mul ((hf.integrable _).sub (hf.integrable _))\n    h\u03be.stronglyMeasurable.aestronglyMeasurable (h\u03bebdd _) ** \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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 h\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C hint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) \u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc ** have hadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k) := by\n  intro m\n  refine' Finset.stronglyMeasurable_sum' _ fun i hi => _\n  rw [Finset.mem_range] at hi\n  exact (h\u03be.stronglyMeasurable_le hi.le).mul\n    ((hf.adapted.stronglyMeasurable_le (Nat.succ_le_of_lt hi)).sub\n      (hf.adapted.stronglyMeasurable_le hi.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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 h\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C hint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) hadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k) \u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc ** refine' submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => _ ** \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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 h\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C hint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) hadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k) i : \u2115 \u22a2 0 \u2264\u1d50[\u03bc] \u03bc[\u2211 k in Finset.range (i + 1), \u03be k * (f (k + 1) - f k) - \u2211 k in Finset.range i, \u03be k * (f (k + 1) - f k)|\u2191\ud835\udca2 i] ** simp only [\u2190 Finset.sum_Ico_eq_sub _ (Nat.le_succ _), Finset.sum_apply, Pi.mul_apply,\n  Pi.sub_apply, Nat.Ico_succ_singleton, Finset.sum_singleton] ** \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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 h\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C hint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) hadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k) i : \u2115 \u22a2 0 \u2264\u1d50[\u03bc] \u03bc[\u03be i * (f (i + 1) - f i)|\u2191\ud835\udca2 i] ** exact EventuallyLE.trans (EventuallyLE.mul_nonneg (eventually_of_forall (hnonneg _))\n  (hf.condexp_sub_nonneg (Nat.le_succ _))) (condexp_stronglyMeasurable_mul (h\u03be _)\n  (((hf.integrable _).sub (hf.integrable _)).bdd_mul\n    h\u03be.stronglyMeasurable.aestronglyMeasurable (h\u03bebdd _))\n  ((hf.integrable _).sub (hf.integrable _))).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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 h\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C hint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) \u22a2 Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k) ** intro m ** \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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 h\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C hint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) m : \u2115 \u22a2 StronglyMeasurable ((fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) m) ** refine' Finset.stronglyMeasurable_sum' _ fun i hi => _ ** \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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 h\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C hint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) m i : \u2115 hi : i \u2208 Finset.range m \u22a2 StronglyMeasurable (\u03be i * (f (i + 1) - f i)) ** rw [Finset.mem_range] at hi ** \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 R : \u211d \u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \ud835\udca2 \u03bc h\u03be : Adapted \ud835\udca2 \u03be hbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R hnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9 h\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C hint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) m i : \u2115 hi : i < m \u22a2 StronglyMeasurable (\u03be i * (f (i + 1) - f i)) ** exact (h\u03be.stronglyMeasurable_le hi.le).mul\n  ((hf.adapted.stronglyMeasurable_le (Nat.succ_le_of_lt hi)).sub\n    (hf.adapted.stronglyMeasurable_le hi.le)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Semiquot.ext_s ** \u03b1 : Type u_1 \u03b2 : Type u_2 q\u2081 q\u2082 : Semiquot \u03b1 \u22a2 q\u2081 = q\u2082 \u2194 q\u2081.s = q\u2082.s ** refine' \u27e8congr_arg _, fun h => _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 q\u2081 q\u2082 : Semiquot \u03b1 h : q\u2081.s = q\u2082.s \u22a2 q\u2081 = q\u2082 ** cases' q\u2081 with _ v\u2081 ** case mk' \u03b1 : Type u_1 \u03b2 : Type u_2 q\u2082 : Semiquot \u03b1 s\u271d : Set \u03b1 v\u2081 : Trunc \u2191s\u271d h : { s := s\u271d, val := v\u2081 }.s = q\u2082.s \u22a2 { s := s\u271d, val := v\u2081 } = q\u2082 ** cases' q\u2082 with _ v\u2082 ** case mk'.mk' \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d\u00b9 : Set \u03b1 v\u2081 : Trunc \u2191s\u271d\u00b9 s\u271d : Set \u03b1 v\u2082 : Trunc \u2191s\u271d h : { s := s\u271d\u00b9, val := v\u2081 }.s = { s := s\u271d, val := v\u2082 }.s \u22a2 { s := s\u271d\u00b9, val := v\u2081 } = { s := s\u271d, val := v\u2082 } ** congr ** case mk'.mk'.h.e_3 \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d\u00b9 : Set \u03b1 v\u2081 : Trunc \u2191s\u271d\u00b9 s\u271d : Set \u03b1 v\u2082 : Trunc \u2191s\u271d h : { s := s\u271d\u00b9, val := v\u2081 }.s = { s := s\u271d, val := v\u2082 }.s \u22a2 HEq v\u2081 v\u2082 ** exact Subsingleton.helim (congrArg Trunc (congrArg Set.Elem h)) v\u2081 v\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.tendsto_filterAt ** \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 : SigmaFinite \u03bc x : \u03b1 \u22a2 Tendsto (fun r => closedBall x r) (\ud835\udcdd[Ioi 0] 0) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x) ** intro s hs ** \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 : SigmaFinite \u03bc x : \u03b1 s : Set (Set \u03b1) hs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x \u22a2 s \u2208 map (fun r => closedBall x r) (\ud835\udcdd[Ioi 0] 0) ** simp only [mem_map] ** \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 : SigmaFinite \u03bc x : \u03b1 s : Set (Set \u03b1) hs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x \u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0 ** obtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 :\n  \u2203 (\u03b5 : \u211d), \u03b5 > 0 \u2227\n    \u2200 a : Set \u03b1, a \u2208 (Besicovitch.vitaliFamily \u03bc).setsAt x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s :=\n  (VitaliFamily.mem_filterAt_iff _).1 hs ** case 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 : SigmaFinite \u03bc x : \u03b1 s : Set (Set \u03b1) hs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 h\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s \u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0 ** have : Ioc (0 : \u211d) \u03b5 \u2208 \ud835\udcdd[>] (0 : \u211d) := Ioc_mem_nhdsWithin_Ioi \u27e8le_rfl, \u03b5pos\u27e9 ** case 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 : SigmaFinite \u03bc x : \u03b1 s : Set (Set \u03b1) hs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 h\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s this : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0 \u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0 ** filter_upwards [this] with _ hr ** 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 : SigmaFinite \u03bc x : \u03b1 s : Set (Set \u03b1) hs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 h\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s this : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0 a\u271d : \u211d hr : a\u271d \u2208 Ioc 0 \u03b5 \u22a2 a\u271d \u2208 (fun r => closedBall x r) \u207b\u00b9' s ** apply h\u03b5 ** case h.a \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 : SigmaFinite \u03bc x : \u03b1 s : Set (Set \u03b1) hs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 h\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s this : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0 a\u271d : \u211d hr : a\u271d \u2208 Ioc 0 \u03b5 \u22a2 (fun r => closedBall x r) a\u271d \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x ** exact mem_image_of_mem _ hr.1 ** case h.a \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 : SigmaFinite \u03bc x : \u03b1 s : Set (Set \u03b1) hs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x \u03b5 : \u211d \u03b5pos : \u03b5 > 0 h\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s this : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0 a\u271d : \u211d hr : a\u271d \u2208 Ioc 0 \u03b5 \u22a2 (fun r => closedBall x r) a\u271d \u2286 closedBall x \u03b5 ** exact closedBall_subset_closedBall hr.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.tendsto_ae_condexp ** \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 g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x)) ** have hle : \u2a06 n, \u2131 n \u2264 m0 := sSup_le fun m \u27e8n, hn\u27e9 => hn \u25b8 \u2131.le _ ** \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 g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x)) ** have hunif : UniformIntegrable (fun n => \u03bc[g|\u2131 n]) 1 \u03bc :=\n  hg.uniformIntegrable_condexp_filtration ** \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 g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x)) ** obtain \u27e8R, hR\u27e9 := hunif.2.2 ** 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x)) ** have hlimint : Integrable (\u2131.limitProcess (fun n => \u03bc[g|\u2131 n]) \u03bc) \u03bc :=\n  (mem\u2112p_limitProcess_of_snorm_bdd hunif.1 hR).integrable le_rfl ** 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x)) ** suffices g =\u1d50[\u03bc] \u2131.limitProcess (fun n x => (\u03bc[g|\u2131 n]) x) \u03bc by\n  filter_upwards [this, (martingale_condexp g \u2131 \u03bc).submartingale.ae_tendsto_limitProcess hR] with\n    x heq ht\n  rwa [heq] ** 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) \u22a2 g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc ** have : \u2200 n s, MeasurableSet[\u2131 n] s \u2192\n    \u222b x in s, g x \u2202\u03bc = \u222b x in s, \u2131.limitProcess (fun n x => (\u03bc[g|\u2131 n]) x) \u03bc x \u2202\u03bc := by\n  intro n s hs\n  rw [\u2190 set_integral_condexp (\u2131.le n) hg hs, \u2190 set_integral_condexp (\u2131.le n) hlimint hs]\n  refine' set_integral_congr_ae (\u2131.le _ _ hs) _\n  filter_upwards [(martingale_condexp g \u2131 \u03bc).ae_eq_condexp_limitProcess hunif n] with x hx _\n  rw [hx] ** 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc \u22a2 g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc ** refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hle (fun s _ _ => hg.integrableOn)\n  (fun s _ _ => hlimint.integrableOn) (fun s hs => _) hgmeas.aeStronglyMeasurable'\n  stronglyMeasurable_limitProcess.aeStronglyMeasurable' ** 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s \u22a2 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** apply @MeasurableSpace.induction_on_inter _ _ _ (\u2a06 n, \u2131 n)\n  (MeasurableSpace.measurableSpace_iSup_eq \u2131) _ _ _ _ _ _ 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this : g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x)) ** filter_upwards [this, (martingale_condexp g \u2131 \u03bc).submartingale.ae_tendsto_limitProcess hR] with\n  x heq ht ** case h \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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this : g =\u1d50[\u03bc] limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x : \u03a9 heq : g x = limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x ht : Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (limitProcess (fun i => \u03bc[g|\u2191\u2131 i]) \u2131 \u03bc x)) \u22a2 Tendsto (fun n => (\u03bc[g|\u2191\u2131 n]) x) atTop (\ud835\udcdd (g x)) ** rwa [heq] ** \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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) \u22a2 \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** intro n s 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) n : \u2115 s : Set \u03a9 hs : MeasurableSet s \u22a2 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** rw [\u2190 set_integral_condexp (\u2131.le n) hg hs, \u2190 set_integral_condexp (\u2131.le n) hlimint 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) n : \u2115 s : Set \u03a9 hs : MeasurableSet s \u22a2 \u222b (x : \u03a9) in s, (\u03bc[g|\u2191\u2131 n]) x \u2202\u03bc = \u222b (x : \u03a9) in s, (\u03bc[limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc|\u2191\u2131 n]) x \u2202\u03bc ** refine' set_integral_congr_ae (\u2131.le _ _ 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) n : \u2115 s : Set \u03a9 hs : MeasurableSet s \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 s \u2192 (\u03bc[g|\u2191\u2131 n]) x = (\u03bc[limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc|\u2191\u2131 n]) x ** filter_upwards [(martingale_condexp g \u2131 \u03bc).ae_eq_condexp_limitProcess hunif n] with x hx _ ** case h \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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) n : \u2115 s : Set \u03a9 hs : MeasurableSet s x : \u03a9 hx : (\u03bc[g|\u2191\u2131 n]) x = (\u03bc[limitProcess (fun i => \u03bc[g|\u2191\u2131 i]) \u2131 \u03bc|\u2191\u2131 n]) x a\u271d : x \u2208 s \u22a2 (\u03bc[g|\u2191\u2131 n]) x = (\u03bc[limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc|\u2191\u2131 n]) x ** rw [hx] ** \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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s \u22a2 IsPiSystem {s | \u2203 n, MeasurableSet s} ** rintro s \u27e8n, hs\u27e9 t \u27e8m, ht\u27e9 - ** case 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s\u271d : Set \u03a9 hs\u271d : MeasurableSet s\u271d s : Set \u03a9 n : \u2115 hs : MeasurableSet s t : Set \u03a9 m : \u2115 ht : MeasurableSet t \u22a2 s \u2229 t \u2208 {s | \u2203 n, MeasurableSet s} ** by_cases hnm : n \u2264 m ** 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s\u271d : Set \u03a9 hs\u271d : MeasurableSet s\u271d s : Set \u03a9 n : \u2115 hs : MeasurableSet s t : Set \u03a9 m : \u2115 ht : MeasurableSet t hnm : n \u2264 m \u22a2 s \u2229 t \u2208 {s | \u2203 n, MeasurableSet s} ** exact \u27e8m, (\u2131.mono hnm _ hs).inter ht\u27e9 ** 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s\u271d : Set \u03a9 hs\u271d : MeasurableSet s\u271d s : Set \u03a9 n : \u2115 hs : MeasurableSet s t : Set \u03a9 m : \u2115 ht : MeasurableSet t hnm : \u00acn \u2264 m \u22a2 s \u2229 t \u2208 {s | \u2203 n, MeasurableSet s} ** exact \u27e8n, hs.inter (\u2131.mono (not_le.1 hnm).le _ ht)\u27e9 ** \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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s \u22a2 \u2191\u2191\u03bc \u2205 < \u22a4 \u2192 \u222b (x : \u03a9) in \u2205, g x \u2202\u03bc = \u222b (x : \u03a9) in \u2205, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** simp only [measure_empty, WithTop.zero_lt_top, Measure.restrict_empty, integral_zero_measure,\n  forall_true_left] ** \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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s \u22a2 \u2200 (t : Set \u03a9),     t \u2208 {s | \u2203 n, MeasurableSet s} \u2192       \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** rintro t \u27e8n, ht\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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s t : Set \u03a9 n : \u2115 ht : MeasurableSet t \u22a2 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** exact this n _ 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s \u22a2 \u2200 (t : Set \u03a9),     MeasurableSet t \u2192       (\u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc) \u2192         \u2191\u2191\u03bc t\u1d9c < \u22a4 \u2192 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** rintro t htmeas 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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s t : Set \u03a9 htmeas : MeasurableSet t ht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc \u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** have hgeq := @integral_add_compl _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ htmeas (hg.trim hle hgmeas) ** \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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s t : Set \u03a9 htmeas : MeasurableSet t ht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc hgeq :   \u222b (x : \u03a9) in t, g x \u2202Measure.trim \u03bc hle + \u222b (x : \u03a9) in t\u1d9c, g x \u2202Measure.trim \u03bc hle =     \u222b (x : \u03a9), g x \u2202Measure.trim \u03bc hle \u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** have hheq := @integral_add_compl _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ htmeas\n  (hlimint.trim hle stronglyMeasurable_limitProcess) ** \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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s t : Set \u03a9 htmeas : MeasurableSet t ht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc hgeq :   \u222b (x : \u03a9) in t, g x \u2202Measure.trim \u03bc hle + \u222b (x : \u03a9) in t\u1d9c, g x \u2202Measure.trim \u03bc hle =     \u222b (x : \u03a9), g x \u2202Measure.trim \u03bc hle hheq :   \u222b (x : \u03a9) in t, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle +       \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle =     \u222b (x : \u03a9), limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle \u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** rw [add_comm, \u2190 eq_sub_iff_add_eq] at hgeq hheq ** \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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s t : Set \u03a9 htmeas : MeasurableSet t ht : \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03a9) in t, g x \u2202\u03bc = \u222b (x : \u03a9) in t, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc hgeq :   \u222b (x : \u03a9) in t\u1d9c, g x \u2202Measure.trim \u03bc hle =     \u222b (x : \u03a9), g x \u2202Measure.trim \u03bc hle - \u222b (x : \u03a9) in t, g x \u2202Measure.trim \u03bc hle hheq :   \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle =     \u222b (x : \u03a9), limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle -       \u222b (x : \u03a9) in t, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc x \u2202Measure.trim \u03bc hle \u22a2 \u222b (x : \u03a9) in t\u1d9c, g x \u2202\u03bc = \u222b (x : \u03a9) in t\u1d9c, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** rw [set_integral_trim hle hgmeas htmeas.compl,\n  set_integral_trim hle stronglyMeasurable_limitProcess htmeas.compl, hgeq, hheq, \u2190\n  set_integral_trim hle hgmeas htmeas, \u2190\n  set_integral_trim hle stronglyMeasurable_limitProcess htmeas, \u2190 integral_trim hle hgmeas, \u2190\n  integral_trim hle stronglyMeasurable_limitProcess, \u2190 integral_univ,\n  this 0 _ MeasurableSet.univ, integral_univ, ht (measure_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\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s \u22a2 \u2200 (f : \u2115 \u2192 Set \u03a9),     Pairwise (Disjoint on f) \u2192       (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192         (\u2200 (i : \u2115),             \u2191\u2191\u03bc (f i) < \u22a4 \u2192               \u222b (x : \u03a9) in f i, g x \u2202\u03bc = \u222b (x : \u03a9) in f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc) \u2192           \u2191\u2191\u03bc (\u22c3 i, f i) < \u22a4 \u2192             \u222b (x : \u03a9) in \u22c3 i, f i, g x \u2202\u03bc = \u222b (x : \u03a9) in \u22c3 i, f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** rintro f hf hfmeas heq - ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s f : \u2115 \u2192 Set \u03a9 hf : Pairwise (Disjoint on f) hfmeas : \u2200 (i : \u2115), MeasurableSet (f i) heq :   \u2200 (i : \u2115),     \u2191\u2191\u03bc (f i) < \u22a4 \u2192 \u222b (x : \u03a9) in f i, g x \u2202\u03bc = \u222b (x : \u03a9) in f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc \u22a2 \u222b (x : \u03a9) in \u22c3 i, f i, g x \u2202\u03bc = \u222b (x : \u03a9) in \u22c3 i, f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc ** rw [integral_iUnion (fun n => hle _ (hfmeas n)) hf hg.integrableOn,\n  integral_iUnion (fun n => hle _ (hfmeas n)) hf hlimint.integrableOn] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R\u271d : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc g : \u03a9 \u2192 \u211d hg : Integrable g hgmeas : StronglyMeasurable g hle : \u2a06 n, \u2191\u2131 n \u2264 m0 hunif : UniformIntegrable (fun n => \u03bc[g|\u2191\u2131 n]) 1 \u03bc R : \u211d\u22650 hR : \u2200 (i : \u2115), snorm ((fun n => \u03bc[g|\u2191\u2131 n]) i) 1 \u03bc \u2264 \u2191R hlimint : Integrable (limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc) this :   \u2200 (n : \u2115) (s : Set \u03a9),     MeasurableSet s \u2192 \u222b (x : \u03a9) in s, g x \u2202\u03bc = \u222b (x : \u03a9) in s, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc s : Set \u03a9 hs : MeasurableSet s f : \u2115 \u2192 Set \u03a9 hf : Pairwise (Disjoint on f) hfmeas : \u2200 (i : \u2115), MeasurableSet (f i) heq :   \u2200 (i : \u2115),     \u2191\u2191\u03bc (f i) < \u22a4 \u2192 \u222b (x : \u03a9) in f i, g x \u2202\u03bc = \u222b (x : \u03a9) in f i, limitProcess (fun n x => (\u03bc[g|\u2191\u2131 n]) x) \u2131 \u03bc x \u2202\u03bc \u22a2 \u2211' (n : \u2115), \u222b (a : \u03a9) in f n, g a \u2202\u03bc = \u2211' (n : \u2115), \u222b (a : \u03a9) in f n, limitProcess (fun n => \u03bc[g|\u2191\u2131 n]) \u2131 \u03bc a \u2202\u03bc ** exact tsum_congr fun n => heq _ (measure_lt_top _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.eq_one_of_mul_eq_one_left ** a b : Int H : 0 \u2264 b H' : a * b = 1 \u22a2 b * ?m.91489 H H' = 1 ** rw [Int.mul_comm, H'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_inter_add_diff\u2080 ** \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 s : Set \u03b1 ht : NullMeasurableSet t \u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) = \u2191\u2191\u03bc s ** refine' le_antisymm _ _ ** case refine'_1 \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 s : Set \u03b1 ht : NullMeasurableSet t \u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s ** rcases exists_measurable_superset \u03bc s with \u27e8s', hsub, hs'm, hs'\u27e9 ** case refine'_1.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 s : Set \u03b1 ht : NullMeasurableSet t s' : Set \u03b1 hsub : s \u2286 s' hs'm : MeasurableSet s' hs' : \u2191\u2191\u03bc s' = \u2191\u2191\u03bc s \u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s ** replace hs'm : NullMeasurableSet s' \u03bc := hs'm.nullMeasurableSet ** case refine'_1.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 s : Set \u03b1 ht : NullMeasurableSet t s' : Set \u03b1 hsub : s \u2286 s' hs' : \u2191\u2191\u03bc s' = \u2191\u2191\u03bc s hs'm : NullMeasurableSet s' \u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s ** calc\n  \u03bc (s \u2229 t) + \u03bc (s \\ t) \u2264 \u03bc (s' \u2229 t) + \u03bc (s' \\ t) :=\n    add_le_add (measure_mono <| inter_subset_inter_left _ hsub)\n      (measure_mono <| diff_subset_diff_left hsub)\n  _ = \u03bc (s' \u2229 t \u222a s' \\ t) :=\n    (measure_union\u2080_aux (hs'm.inter ht) (hs'm.diff ht) <|\n        (@disjoint_inf_sdiff _ s' t _).aedisjoint).symm\n  _ = \u03bc s' := (congr_arg \u03bc (inter_union_diff _ _))\n  _ = \u03bc s := hs' ** case refine'_2 \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 s : Set \u03b1 ht : NullMeasurableSet t \u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) ** calc\n  \u03bc s = \u03bc (s \u2229 t \u222a s \\ t) := by rw [inter_union_diff]\n  _ \u2264 \u03bc (s \u2229 t) + \u03bc (s \\ t) := measure_union_le _ _ ** \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 s : Set \u03b1 ht : NullMeasurableSet t \u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 t \u222a s \\ t) ** rw [inter_union_diff] ** Qed",
        "informal": ""
    },
    {
        "formal": "TorusIntegrable.torusIntegrable_zero_radius ** n : \u2115 E : Type u_1 inst\u271d : NormedAddCommGroup E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R : Fin n \u2192 \u211d f : (Fin n \u2192 \u2102) \u2192 E c : Fin n \u2192 \u2102 \u22a2 TorusIntegrable f c 0 ** rw [TorusIntegrable, torusMap_zero_radius] ** n : \u2115 E : Type u_1 inst\u271d : NormedAddCommGroup E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R : Fin n \u2192 \u211d f : (Fin n \u2192 \u2102) \u2192 E c : Fin n \u2192 \u2102 \u22a2 IntegrableOn (fun \u03b8 => f (const (Fin n \u2192 \u211d) c \u03b8)) (Icc 0 fun x => 2 * \u03c0) ** apply torusIntegrable_const (f c) c 0 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.quasiMeasurePreserving_div_left_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 g : G \u22a2 QuasiMeasurePreserving fun h => g / h ** exact\n  (quasiMeasurePreserving_div_left \u03bc.inv g).mono (inv_absolutelyContinuous \u03bc.inv)\n    (absolutelyContinuous_inv \u03bc.inv) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.add_measure ** \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 h\u03bc : Integrable f h\u03bd : Integrable f \u22a2 Integrable f ** simp_rw [\u2190 mem\u2112p_one_iff_integrable] at h\u03bc h\u03bd \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\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192 \u03b2 h\u03bc : Mem\u2112p f 1 h\u03bd : Mem\u2112p f 1 \u22a2 Mem\u2112p f 1 ** refine' \u27e8h\u03bc.aestronglyMeasurable.add_measure h\u03bd.aestronglyMeasurable, _\u27e9 ** \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 h\u03bc : Mem\u2112p f 1 h\u03bd : Mem\u2112p f 1 \u22a2 snorm f 1 (\u03bc + \u03bd) < \u22a4 ** rw [snorm_one_add_measure, ENNReal.add_lt_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 \u03b2 h\u03bc : Mem\u2112p f 1 h\u03bd : Mem\u2112p f 1 \u22a2 snorm f 1 \u03bc < \u22a4 \u2227 snorm f 1 \u03bd < \u22a4 ** exact \u27e8h\u03bc.snorm_lt_top, h\u03bd.snorm_lt_top\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_to_prop ** \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 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 Prop h : MeasurableSet (f \u207b\u00b9' {True}) \u22a2 Measurable f ** refine' measurable_to_countable' fun x => _ ** \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 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 Prop h : MeasurableSet (f \u207b\u00b9' {True}) x : Prop \u22a2 MeasurableSet (f \u207b\u00b9' {x}) ** by_cases hx : x ** case pos \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 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 Prop h : MeasurableSet (f \u207b\u00b9' {True}) x : Prop hx : x \u22a2 MeasurableSet (f \u207b\u00b9' {x}) ** simpa [hx] using h ** case neg \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 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 Prop h : MeasurableSet (f \u207b\u00b9' {True}) x : Prop hx : \u00acx \u22a2 MeasurableSet (f \u207b\u00b9' {x}) ** simpa only [hx, \u2190 preimage_compl, Prop.compl_singleton, not_true, preimage_singleton_false]\n  using h.compl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.powerset_insert ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 \u22a2 \ud835\udcab insert a s = \ud835\udcab s \u222a insert a '' \ud835\udcab s ** ext t ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 \u22a2 t \u2208 \ud835\udcab insert a s \u2194 t \u2208 \ud835\udcab s \u222a insert a '' \ud835\udcab s ** simp_rw [mem_union, mem_image, mem_powerset_iff] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 \u22a2 t \u2286 insert a s \u2194 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t ** constructor ** case h.mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 \u22a2 t \u2286 insert a s \u2192 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t ** intro h ** case h.mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 h : t \u2286 insert a s \u22a2 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t ** by_cases hs : a \u2208 t ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 h : t \u2286 insert a s hs : a \u2208 t \u22a2 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t ** right ** case pos.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 h : t \u2286 insert a s hs : a \u2208 t \u22a2 \u2203 x, x \u2286 s \u2227 insert a x = t ** refine' \u27e8t \\ {a}, _, _\u27e9 ** case pos.h.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 h : t \u2286 insert a s hs : a \u2208 t \u22a2 t \\ {a} \u2286 s ** rw [diff_singleton_subset_iff] ** case pos.h.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 h : t \u2286 insert a s hs : a \u2208 t \u22a2 t \u2286 insert a s ** assumption ** case pos.h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 h : t \u2286 insert a s hs : a \u2208 t \u22a2 insert a (t \\ {a}) = t ** rw [insert_diff_singleton, insert_eq_of_mem hs] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 h : t \u2286 insert a s hs : \u00aca \u2208 t \u22a2 t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t ** left ** case neg.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 h : t \u2286 insert a s hs : \u00aca \u2208 t \u22a2 t \u2286 s ** exact (subset_insert_iff_of_not_mem hs).mp h ** case h.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 \u22a2 (t \u2286 s \u2228 \u2203 x, x \u2286 s \u2227 insert a x = t) \u2192 t \u2286 insert a s ** rintro (h | \u27e8s', h\u2081, rfl\u27e9) ** case h.mpr.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 t : Set \u03b1 h : t \u2286 s \u22a2 t \u2286 insert a s ** exact subset_trans h (subset_insert a s) ** case h.mpr.inr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 s : Set \u03b1 a : \u03b1 s' : Set \u03b1 h\u2081 : s' \u2286 s \u22a2 insert a s' \u2286 insert a s ** exact insert_subset_insert h\u2081 ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.le_iff_cmp ** \u03b1 : Type u_1 m n : Num \u22a2 cmp n m = Ordering.lt \u2194 cmp m n = Ordering.gt ** rw [\u2190 cmp_swap] ** \u03b1 : Type u_1 m n : Num \u22a2 Ordering.swap (cmp m n) = Ordering.lt \u2194 cmp m n = Ordering.gt ** cases cmp m n <;> exact by decide ** \u03b1 : Type u_1 m n : Num \u22a2 Ordering.swap Ordering.gt = Ordering.lt \u2194 Ordering.gt = Ordering.gt ** decide ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.map_top_of_surjective ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m : OuterMeasure \u03b1 f : \u03b1 \u2192 \u03b2 hf : Surjective f \u22a2 \u2191(map f) \u22a4 = \u22a4 ** rw [map_top, hf.range_eq, restrict_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.zero_trim ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hm : m \u2264 m0 \u22a2 Measure.trim 0 hm = 0 ** simp [Measure.trim, @OuterMeasure.toMeasure_zero _ m] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm_top ** \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0) \u22a2 TendstoInMeasure \u03bc f l g ** intro \u03b4 h\u03b4 ** \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0) \u03b4 : \u211d h\u03b4 : 0 < \u03b4 \u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0) ** simp only [snorm_exponent_top, snormEssSup] at hfg ** \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 hfg : Tendsto (fun n => essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc) l (\ud835\udcdd 0) \u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \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\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 hfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5 \u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5 ** intro \u03b5 h\u03b5 ** \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 hfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 \u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5 ** specialize hfg (ENNReal.ofReal \u03b4 / 2)\n    (ENNReal.div_pos_iff.2 \u27e8(ENNReal.ofReal_pos.2 h\u03b4).ne.symm, ENNReal.two_ne_top\u27e9) ** \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 hfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 \u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5 ** refine' hfg.mono fun n hn => _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 hfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 n : \u03b9 hn : essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 \u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5 ** simp only [true_and_iff, gt_iff_lt, ge_iff_le, zero_tsub, zero_le, zero_add, Set.mem_Icc,\n  Pi.sub_apply] at * ** \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 \u03b5 : \u211d\u22650\u221e hfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 n : \u03b9 hn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 h\u03b5 : 0 < \u03b5 \u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5 ** have : essSup (fun x : \u03b1 => (\u2016f n x - g x\u2016\u208a : \u211d\u22650\u221e)) \u03bc < ENNReal.ofReal \u03b4 :=\n  lt_of_le_of_lt hn\n    (ENNReal.half_lt_self (ENNReal.ofReal_pos.2 h\u03b4).ne.symm ENNReal.ofReal_lt_top.ne) ** \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 \u03b5 : \u211d\u22650\u221e hfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 n : \u03b9 hn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 h\u03b5 : 0 < \u03b5 this : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4 \u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5 ** refine' ((le_of_eq _).trans (ae_lt_of_essSup_lt this).le).trans h\u03b5.le ** \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 \u03b5 : \u211d\u22650\u221e hfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 n : \u03b9 hn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 h\u03b5 : 0 < \u03b5 this : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4 \u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} = \u2191\u2191\u03bc {x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c ** congr with x ** case e_a.h \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 \u03b5 : \u211d\u22650\u221e hfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 n : \u03b9 hn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 h\u03b5 : 0 < \u03b5 this : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4 x : \u03b1 \u22a2 x \u2208 {x | \u03b4 \u2264 dist (f n x) (g x)} \u2194 x \u2208 {x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c ** simp only [ENNReal.ofReal_le_iff_le_toReal ENNReal.coe_lt_top.ne, ENNReal.coe_toReal, not_lt,\n  coe_nnnorm, Set.mem_setOf_eq, Set.mem_compl_iff] ** case e_a.h \u03b1 : Type u_1 \u03b9 : Type u_2 E\u271d : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E\u271d p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d g\u271d : \u03b1 \u2192 E\u271d E : Type u_4 inst\u271d : NormedAddCommGroup E f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E l : Filter \u03b9 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 \u03b5 : \u211d\u22650\u221e hfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 n : \u03b9 hn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2 h\u03b5 : 0 < \u03b5 this : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4 x : \u03b1 \u22a2 \u03b4 \u2264 dist (f n x) (g x) \u2194 \u03b4 \u2264 \u2016f n x - g x\u2016 ** rw [\u2190 dist_eq_norm (f n x) (g x)] ** Qed",
        "informal": ""
    },
    {
        "formal": "IsUnifLocDoublingMeasure.ae_tendsto_average ** \u03b1 : Type u_1 inst\u271d\u2078 : MetricSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 inst\u271d\u00b3 : IsLocallyFiniteMeasure \u03bc E : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u03b1 \u2192 E hf : LocallyIntegrable f K : \u211d \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 {\u03b9 : Type u_3} {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, x \u2208 closedBall (w j) (K * \u03b4 j)) \u2192           Tendsto (fun j => \u2a0d (y : \u03b1) in closedBall (w j) (\u03b4 j), f y \u2202\u03bc) l (\ud835\udcdd (f x)) ** filter_upwards [(vitaliFamily \u03bc K).ae_tendsto_average hf] with x hx \u03b9 l w \u03b4 \u03b4lim xmem using\n  hx.comp (tendsto_closedBall_filterAt \u03bc _ _ \u03b4lim xmem) ** Qed",
        "informal": ""
    },
    {
        "formal": "Function.Periodic.intervalIntegral_add_eq_of_pos ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T hT : 0 < T t s : \u211d \u22a2 \u222b (x : \u211d) in t..t + T, f x = \u222b (x : \u211d) in s..s + T, f x ** simp only [integral_of_le, hT.le, le_add_iff_nonneg_right] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T hT : 0 < T t s : \u211d \u22a2 \u222b (x : \u211d) in Ioc t (t + T), f x = \u222b (x : \u211d) in Ioc s (s + T), f x ** haveI : VAddInvariantMeasure (AddSubgroup.zmultiples T) \u211d volume :=\n  \u27e8fun c s _ => measure_preimage_add _ _ _\u27e9 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T hT : 0 < T t s : \u211d this : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume \u22a2 \u222b (x : \u211d) in Ioc t (t + T), f x = \u222b (x : \u211d) in Ioc s (s + T), f x ** apply IsAddFundamentalDomain.set_integral_eq (G := AddSubgroup.zmultiples T) ** case hs E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T hT : 0 < T t s : \u211d this : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume \u22a2 IsAddFundamentalDomain { x // x \u2208 zmultiples T } (Ioc t (t + T))  case ht E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T hT : 0 < T t s : \u211d this : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume \u22a2 IsAddFundamentalDomain { x // x \u2208 zmultiples T } (Ioc s (s + T))  case hf E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 E T : \u211d hf : Periodic f T hT : 0 < T t s : \u211d this : VAddInvariantMeasure { x // x \u2208 zmultiples T } \u211d volume \u22a2 \u2200 (g : { x // x \u2208 zmultiples T }) (x : \u211d), f (g +\u1d65 x) = f x ** exacts [isAddFundamentalDomain_Ioc hT t, isAddFundamentalDomain_Ioc hT s, hf.map_vadd_zmultiples] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.biUnion_diff_biUnion_eq ** \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 s t : Set \u03b9 f : \u03b9 \u2192 Set \u03b1 h : PairwiseDisjoint (s \u222a t) f \u22a2 (\u22c3 i \u2208 s, f i) \\ \u22c3 i \u2208 t, f i = \u22c3 i \u2208 s \\ t, f i ** refine'\n  (biUnion_diff_biUnion_subset f s t).antisymm\n    (iUnion\u2082_subset fun i hi a ha => (mem_diff _).2 \u27e8mem_biUnion hi.1 ha, _\u27e9) ** \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 s t : Set \u03b9 f : \u03b9 \u2192 Set \u03b1 h : PairwiseDisjoint (s \u222a t) f i : \u03b9 hi : i \u2208 s \\ t a : \u03b1 ha : a \u2208 f i \u22a2 \u00aca \u2208 \u22c3 x \u2208 t, f x ** rw [mem_iUnion\u2082] ** \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 s t : Set \u03b9 f : \u03b9 \u2192 Set \u03b1 h : PairwiseDisjoint (s \u222a t) f i : \u03b9 hi : i \u2208 s \\ t a : \u03b1 ha : a \u2208 f i \u22a2 \u00ac\u2203 i j, a \u2208 f i ** rintro \u27e8j, hj, haj\u27e9 ** case intro.intro \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 s t : Set \u03b9 f : \u03b9 \u2192 Set \u03b1 h : PairwiseDisjoint (s \u222a t) f i : \u03b9 hi : i \u2208 s \\ t a : \u03b1 ha : a \u2208 f i j : \u03b9 hj : j \u2208 t haj : a \u2208 f j \u22a2 False ** exact (h (Or.inl hi.1) (Or.inr hj) (ne_of_mem_of_not_mem hj hi.2).symm).le_bot \u27e8ha, haj\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.ofInt'_toZNum ** \u03b1 : Type u_1 n : \u2115 \u22a2 toZNum \u2191(n + 1) = ZNum.ofInt' \u2191(n + 1) ** rw [Nat.cast_succ, Num.add_one, toZNum_succ, ofInt'_toZNum n, Nat.cast_succ, succ_ofInt',\n  ZNum.add_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.offDiag_insert ** \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 \u00d7 \u03b1 a : \u03b1 ha : \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, offDiag_singleton, union_empty, union_right_comm] ** \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 \u00d7 \u03b1 a : \u03b1 ha : \u00aca \u2208 s \u22a2 Disjoint s {a} ** rw [disjoint_left] ** \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 \u00d7 \u03b1 a : \u03b1 ha : \u00aca \u2208 s \u22a2 \u2200 \u2983a_1 : \u03b1\u2984, a_1 \u2208 s \u2192 \u00aca_1 \u2208 {a} ** rintro b hb (rfl : b = a) ** \u03b1 : Type u_1 s t : Set \u03b1 x : \u03b1 \u00d7 \u03b1 b : \u03b1 hb : b \u2208 s ha : \u00acb \u2208 s \u22a2 False ** exact ha hb ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.Subtype.volume_univ ** \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 s : Set \u03b1 inst\u271d : MeasureSpace \u03b1 p : \u03b1 \u2192 Prop hs : NullMeasurableSet s \u22a2 \u2191\u2191volume univ = \u2191\u2191volume s ** rw [Subtype.volume_def, comap_apply\u2080 _ _ _ _ MeasurableSet.univ.nullMeasurableSet] ** \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 s : Set \u03b1 inst\u271d : MeasureSpace \u03b1 p : \u03b1 \u2192 Prop hs : NullMeasurableSet s \u22a2 \u2191\u2191volume (Subtype.val '' univ) = \u2191\u2191volume s ** congr ** case e_a \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 s : Set \u03b1 inst\u271d : MeasureSpace \u03b1 p : \u03b1 \u2192 Prop hs : NullMeasurableSet s \u22a2 Subtype.val '' univ = s ** simp only [image_univ, Subtype.range_coe_subtype, setOf_mem_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 s : Set \u03b1 inst\u271d : MeasureSpace \u03b1 p : \u03b1 \u2192 Prop hs : NullMeasurableSet s \u22a2 Injective Subtype.val ** exact Subtype.coe_injective ** \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 s : Set \u03b1 inst\u271d : MeasureSpace \u03b1 p : \u03b1 \u2192 Prop hs : NullMeasurableSet s \u22a2 \u2200 (s_1 : Set \u2191s), MeasurableSet s_1 \u2192 NullMeasurableSet (Subtype.val '' s_1) ** exact fun t => MeasurableSet.nullMeasurableSet_subtype_coe hs ** Qed",
        "informal": ""
    },
    {
        "formal": "List.map_get_sublist ** \u03b1 : Type u_1 l : List \u03b1 is : List (Fin (length l)) h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is \u22a2 map (get l) is <+ l ** suffices \u2200 n l', l' = l.drop n \u2192 (\u2200 i \u2208 is, n \u2264 i) \u2192 map (get l) is <+ l'\n  from this 0 l (by simp) (by simp) ** \u03b1 : Type u_1 l : List \u03b1 is : List (Fin (length l)) h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is \u22a2 \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i) \u2192 map (get l) is <+ l' ** intro n l' hl' his ** \u03b1 : Type u_1 l : List \u03b1 is : List (Fin (length l)) h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is n : Nat l' : List \u03b1 hl' : l' = drop n l his : \u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i \u22a2 map (get l) is <+ l' ** induction is generalizing n l' with\n| nil => simp\n| cons hd tl IH =>\n  simp; cases hl'\n  have := IH h.of_cons (hd+1) _ rfl (pairwise_cons.mp h).1\n  specialize his hd (.head _)\n  have := get_cons_drop .. \u25b8 this.cons\u2082 (get l hd)\n  have := Sublist.append (nil_sublist (take hd l |>.drop n)) this\n  rwa [nil_append, \u2190 (drop_append_of_le_length ?_), take_append_drop] at this\n  simp [Nat.min_eq_left (Nat.le_of_lt hd.isLt), his] ** \u03b1 : Type u_1 l : List \u03b1 is : List (Fin (length l)) h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is this : \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i) \u2192 map (get l) is <+ l' \u22a2 l = drop 0 l ** simp ** \u03b1 : Type u_1 l : List \u03b1 is : List (Fin (length l)) h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) is this : \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 is \u2192 n \u2264 \u2191i) \u2192 map (get l) is <+ l' \u22a2 \u2200 (i : Fin (length l)), i \u2208 is \u2192 0 \u2264 \u2191i ** simp ** case nil \u03b1 : Type u_1 l : List \u03b1 h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) [] n : Nat l' : List \u03b1 hl' : l' = drop n l his : \u2200 (i : Fin (length l)), i \u2208 [] \u2192 n \u2264 \u2191i \u22a2 map (get l) [] <+ l' ** simp ** case cons \u03b1 : Type u_1 l : List \u03b1 hd : Fin (length l) tl : List (Fin (length l)) IH :   Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192     \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l' h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl) n : Nat l' : List \u03b1 hl' : l' = drop n l his : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i \u22a2 map (get l) (hd :: tl) <+ l' ** simp ** case cons \u03b1 : Type u_1 l : List \u03b1 hd : Fin (length l) tl : List (Fin (length l)) IH :   Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192     \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l' h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl) n : Nat l' : List \u03b1 hl' : l' = drop n l his : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i \u22a2 get l hd :: map (get l) tl <+ l' ** cases hl' ** case cons.refl \u03b1 : Type u_1 l : List \u03b1 hd : Fin (length l) tl : List (Fin (length l)) IH :   Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192     \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l' h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl) n : Nat his : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i \u22a2 get l hd :: map (get l) tl <+ drop n l ** have := IH h.of_cons (hd+1) _ rfl (pairwise_cons.mp h).1 ** case cons.refl \u03b1 : Type u_1 l : List \u03b1 hd : Fin (length l) tl : List (Fin (length l)) IH :   Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192     \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l' h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl) n : Nat his : \u2200 (i : Fin (length l)), i \u2208 hd :: tl \u2192 n \u2264 \u2191i this : map (get l) tl <+ drop (\u2191hd + 1) l \u22a2 get l hd :: map (get l) tl <+ drop n l ** specialize his hd (.head _) ** case cons.refl \u03b1 : Type u_1 l : List \u03b1 hd : Fin (length l) tl : List (Fin (length l)) IH :   Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192     \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l' h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl) n : Nat this : map (get l) tl <+ drop (\u2191hd + 1) l his : n \u2264 \u2191hd \u22a2 get l hd :: map (get l) tl <+ drop n l ** have := get_cons_drop .. \u25b8 this.cons\u2082 (get l hd) ** case cons.refl \u03b1 : Type u_1 l : List \u03b1 hd : Fin (length l) tl : List (Fin (length l)) IH :   Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192     \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l' h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl) n : Nat this\u271d : map (get l) tl <+ drop (\u2191hd + 1) l his : n \u2264 \u2191hd this : get l hd :: map (get l) tl <+ drop (\u2191hd) l \u22a2 get l hd :: map (get l) tl <+ drop n l ** have := Sublist.append (nil_sublist (take hd l |>.drop n)) this ** case cons.refl \u03b1 : Type u_1 l : List \u03b1 hd : Fin (length l) tl : List (Fin (length l)) IH :   Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192     \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l' h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl) n : Nat this\u271d\u00b9 : map (get l) tl <+ drop (\u2191hd + 1) l his : n \u2264 \u2191hd this\u271d : get l hd :: map (get l) tl <+ drop (\u2191hd) l this : [] ++ get l hd :: map (get l) tl <+ drop n (take (\u2191hd) l) ++ drop (\u2191hd) l \u22a2 get l hd :: map (get l) tl <+ drop n l ** rwa [nil_append, \u2190 (drop_append_of_le_length ?_), take_append_drop] at this ** \u03b1 : Type u_1 l : List \u03b1 hd : Fin (length l) tl : List (Fin (length l)) IH :   Pairwise (fun x x_1 => \u2191x < \u2191x_1) tl \u2192     \u2200 (n : Nat) (l' : List \u03b1), l' = drop n l \u2192 (\u2200 (i : Fin (length l)), i \u2208 tl \u2192 n \u2264 \u2191i) \u2192 map (get l) tl <+ l' h : Pairwise (fun x x_1 => \u2191x < \u2191x_1) (hd :: tl) n : Nat this\u271d\u00b9 : map (get l) tl <+ drop (\u2191hd + 1) l his : n \u2264 \u2191hd this\u271d : get l hd :: map (get l) tl <+ drop (\u2191hd) l this : get l hd :: map (get l) tl <+ drop n (take (\u2191hd) l) ++ drop (\u2191hd) l \u22a2 n \u2264 length (take (\u2191hd) l) ** simp [Nat.min_eq_left (Nat.le_of_lt hd.isLt), his] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2081 ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i) ** simp only [\u2190 set_integral_congr_set_ae (Box.coe_ae_eq_Icc _)] ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =     \u2211 x : Fin (n + 1),       ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -         \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x) ** have A := (Hi.mono_set Box.coe_subset_Icc).hasBoxIntegral \u22a5 rfl ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =     \u2211 x : Fin (n + 1),       ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -         \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x) ** have B :=\n  hasIntegral_GP_divergence_of_forall_hasDerivWithinAt I f f' (s \u2229 Box.Icc I)\n    (hs.mono (inter_subset_left _ _)) (fun x hx => Hc _ hx.2) fun x hx =>\n    Hd _ \u27e8hx.1, fun h => hx.2 \u27e8h, hx.1\u27e9\u27e9 ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) B :   HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume     (\u2211 i : Fin (n + 1),       (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)           BoxAdditiveMap.volume -         BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)           BoxAdditiveMap.volume)) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =     \u2211 x : Fin (n + 1),       ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -         \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x) ** rw [continuousOn_pi] at Hc ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) B :   HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume     (\u2211 i : Fin (n + 1),       (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)           BoxAdditiveMap.volume -         BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)           BoxAdditiveMap.volume)) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =     \u2211 x : Fin (n + 1),       ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -         \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x) ** refine' (A.unique B).trans (sum_congr rfl fun i _ => _) ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) B :   HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume     (\u2211 i : Fin (n + 1),       (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)           BoxAdditiveMap.volume -         BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)           BoxAdditiveMap.volume)) i : Fin (n + 1) x\u271d : i \u2208 Finset.univ \u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)         BoxAdditiveMap.volume -       BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)         BoxAdditiveMap.volume =     (\u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -       \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i ** refine' congr_arg\u2082 Sub.sub _ _ ** case refine'_1 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) B :   HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume     (\u2211 i : Fin (n + 1),       (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)           BoxAdditiveMap.volume -         BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)           BoxAdditiveMap.volume)) i : Fin (n + 1) x\u271d : i \u2208 Finset.univ \u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)       BoxAdditiveMap.volume =     \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i ** have := Box.continuousOn_face_Icc (Hc i) (Set.right_mem_Icc.2 (I.lower_le_upper i)) ** case refine'_1 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) B :   HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume     (\u2211 i : Fin (n + 1),       (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)           BoxAdditiveMap.volume -         BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)           BoxAdditiveMap.volume)) i : Fin (n + 1) x\u271d : i \u2208 Finset.univ this : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) (\u2191Box.Icc (Box.face I i)) \u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)       BoxAdditiveMap.volume =     \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i ** have := (this.integrableOn_compact (\u03bc := volume) (Box.isCompact_Icc _)).mono_set\n  Box.coe_subset_Icc ** case refine'_1 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) B :   HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume     (\u2211 i : Fin (n + 1),       (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)           BoxAdditiveMap.volume -         BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)           BoxAdditiveMap.volume)) i : Fin (n + 1) x\u271d : i \u2208 Finset.univ this\u271d : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) (\u2191Box.Icc (Box.face I i)) this : IntegrableOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) \u2191(Box.face I i) \u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)       BoxAdditiveMap.volume =     \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i ** exact (this.hasBoxIntegral \u22a5 rfl).integral_eq ** case refine'_2 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) B :   HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume     (\u2211 i : Fin (n + 1),       (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)           BoxAdditiveMap.volume -         BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)           BoxAdditiveMap.volume)) i : Fin (n + 1) x\u271d : i \u2208 Finset.univ \u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)       BoxAdditiveMap.volume =     \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i ** have := Box.continuousOn_face_Icc (Hc i) (Set.left_mem_Icc.2 (I.lower_le_upper i)) ** case refine'_2 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) B :   HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume     (\u2211 i : Fin (n + 1),       (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)           BoxAdditiveMap.volume -         BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)           BoxAdditiveMap.volume)) i : Fin (n + 1) x\u271d : i \u2208 Finset.univ this : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) (\u2191Box.Icc (Box.face I i)) \u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)       BoxAdditiveMap.volume =     \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i ** have := (this.integrableOn_compact (\u03bc := volume) (Box.isCompact_Icc _)).mono_set\n  Box.coe_subset_Icc ** case refine'_2 E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 I : Box (Fin (n + 1)) 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 : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I) A :   HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))     (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) B :   HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume     (\u2211 i : Fin (n + 1),       (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)           BoxAdditiveMap.volume -         BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)           BoxAdditiveMap.volume)) i : Fin (n + 1) x\u271d : i \u2208 Finset.univ this\u271d : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) (\u2191Box.Icc (Box.face I i)) this : IntegrableOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) \u2191(Box.face I i) \u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)       BoxAdditiveMap.volume =     \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i ** exact (this.hasBoxIntegral \u22a5 rfl).integral_eq ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.mkMetric_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\u221e hc : c \u2260 \u22a4 hc' : c \u2260 0 \u22a2 mkMetric (c \u2022 m) = c \u2022 mkMetric m ** simp only [mkMetric, mkMetric', mkMetric'.pre, inducedOuterMeasure, ENNReal.smul_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 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 hc' : c \u2260 0 \u22a2 \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => (c \u2022 m) (diam s)) =     c \u2022 \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => m (diam s)) ** simp_rw [smul_iSup, smul_boundedBy hc, smul_extend _ hc', Pi.smul_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.get?_eq_some ** \u03b1\u271d : Type u_1 a : \u03b1\u271d l : List \u03b1\u271d n : Nat e : get? l n = some a hn : length l \u2264 n \u22a2 False ** cases get?_len_le hn \u25b8 e ** \u03b1\u271d : Type u_1 a : \u03b1\u271d l : List \u03b1\u271d n : Nat e : get? l n = some a this : n < length l \u22a2 get l { val := n, isLt := this } = a ** rwa [get?_eq_get this, Option.some.injEq] at e ** Qed",
        "informal": ""
    },
    {
        "formal": "VitaliFamily.aemeasurable_limRatio ** \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 h\u03c1 : \u03c1 \u226a \u03bc \u22a2 AEMeasurable (limRatio v \u03c1) ** apply ENNReal.aemeasurable_of_exist_almost_disjoint_supersets _ _ fun p q hpq => ?_ ** \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 h\u03c1 : \u03c1 \u226a \u03bc p q : \u211d\u22650 hpq : p < q \u22a2 \u2203 u v_1,     MeasurableSet u \u2227       MeasurableSet v_1 \u2227 {x | limRatio v \u03c1 x < \u2191p} \u2286 u \u2227 {x | \u2191q < limRatio v \u03c1 x} \u2286 v_1 \u2227 \u2191\u2191\u03bc (u \u2229 v_1) = 0 ** exact v.exists_measurable_supersets_limRatio h\u03c1 hpq ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.diag_singleton ** \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 \u22a2 diag {a} = {(a, a)} ** rw [\u2190 product_sdiff_offDiag, offDiag_singleton, sdiff_empty, singleton_product_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.piecewise_insert ** \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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s) inst\u271d\u00b9 : DecidableEq \u03b1 j : \u03b1 inst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s) \u22a2 piecewise (insert j s) f g = update (piecewise s f g) j (f j) ** classical simp only [\u2190 piecewise_coe, coe_insert, \u2190 Set.piecewise_insert] ** \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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s) inst\u271d\u00b9 : DecidableEq \u03b1 j : \u03b1 inst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s) \u22a2 Set.piecewise (\u2191(insert j s)) f g = Set.piecewise (insert j \u2191s) f g ** ext ** case h \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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s) inst\u271d\u00b9 : DecidableEq \u03b1 j : \u03b1 inst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s) x\u271d : \u03b1 \u22a2 Set.piecewise (\u2191(insert j s)) f g x\u271d = Set.piecewise (insert j \u2191s) f g x\u271d ** congr ** case h.e_s \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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s) inst\u271d\u00b9 : DecidableEq \u03b1 j : \u03b1 inst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s) x\u271d : \u03b1 \u22a2 \u2191(insert j s) = insert j \u2191s ** simp ** \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\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s) inst\u271d\u00b9 : DecidableEq \u03b1 j : \u03b1 inst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s) \u22a2 piecewise (insert j s) f g = update (piecewise s f g) j (f j) ** simp only [\u2190 piecewise_coe, coe_insert, \u2190 Set.piecewise_insert] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E ht : IsSeparable t f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t \u22a2 f =\u1d50[\u03bc] 0 ** rcases ht with \u27e8d, d_count, hd\u27e9 ** case intro.intro \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d \u22a2 f =\u1d50[\u03bc] 0 ** haveI : Encodable d := d_count.toEncodable ** case intro.intro \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d \u22a2 f =\u1d50[\u03bc] 0 ** have : \u2200 x : d, \u2203 g : E \u2192L[\ud835\udd5c] \ud835\udd5c, \u2016g\u2016 \u2264 1 \u2227 g x = \u2016(x : E)\u2016 :=\n  fun x => exists_dual_vector'' \ud835\udd5c (x : E) ** case intro.intro \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this\u271d : Encodable \u2191d this : \u2200 (x : \u2191d), \u2203 g, \u2016g\u2016 \u2264 1 \u2227 \u2191g \u2191x = \u2191\u2016\u2191x\u2016 \u22a2 f =\u1d50[\u03bc] 0 ** choose s hs using this ** case intro.intro \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 \u22a2 f =\u1d50[\u03bc] 0 ** have A : \u2200 a : E, a \u2208 t \u2192 (\u2200 x, \u27eaa, s x\u27eb = (0 : \ud835\udd5c)) \u2192 a = 0 := by\n  intro a hat ha\n  contrapose! ha\n  have a_pos : 0 < \u2016a\u2016 := by simp only [ha, norm_pos_iff, Ne.def, not_false_iff]\n  have a_mem : a \u2208 closure d := hd hat\n  obtain \u27e8x, hx\u27e9 : \u2203 x : d, dist a x < \u2016a\u2016 / 2 := by\n    rcases Metric.mem_closure_iff.1 a_mem (\u2016a\u2016 / 2) (half_pos a_pos) with \u27e8x, h'x, hx\u27e9\n    exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9\n  use x\n  have I : \u2016a\u2016 / 2 < \u2016(x : E)\u2016 := by\n    have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := norm_le_insert' _ _\n    have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [dist_eq_norm] at hx\n    linarith\n  intro h\n  apply lt_irrefl \u2016s x x\u2016\n  calc\n    \u2016s x x\u2016 = \u2016s x (x - a)\u2016 := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]\n    _ \u2264 1 * \u2016(x : E) - a\u2016 := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)\n    _ < \u2016a\u2016 / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx\n    _ < \u2016(x : E)\u2016 := I\n    _ = \u2016s x x\u2016 := by rw [(hs x).2, IsROrC.norm_coe_norm] ** case intro.intro \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 A : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0 \u22a2 f =\u1d50[\u03bc] 0 ** have hfs : \u2200 y : d, \u2200\u1d50 x \u2202\u03bc, \u27eaf x, s y\u27eb = (0 : \ud835\udd5c) := fun y => hf (s y) ** case intro.intro \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 A : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0 hfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0 \u22a2 f =\u1d50[\u03bc] 0 ** have hf' : \u2200\u1d50 x \u2202\u03bc, \u2200 y : d, \u27eaf x, s y\u27eb = (0 : \ud835\udd5c) := by rwa [ae_all_iff] ** case intro.intro \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 A : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0 hfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0 hf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0 \u22a2 f =\u1d50[\u03bc] 0 ** filter_upwards [hf', h't] with x hx h'x ** case h \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 A : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0 hfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0 hf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0 x : \u03b1 hx : \u2200 (y : \u2191d), \u2191(s y) (f x) = 0 h'x : f x \u2208 t \u22a2 f x = OfNat.ofNat 0 x ** exact A (f x) h'x hx ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 \u22a2 \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0 ** intro a hat ha ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : \u2200 (x : \u2191d), \u2191(s x) a = 0 \u22a2 a = 0 ** contrapose! ha ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 \u22a2 \u2203 x, \u2191(s x) a \u2260 0 ** have a_pos : 0 < \u2016a\u2016 := by simp only [ha, norm_pos_iff, Ne.def, not_false_iff] ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 \u22a2 \u2203 x, \u2191(s x) a \u2260 0 ** have a_mem : a \u2208 closure d := hd hat ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d \u22a2 \u2203 x, \u2191(s x) a \u2260 0 ** obtain \u27e8x, hx\u27e9 : \u2203 x : d, dist a x < \u2016a\u2016 / 2 := by\n  rcases Metric.mem_closure_iff.1 a_mem (\u2016a\u2016 / 2) (half_pos a_pos) with \u27e8x, h'x, hx\u27e9\n  exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9 ** case intro \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 \u22a2 \u2203 x, \u2191(s x) a \u2260 0 ** use x ** case h \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 \u22a2 \u2191(s x) a \u2260 0 ** have I : \u2016a\u2016 / 2 < \u2016(x : E)\u2016 := by\n  have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := norm_le_insert' _ _\n  have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [dist_eq_norm] at hx\n  linarith ** case h \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 I : \u2016a\u2016 / 2 < \u2016\u2191x\u2016 \u22a2 \u2191(s x) a \u2260 0 ** intro h ** case h \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 I : \u2016a\u2016 / 2 < \u2016\u2191x\u2016 h : \u2191(s x) a = 0 \u22a2 False ** apply lt_irrefl \u2016s x x\u2016 ** case h \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 I : \u2016a\u2016 / 2 < \u2016\u2191x\u2016 h : \u2191(s x) a = 0 \u22a2 \u2016\u2191(s x) \u2191x\u2016 < \u2016\u2191(s x) \u2191x\u2016 ** calc\n  \u2016s x x\u2016 = \u2016s x (x - a)\u2016 := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]\n  _ \u2264 1 * \u2016(x : E) - a\u2016 := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)\n  _ < \u2016a\u2016 / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx\n  _ < \u2016(x : E)\u2016 := I\n  _ = \u2016s x x\u2016 := by rw [(hs x).2, IsROrC.norm_coe_norm] ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 \u22a2 0 < \u2016a\u2016 ** simp only [ha, norm_pos_iff, Ne.def, not_false_iff] ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d \u22a2 \u2203 x, dist a \u2191x < \u2016a\u2016 / 2 ** rcases Metric.mem_closure_iff.1 a_mem (\u2016a\u2016 / 2) (half_pos a_pos) with \u27e8x, h'x, hx\u27e9 ** case intro.intro \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : E h'x : x \u2208 d hx : dist a x < \u2016a\u2016 / 2 \u22a2 \u2203 x, dist a \u2191x < \u2016a\u2016 / 2 ** exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9 ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 \u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016 ** have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := norm_le_insert' _ _ ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this\u271d : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 this : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016 \u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016 ** have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [dist_eq_norm] at hx ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this\u271d\u00b9 : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 this\u271d : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016 this : \u2016a - \u2191x\u2016 < \u2016a\u2016 / 2 \u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016 ** linarith ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this\u271d : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 this : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016 \u22a2 \u2016a - \u2191x\u2016 < \u2016a\u2016 / 2 ** rwa [dist_eq_norm] at hx ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 I : \u2016a\u2016 / 2 < \u2016\u2191x\u2016 h : \u2191(s x) a = 0 \u22a2 \u2016\u2191(s x) \u2191x\u2016 = \u2016\u2191(s x) (\u2191x - a)\u2016 ** simp only [h, sub_zero, ContinuousLinearMap.map_sub] ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 I : \u2016a\u2016 / 2 < \u2016\u2191x\u2016 h : \u2191(s x) a = 0 \u22a2 1 * \u2016\u2191x - a\u2016 < \u2016a\u2016 / 2 ** rw [one_mul] ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 I : \u2016a\u2016 / 2 < \u2016\u2191x\u2016 h : \u2191(s x) a = 0 \u22a2 \u2016\u2191x - a\u2016 < \u2016a\u2016 / 2 ** rwa [dist_eq_norm'] at hx ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 a : E hat : a \u2208 t ha : a \u2260 0 a_pos : 0 < \u2016a\u2016 a_mem : a \u2208 closure d x : \u2191d hx : dist a \u2191x < \u2016a\u2016 / 2 I : \u2016a\u2016 / 2 < \u2016\u2191x\u2016 h : \u2191(s x) a = 0 \u22a2 \u2016\u2191x\u2016 = \u2016\u2191(s x) \u2191x\u2016 ** rw [(hs x).2, IsROrC.norm_coe_norm] ** \u03b1 : Type u_1 E : Type u_2 \ud835\udd5c : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E t : Set E f : \u03b1 \u2192 E hf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0 h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t d : Set E d_count : Set.Countable d hd : t \u2286 closure d this : Encodable \u2191d s : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c hs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016 A : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0 hfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0 ** rwa [ae_all_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.restrict_biInf ** \u03b1 : Type u_1 \u03b9 : Type u_2 I : Set \u03b9 hI : Set.Nonempty I s : Set \u03b1 m : \u03b9 \u2192 OuterMeasure \u03b1 \u22a2 \u2191(restrict s) (\u2a05 i \u2208 I, m i) = \u2a05 i \u2208 I, \u2191(restrict s) (m i) ** haveI := hI.to_subtype ** \u03b1 : Type u_1 \u03b9 : Type u_2 I : Set \u03b9 hI : Set.Nonempty I s : Set \u03b1 m : \u03b9 \u2192 OuterMeasure \u03b1 this : Nonempty \u2191I \u22a2 \u2191(restrict s) (\u2a05 i \u2208 I, m i) = \u2a05 i \u2208 I, \u2191(restrict s) (m i) ** rw [\u2190 iInf_subtype'', \u2190 iInf_subtype''] ** \u03b1 : Type u_1 \u03b9 : Type u_2 I : Set \u03b9 hI : Set.Nonempty I s : Set \u03b1 m : \u03b9 \u2192 OuterMeasure \u03b1 this : Nonempty \u2191I \u22a2 \u2191(restrict s) (\u2a05 i, m \u2191i) = \u2a05 i, \u2191(restrict s) (m \u2191i) ** exact restrict_iInf _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.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 \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : MonoidHomClass F \u03b2 \u03b3 s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b) g : F \u22a2 \u2191g (noncommProd s f comm) =     noncommProd s (fun i => \u2191g (f i))       (_ : \u2200 (x : \u03b1), x \u2208 \u2191s \u2192 \u2200 (y : \u03b1), y \u2208 \u2191s \u2192 x \u2260 y \u2192 Commute (\u2191g (f x)) (\u2191g (f y))) ** simp [noncommProd, Multiset.noncommProd_map] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hausdorffMeasure_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[1] = volume ** rw [\u2190 (volume_preserving_funUnique Unit \u211d).map_eq,\n  \u2190 (hausdorffMeasure_measurePreserving_funUnique Unit \u211d 1).map_eq,\n  \u2190 hausdorffMeasure_pi_real, Fintype.card_unit, Nat.cast_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.cast_neg_one ** R : Type u_1 inst\u271d : Ring R n : \u2115 \u22a2 \u2191(-1) = \u2191n - 1 ** cases' n with n ** case zero R : Type u_1 inst\u271d : Ring R \u22a2 \u2191(-1) = \u2191Nat.zero - 1 ** dsimp [ZMod, ZMod.cast] ** case zero R : Type u_1 inst\u271d : Ring R \u22a2 \u2191(-1) = \u21910 - 1 ** simp ** case succ R : Type u_1 inst\u271d : Ring R n : \u2115 \u22a2 \u2191(-1) = \u2191(Nat.succ n) - 1 ** rw [\u2190 nat_cast_val, val_neg_one, Nat.cast_succ, add_sub_cancel] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.inf_gt_condCdfRat ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 t : \u211a \u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t ** by_cases ha : a \u2208 condCdfSet \u03c1 ** 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 t : \u211a ha : a \u2208 condCdfSet \u03c1 \u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t ** simp_rw [condCdfRat_of_mem \u03c1 a ha] ** 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 t : \u211a ha : a \u2208 condCdfSet \u03c1 \u22a2 \u2a05 r, ENNReal.toReal (preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a) ** have ha' := hasCondCdf_of_mem_condCdfSet ha ** 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 t : \u211a ha : a \u2208 condCdfSet \u03c1 ha' : HasCondCdf \u03c1 a \u22a2 \u2a05 r, ENNReal.toReal (preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a) ** rw [\u2190 ENNReal.toReal_iInf] ** 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 t : \u211a ha : a \u2208 condCdfSet \u03c1 ha' : HasCondCdf \u03c1 a \u22a2 ENNReal.toReal (\u2a05 r, preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a) ** suffices \u2a05 i : \u21a5(Ioi t), preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a by rw [this] ** 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 t : \u211a ha : a \u2208 condCdfSet \u03c1 ha' : HasCondCdf \u03c1 a \u22a2 \u2a05 i, preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a ** rw [\u2190 ha'.iInf_rat_gt_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 t : \u211a ha : a \u2208 condCdfSet \u03c1 ha' : HasCondCdf \u03c1 a this : \u2a05 i, preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a \u22a2 ENNReal.toReal (\u2a05 r, preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a) ** rw [this] ** 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 t : \u211a ha : a \u2208 condCdfSet \u03c1 ha' : HasCondCdf \u03c1 a \u22a2 \u2200 (i : \u2191(Ioi t)), preCdf \u03c1 (\u2191i) a \u2260 \u22a4 ** exact fun r => ((ha'.le_one r).trans_lt ENNReal.one_lt_top).ne ** 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 \u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t ** simp_rw [condCdfRat_of_not_mem \u03c1 a ha] ** 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 \u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = if t < 0 then 0 else 1 ** have h_bdd : BddBelow (range fun r : \u21a5(Ioi t) => ite ((r : \u211a) < 0) (0 : \u211d) 1) := by\n  refine' \u27e80, fun x hx => _\u27e9\n  obtain \u27e8y, rfl\u27e9 := mem_range.mpr hx\n  dsimp only\n  split_ifs\n  exacts [le_rfl, zero_le_one] ** 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) \u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = if t < 0 then 0 else 1 ** split_ifs with h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 \u22a2 BddBelow (range fun r => if \u2191r < 0 then 0 else 1) ** refine' \u27e80, fun x hx => _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 x : \u211d hx : x \u2208 range fun r => if \u2191r < 0 then 0 else 1 \u22a2 0 \u2264 x ** obtain \u27e8y, rfl\u27e9 := mem_range.mpr hx ** 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 y : \u2191(Ioi t) hx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1 \u22a2 0 \u2264 (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y ** dsimp only ** 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 y : \u2191(Ioi t) hx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1 \u22a2 0 \u2264 if \u2191y < 0 then 0 else 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 y : \u2191(Ioi t) hx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1 h\u271d : \u2191y < 0 \u22a2 0 \u2264 0  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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 y : \u2191(Ioi t) hx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1 h\u271d : \u00ac\u2191y < 0 \u22a2 0 \u2264 1 ** exacts [le_rfl, 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 \u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = 0 ** refine' le_antisymm _ (le_ciInf fun x => _) ** case pos.refine'_1.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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 q : \u211a htq : t < q hq_neg : q < 0 \u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) \u2264 0 ** refine' (ciInf_le h_bdd \u27e8q, htq\u27e9).trans _ ** case pos.refine'_1.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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 q : \u211a htq : t < q hq_neg : q < 0 \u22a2 (if \u2191{ val := q, property := htq } < 0 then 0 else 1) \u2264 0 ** rw [if_pos] ** case pos.refine'_1.intro.intro.hc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 q : \u211a htq : t < q hq_neg : q < 0 \u22a2 \u2191{ val := q, property := htq } < 0 ** rwa [Subtype.coe_mk] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 \u22a2 \u2203 q, t < q \u2227 q < 0 ** refine' \u27e8t / 2, _, _\u27e9 ** 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 \u22a2 t < t / 2 ** linarith ** 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 \u22a2 t / 2 < 0 ** linarith ** case pos.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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 x : \u2191(Ioi t) \u22a2 0 \u2264 if \u2191x < 0 then 0 else 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 x : \u2191(Ioi t) h\u271d : \u2191x < 0 \u22a2 0 \u2264 0  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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : t < 0 x : \u2191(Ioi t) h\u271d : \u00ac\u2191x < 0 \u22a2 0 \u2264 1 ** exacts [le_rfl, zero_le_one] ** 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : \u00act < 0 \u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) = 1 ** refine' le_antisymm _ _ ** case neg.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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : \u00act < 0 \u22a2 (\u2a05 r, if \u2191r < 0 then 0 else 1) \u2264 1 ** refine' (ciInf_le h_bdd \u27e8t + 1, lt_add_one t\u27e9).trans _ ** case neg.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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : \u00act < 0 \u22a2 (if \u2191{ val := t + 1, property := (_ : t < t + 1) } < 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : \u00act < 0 h\u271d : \u2191{ val := t + 1, property := (_ : t < t + 1) } < 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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : \u00act < 0 h\u271d : \u00ac\u2191{ val := t + 1, property := (_ : t < t + 1) } < 0 \u22a2 1 \u2264 1 ** exacts [zero_le_one, le_rfl] ** case neg.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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : \u00act < 0 \u22a2 1 \u2264 \u2a05 r, if \u2191r < 0 then 0 else 1 ** refine' le_ciInf fun x => _ ** case neg.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 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : \u00act < 0 x : \u2191(Ioi t) \u22a2 1 \u2264 if \u2191x < 0 then 0 else 1 ** rw [if_neg] ** case neg.refine'_2.hnc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : \u00act < 0 x : \u2191(Ioi t) \u22a2 \u00ac\u2191x < 0 ** rw [not_lt] at h \u22a2 ** case neg.refine'_2.hnc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 t : \u211a ha : \u00aca \u2208 condCdfSet \u03c1 h_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1) h : 0 \u2264 t x : \u2191(Ioi t) \u22a2 0 \u2264 \u2191x ** exact h.trans (mem_Ioi.mp x.prop).le ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.sdiff_erase ** \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 h : a \u2208 s \u22a2 s \\ erase t a = insert a (s \\ t) ** rw [\u2190 sdiff_singleton_eq_erase, sdiff_sdiff_eq_sdiff_union (singleton_subset_iff.2 h), insert_eq,\n  union_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "volume_regionBetween_eq_integral' ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc f_int : IntegrableOn f s g_int : IntegrableOn g s hs : MeasurableSet s hfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = ENNReal.ofReal (\u222b (y : \u03b1) in s, (g - f) y \u2202\u03bc) ** have h : g - f =\u1d50[\u03bc.restrict s] fun x => Real.toNNReal (g x - f x) :=\n  hfg.mono fun x hx => (Real.coe_toNNReal _ <| sub_nonneg.2 hx).symm ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc f_int : IntegrableOn f s g_int : IntegrableOn g s hs : MeasurableSet s hfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g h : g - f =\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2191(Real.toNNReal (g x - f x)) \u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = ENNReal.ofReal (\u222b (y : \u03b1) in s, (g - f) y \u2202\u03bc) ** rw [volume_regionBetween_eq_lintegral f_int.aemeasurable g_int.aemeasurable hs,\n  integral_congr_ae h, lintegral_congr_ae,\n  lintegral_coe_eq_integral _ ((integrable_congr h).mp (g_int.sub f_int))] ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 inst\u271d : SigmaFinite \u03bc f_int : IntegrableOn f s g_int : IntegrableOn g s hs : MeasurableSet s hfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g h : g - f =\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2191(Real.toNNReal (g x - f x)) \u22a2 (fun y => ENNReal.ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun a => \u2191(Real.toNNReal (g a - f a)) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_nonneg_of_ae ** \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 : 0 \u2264\u1d50[\u03bc] f \u22a2 0 \u2264 \u222b (a : \u03b1), f a \u2202\u03bc ** have A : CompleteSpace \u211d := by infer_instance ** \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 : 0 \u2264\u1d50[\u03bc] f A : CompleteSpace \u211d \u22a2 0 \u2264 \u222b (a : \u03b1), f a \u2202\u03bc ** simp only [integral_def, A, L1.integral_def, dite_true, ge_iff_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 hf : 0 \u2264\u1d50[\u03bc] f A : CompleteSpace \u211d \u22a2 0 \u2264 if hf : Integrable fun a => f a then \u2191L1.integralCLM (Integrable.toL1 (fun a => f a) hf) else 0 ** exact setToFun_nonneg (dominatedFinMeasAdditive_weightedSMul \u03bc)\n  (fun s _ _ => weightedSMul_nonneg s) hf ** \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 : 0 \u2264\u1d50[\u03bc] f \u22a2 CompleteSpace \u211d ** infer_instance ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.sdiff_insert ** \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 \u22a2 s \\ insert x t = erase (s \\ t) x ** simp_rw [\u2190 sdiff_singleton_eq_erase, insert_eq, sdiff_sdiff_left', sdiff_union_distrib,\n  inter_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.isCaratheodory_iUnion_nat ** \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) \u22a2 IsCaratheodory m (\u22c3 i, s i) ** apply (isCaratheodory_iff_le' m).mpr ** \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) \u22a2 \u2200 (t : Set \u03b1), \u2191m (t \u2229 \u22c3 i, s i) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t ** intro t ** \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) t : Set \u03b1 hp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) \u22a2 \u2191m (t \u2229 \u22c3 i, s i) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t ** refine' le_trans (add_le_add_right hp _) _ ** \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) t : Set \u03b1 hp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) \u22a2 (\u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t ** rw [ENNReal.iSup_add] ** \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) t : Set \u03b1 hp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) \u22a2 \u2a06 b, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < b), s i) + \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m t ** refine'\n  iSup_le fun n =>\n    le_trans (add_le_add_left _ _) (ge_of_eq (isCaratheodory_iUnion_lt m (fun i _ => h i) _)) ** \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) t : Set \u03b1 hp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) n : \u2115 \u22a2 \u2191m (t \\ \u22c3 i, s i) \u2264 \u2191m (t \\ \u22c3 i, \u22c3 (_ : i < n), s i) ** refine' m.mono (diff_subset_diff_right _) ** \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) t : Set \u03b1 hp : \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) n : \u2115 \u22a2 \u22c3 i, \u22c3 (_ : i < n), s i \u2286 \u22c3 i, s i ** exact iUnion\u2082_subset fun i _ => subset_iUnion _ i ** \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) t : Set \u03b1 \u22a2 \u2191m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) ** convert m.iUnion fun i => t \u2229 s i using 1 ** case h.e'_3 \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) t : Set \u03b1 \u22a2 \u2191m (t \u2229 \u22c3 i, s i) = \u2191m (\u22c3 i, t \u2229 s i) ** simp [inter_iUnion] ** case h.e'_4 \u03b1 : Type u m : OuterMeasure \u03b1 s\u271d s\u2081 s\u2082 : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), IsCaratheodory m (s i) hd : Pairwise (Disjoint on s) t : Set \u03b1 \u22a2 \u2a06 n, \u2191m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) = \u2211' (i : \u2115), \u2191m (t \u2229 s i) ** simp [ENNReal.tsum_eq_iSup_nat, isCaratheodory_sum m h hd] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_neg_uIcc ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 -[[a, b]] = [[-a, -b]] ** simp only [\u2190 Icc_min_max, preimage_neg_Icc, min_neg_neg, max_neg_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.isometry_lpMeasSubgroupToLpTrim ** \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 hp : Fact (1 \u2264 p) hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 dist (lpMeasSubgroupToLpTrim F p \u03bc hm f) (lpMeasSubgroupToLpTrim F p \u03bc hm g) = dist f g ** rw [dist_eq_norm, \u2190 lpMeasSubgroupToLpTrim_sub, lpMeasSubgroupToLpTrim_norm_map,\n  dist_eq_norm] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul ** \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 : \u03a9 \u2192 \u03b2 g : \u03a9 \u2192 \u03b2' m\u03b2 : MeasurableSpace \u03b2 m\u03b2' : MeasurableSpace \u03b2' \u22a2 IndepFun f g \u2194     \u2200 (s : Set \u03b2) (t : Set \u03b2'),       MeasurableSet s \u2192 MeasurableSet t \u2192 \u2191\u2191\u03bc (f \u207b\u00b9' s \u2229 g \u207b\u00b9' t) = \u2191\u2191\u03bc (f \u207b\u00b9' s) * \u2191\u2191\u03bc (g \u207b\u00b9' t) ** simp only [IndepFun, kernel.indepFun_iff_measure_inter_preimage_eq_mul, ae_dirac_eq,\n  Filter.eventually_pure, kernel.const_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.sum_add_sum_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 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 s : Set \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 \u22a2 ((sum fun i => \u03bc \u2191i) + sum fun i => \u03bc \u2191i) = sum \u03bc ** ext1 t ht ** 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\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 s : Set \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 t : Set \u03b1 ht : MeasurableSet t \u22a2 \u2191\u2191((sum fun i => \u03bc \u2191i) + sum fun i => \u03bc \u2191i) t = \u2191\u2191(sum \u03bc) t ** simp only [add_apply, sum_apply _ ht] ** 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\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 s : Set \u03b9 \u03bc : \u03b9 \u2192 Measure \u03b1 t : Set \u03b1 ht : MeasurableSet t \u22a2 \u2211' (i : \u2191s), \u2191\u2191(\u03bc \u2191i) t + \u2211' (i : \u2191s\u1d9c), \u2191\u2191(\u03bc \u2191i) t = \u2211' (i : \u03b9), \u2191\u2191(\u03bc i) t ** exact tsum_add_tsum_compl (f := fun i => \u03bc i t) ENNReal.summable ENNReal.summable ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.NullMeasurableSet.toMeasurable_ae_eq ** \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 h : NullMeasurableSet s \u22a2 toMeasurable \u03bc s =\u1d50[\u03bc] s ** rw [toMeasurable_def, dif_pos] ** \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 h : NullMeasurableSet s \u22a2 Exists.choose ?hc =\u1d50[\u03bc] s  case hc \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 h : NullMeasurableSet s \u22a2 \u2203 t, t \u2287 s \u2227 MeasurableSet t \u2227 t =\u1d50[\u03bc] s ** exact (exists_measurable_superset_ae_eq h).choose_spec.2.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Martingale.stoppedValue_ae_eq_condexp_of_le_const_of_countable_range ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_countable_range : Set.Countable (Set.range \u03c4) inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) \u22a2 stoppedValue f \u03c4 =\u1d50[\u03bc] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4] ** have : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i} := by\n  ext1 x\n  simp only [Set.mem_univ, Set.mem_range, true_and_iff, Set.iUnion_exists, Set.iUnion_iUnion_eq',\n    Set.mem_iUnion, Set.mem_setOf_eq, exists_apply_eq_apply'] ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_countable_range : Set.Countable (Set.range \u03c4) inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) this : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i} \u22a2 stoppedValue f \u03c4 =\u1d50[\u03bc] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4] ** nth_rw 1 [\u2190 @Measure.restrict_univ \u03a9 _ \u03bc] ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_countable_range : Set.Countable (Set.range \u03c4) inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) this : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i} \u22a2 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc Set.univ] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4] ** rw [this, ae_eq_restrict_biUnion_iff _ h_countable_range] ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_countable_range : Set.Countable (Set.range \u03c4) inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) this : Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i} \u22a2 \u2200 (i : \u03b9),     i \u2208 Set.range \u03c4 \u2192 stoppedValue f \u03c4 =\u1d50[Measure.restrict \u03bc {x | \u03c4 x = i}] \u03bc[f n|IsStoppingTime.measurableSpace h\u03c4] ** exact fun i _ => stoppedValue_ae_eq_restrict_eq h _ h\u03c4_le i ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_countable_range : Set.Countable (Set.range \u03c4) inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) \u22a2 Set.univ = \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i} ** ext1 x ** case h \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u2075 : LinearOrder \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : OrderTopology \u03b9 inst\u271d\u00b2 : FirstCountableTopology \u03b9 \u2131 : Filtration \u03b9 m inst\u271d\u00b9 : SigmaFiniteFiltration \u03bc \u2131 \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\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_countable_range : Set.Countable (Set.range \u03c4) inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c4 \u2264 m)) x : \u03a9 \u22a2 x \u2208 Set.univ \u2194 x \u2208 \u22c3 i \u2208 Set.range \u03c4, {x | \u03c4 x = i} ** simp only [Set.mem_univ, Set.mem_range, true_and_iff, Set.iUnion_exists, Set.iUnion_iUnion_eq',\n  Set.mem_iUnion, Set.mem_setOf_eq, exists_apply_eq_apply'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpL1_mono ** \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 hm : m \u2264 m0 inst\u271d\u2074 : SigmaFinite (Measure.trim \u03bc hm) f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_8 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 \u2191\u2191(condexpL1 hm \u03bc f) \u2264\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc g) ** rw [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 hm : m \u2264 m0 inst\u271d\u2074 : SigmaFinite (Measure.trim \u03bc hm) f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_8 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 condexpL1 hm \u03bc f \u2264 condexpL1 hm \u03bc g ** have h_nonneg : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 \u2200 x : E, 0 \u2264 x \u2192 0 \u2264 condexpInd E hm \u03bc s x :=\n  fun s hs h\u03bcs x hx => condexpInd_nonneg hs h\u03bcs.ne x hx ** \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 hm : m \u2264 m0 inst\u271d\u2074 : SigmaFinite (Measure.trim \u03bc hm) f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_8 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 h_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : E), 0 \u2264 x \u2192 0 \u2264 \u2191(condexpInd E hm \u03bc s) x \u22a2 condexpL1 hm \u03bc f \u2264 condexpL1 hm \u03bc g ** exact setToFun_mono (dominatedFinMeasAdditive_condexpInd E hm \u03bc) h_nonneg hf hg hfg ** Qed",
        "informal": ""
    },
    {
        "formal": "WithBot.image_coe_Ioo ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ioo a b = Ioo \u2191a \u2191b ** rw [\u2190 preimage_coe_Ioo, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Subset.trans Ioo_subset_Ioi_self <| Ioi_subset_Ioi bot_le)] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.strong_law_aux6 ** \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 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** have H : \u2200 n : \u2115, (0 : \u211d) < \u230ac ^ n\u230b\u208a := by\n  intro n\n  refine' zero_lt_one.trans_le _\n  simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le 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 H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** filter_upwards [strong_law_aux4 X hint hindep hident hnonneg c_one,\n  strong_law_aux5 X hint hident hnonneg] with \u03c9 h\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 H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a h'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n \u22a2 Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** rw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 Asymptotics.isLittleO_one_iff \u211d] ** 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 H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a h'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n \u22a2 (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a) =o[atTop] fun _x => 1 ** have L : (fun n : \u2115 => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u230ac ^ n\u230b\u208a * \ud835\udd3c[X 0]) =o[atTop] fun n =>\n    (\u230ac ^ n\u230b\u208a : \u211d) := by\n  have A : Tendsto (fun n : \u2115 => \u230ac ^ n\u230b\u208a) atTop atTop :=\n    tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt c_one)\n  convert h\u03c9.sub (h'\u03c9.comp_tendsto A) using 1\n  ext1 n\n  simp only [Function.comp_apply, sub_sub_sub_cancel_left] ** 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 H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a h'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n L : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a \u22a2 (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a) =o[atTop] fun _x => 1 ** convert L.mul_isBigO (isBigO_refl (fun n : \u2115 => (\u230ac ^ n\u230b\u208a : \u211d)\u207b\u00b9) atTop) using 1 <;>\n(ext1 n; field_simp [(H n).ne']) ** \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 \u22a2 \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a ** 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 n : \u2115 \u22a2 0 < \u2191\u230ac ^ n\u230b\u208a ** 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 n : \u2115 \u22a2 1 \u2264 \u2191\u230ac ^ n\u230b\u208a ** simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le 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 H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a h'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n \u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a ** have A : Tendsto (fun n : \u2115 => \u230ac ^ n\u230b\u208a) atTop atTop :=\n  tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt 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 H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a h'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n A : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop \u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a ** convert h\u03c9.sub (h'\u03c9.comp_tendsto 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 c : \u211d c_one : 1 < c H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a h'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n A : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop \u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) = fun x =>     (\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ x\u230b\u208a * \u222b (a : \u03a9), X 0 a) -       ((fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) \u2218 fun n => \u230ac ^ n\u230b\u208a) x ** 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 c : \u211d c_one : 1 < c H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a h'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n A : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop n : \u2115 \u22a2 \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a =     (\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) -       ((fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) \u2218 fun n => \u230ac ^ n\u230b\u208a) n ** simp only [Function.comp_apply, sub_sub_sub_cancel_left] ** case h.e'_8 \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 H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a h'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n L : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a \u22a2 (fun _x => 1) = fun x => \u2191\u230ac ^ x\u230b\u208a * (\u2191\u230ac ^ x\u230b\u208a)\u207b\u00b9 ** ext1 n ** case h.e'_8.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 H : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a h'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n L : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a n : \u2115 \u22a2 1 = \u2191\u230ac ^ n\u230b\u208a * (\u2191\u230ac ^ n\u230b\u208a)\u207b\u00b9 ** field_simp [(H n).ne'] ** 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\u271d : \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 p : \u211d\u22650\u221e \u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc) ** rcases eq_or_ne p 0 with (rfl | hp) ** case inr \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 f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc p : \u211d\u22650\u221e hp : p \u2260 0 \u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc) ** rcases lt_or_le p 1 with (h'p | h'p) ** case inl \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 \u22a2 snorm (f + g) 0 \u03bc \u2264 LpAddConst 0 * (snorm f 0 \u03bc + snorm g 0 \u03bc) ** simp only [snorm_exponent_zero, add_zero, mul_zero, le_zero_iff] ** case inr.inl \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 f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc p : \u211d\u22650\u221e hp : p \u2260 0 h'p : p < 1 \u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc) ** simp only [snorm_eq_snorm' hp (h'p.trans ENNReal.one_lt_top).ne] ** case inr.inl \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 f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc p : \u211d\u22650\u221e hp : p \u2260 0 h'p : p < 1 \u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 LpAddConst p * (snorm' f (ENNReal.toReal p) \u03bc + snorm' g (ENNReal.toReal p) \u03bc) ** convert snorm'_add_le_of_le_one hf ENNReal.toReal_nonneg _ ** case h.e'_4.h.e'_5 \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 f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc p : \u211d\u22650\u221e hp : p \u2260 0 h'p : p < 1 \u22a2 LpAddConst p = 2 ^ (1 / ENNReal.toReal p - 1) ** have : p \u2208 Set.Ioo (0 : \u211d\u22650\u221e) 1 := \u27e8hp.bot_lt, h'p\u27e9 ** case h.e'_4.h.e'_5 \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 f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc p : \u211d\u22650\u221e hp : p \u2260 0 h'p : p < 1 this : p \u2208 Set.Ioo 0 1 \u22a2 LpAddConst p = 2 ^ (1 / ENNReal.toReal p - 1) ** simp only [LpAddConst, if_pos this] ** case inr.inl.convert_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 : \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 p : \u211d\u22650\u221e hp : p \u2260 0 h'p : p < 1 \u22a2 ENNReal.toReal p \u2264 1 ** simpa using ENNReal.toReal_mono ENNReal.one_ne_top h'p.le ** case inr.inr \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 f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc p : \u211d\u22650\u221e hp : p \u2260 0 h'p : 1 \u2264 p \u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc) ** simp [LpAddConst_of_one_le h'p] ** case inr.inr \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 f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc p : \u211d\u22650\u221e hp : p \u2260 0 h'p : 1 \u2264 p \u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc ** exact snorm_add_le hf hg h'p ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.condexpKernel_ae ** \u03a9 : Type u_1 F : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03a9 m m\u03a9 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : NormedAddCommGroup F f : \u03a9 \u2192 F hf_int : Integrable f \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Integrable f ** exact Integrable.condDistrib_ae\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": "Int.measurable_ceil ** \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 : OpensMeasurableSpace R x : R \u22a2 MeasurableSet (ceil \u207b\u00b9' {\u2308x\u2309}) ** simpa only [Int.preimage_ceil_singleton] using measurableSet_Ioc ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.indep_limsup_atBot_self ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s \u22a2 Indep (limsup s atBot) (limsup s atBot) ** let ns : \u03b9 \u2192 Set \u03b9 := Set.Ici ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici \u22a2 Indep (limsup s atBot) (limsup s atBot) ** have hnsp : \u2200 i, BddBelow (ns i) := fun i => bddBelow_Ici ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) \u22a2 Indep (limsup s atBot) (limsup s atBot) ** refine' indep_limsup_self h_le h_indep _ _ hnsp _ ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) \u22a2 \u2200 (t : Set \u03b9), BddBelow t \u2192 t\u1d9c \u2208 atBot ** simp only [mem_atBot_sets, ge_iff_le, Set.mem_compl_iff, BddBelow, lowerBounds, Set.Nonempty] ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) \u22a2 \u2200 (t : Set \u03b9), (\u2203 x, x \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a}) \u2192 \u2203 a, \u2200 (b : \u03b9), b \u2264 a \u2192 \u00acb \u2208 t ** rintro t \u27e8a, ha\u27e9 ** case refine'_1.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a} \u22a2 \u2203 a, \u2200 (b : \u03b9), b \u2264 a \u2192 \u00acb \u2208 t ** obtain \u27e8b, hb\u27e9 : \u2203 b, b < a := exists_lt a ** case refine'_1.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a} b : \u03b9 hb : b < a \u22a2 \u2203 a, \u2200 (b : \u03b9), b \u2264 a \u2192 \u00acb \u2208 t ** refine' \u27e8b, fun c hc hct => _\u27e9 ** case refine'_1.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a} b : \u03b9 hb : b < a c : \u03b9 hc : c \u2264 b hct : c \u2208 t \u22a2 False ** suffices : \u2200 i \u2208 t, c < i ** case refine'_1.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a} b : \u03b9 hb : b < a c : \u03b9 hc : c \u2264 b hct : c \u2208 t this : \u2200 (i : \u03b9), i \u2208 t \u2192 c < i \u22a2 False  case this \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a} b : \u03b9 hb : b < a c : \u03b9 hc : c \u2264 b hct : c \u2208 t \u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 c < i ** exact lt_irrefl c (this c hct) ** case this \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) t : Set \u03b9 a : \u03b9 ha : a \u2208 {x | \u2200 \u2983a : \u03b9\u2984, a \u2208 t \u2192 x \u2264 a} b : \u03b9 hb : b < a c : \u03b9 hc : c \u2264 b hct : c \u2208 t \u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 c < i ** exact fun i hi => hc.trans_lt (hb.trans_le (ha hi)) ** case refine'_2 \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) \u22a2 Directed (fun x x_1 => x \u2264 x_1) fun a => ns a ** exact directed_of_inf fun i j hij k hki => hij.trans hki ** case refine'_3 \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 inst\u271d\u00b2 : SemilatticeInf \u03b9 inst\u271d\u00b9 : NoMinOrder \u03b9 inst\u271d : Nonempty \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s ns : \u03b9 \u2192 Set \u03b9 := Set.Ici hnsp : \u2200 (i : \u03b9), BddBelow (ns i) \u22a2 \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a ** exact fun n => \u27e8n, le_rfl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurableSet_exists_tendsto ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) \u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)} ** rcases l.eq_or_neBot with rfl | hl ** case inr \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l \u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)} ** letI := upgradePolishSpace \u03b3 ** case inr \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 \u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)} ** rcases l.exists_antitone_basis with \u27e8u, hu\u27e9 ** case inr.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 u : \u2115 \u2192 Set \u03b9 hu : HasAntitoneBasis l u \u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)} ** simp_rw [\u2190 cauchy_map_iff_exists_tendsto] ** case inr.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 u : \u2115 \u2192 Set \u03b9 hu : HasAntitoneBasis l u \u22a2 MeasurableSet {x | Cauchy (map (fun n => f n x) l)} ** change MeasurableSet { x | _ \u2227 _ } ** case inr.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 u : \u2115 \u2192 Set \u03b9 hu : HasAntitoneBasis l u \u22a2 MeasurableSet {x | NeBot (map (fun n => f n x) l) \u2227 map (fun n => f n x) l \u00d7\u02e2 map (fun n => f n x) l \u2264 uniformity \u03b3} ** have :\n  \u2200 x,\n    (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l).HasAntitoneBasis fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) :=\n  fun x => hu.map.prod hu.map ** case inr.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 u : \u2115 \u2192 Set \u03b9 hu : HasAntitoneBasis l u this :   \u2200 (x : \u03b2),     HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>       ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) \u22a2 MeasurableSet {x | NeBot (map (fun n => f n x) l) \u2227 map (fun n => f n x) l \u00d7\u02e2 map (fun n => f n x) l \u2264 uniformity \u03b3} ** simp_rw [and_iff_right (hl.map _),\n  Filter.HasBasis.le_basis_iff (this _).toHasBasis Metric.uniformity_basis_dist_inv_nat_succ,\n  Set.setOf_forall] ** case inr.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 u : \u2115 \u2192 Set \u03b9 hu : HasAntitoneBasis l u this :   \u2200 (x : \u03b2),     HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>       ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) \u22a2 MeasurableSet     (\u22c2 i,       \u22c2 (_ : True),         {x |           \u2203 i_2, True \u2227 ((fun n => f n x) '' u i_2) \u00d7\u02e2 ((fun n => f n x) '' u i_2) \u2286 {p | dist p.1 p.2 < 1 / (\u2191i + 1)}}) ** refine' MeasurableSet.biInter Set.countable_univ fun K _ => _ ** case inr.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 u : \u2115 \u2192 Set \u03b9 hu : HasAntitoneBasis l u this :   \u2200 (x : \u03b2),     HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>       ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) K : \u2115 x\u271d : K \u2208 fun i => True \u22a2 MeasurableSet     {x | \u2203 i, True \u2227 ((fun n => f n x) '' u i) \u00d7\u02e2 ((fun n => f n x) '' u i) \u2286 {p | dist p.1 p.2 < 1 / (\u2191K + 1)}} ** simp_rw [Set.setOf_exists, true_and] ** case inr.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 u : \u2115 \u2192 Set \u03b9 hu : HasAntitoneBasis l u this :   \u2200 (x : \u03b2),     HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>       ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) K : \u2115 x\u271d : K \u2208 fun i => True \u22a2 MeasurableSet (\u22c3 i, {x | ((fun n => f n x) '' u i) \u00d7\u02e2 ((fun n => f n x) '' u i) \u2286 {p | dist p.1 p.2 < 1 / (\u2191K + 1)}}) ** refine' MeasurableSet.iUnion fun N => _ ** case inr.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 u : \u2115 \u2192 Set \u03b9 hu : HasAntitoneBasis l u this :   \u2200 (x : \u03b2),     HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>       ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) K : \u2115 x\u271d : K \u2208 fun i => True N : \u2115 \u22a2 MeasurableSet {x | ((fun n => f n x) '' u N) \u00d7\u02e2 ((fun n => f n x) '' u N) \u2286 {p | dist p.1 p.2 < 1 / (\u2191K + 1)}} ** simp_rw [prod_image_image_eq, image_subset_iff, prod_subset_iff, Set.setOf_forall] ** case inr.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 l : Filter \u03b9 inst\u271d : IsCountablyGenerated l f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) hl : NeBot l this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 u : \u2115 \u2192 Set \u03b9 hu : HasAntitoneBasis l u this :   \u2200 (x : \u03b2),     HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>       ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) K : \u2115 x\u271d : K \u2208 fun i => True N : \u2115 \u22a2 MeasurableSet     (\u22c2 i \u2208 u N, \u22c2 i_1 \u2208 u N, {x | (i, i_1) \u2208 (fun p => (f p.1 x, f p.2 x)) \u207b\u00b9' {p | dist p.1 p.2 < 1 / (\u2191K + 1)}}) ** exact\n  MeasurableSet.biInter (to_countable (u N)) fun i _ =>\n    MeasurableSet.biInter (to_countable (u N)) fun j _ =>\n      measurableSet_lt (Measurable.dist (hf i) (hf j)) measurable_const ** case inl \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2076 : T2Space \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 s : Set \u03b3 f\u271d : \u03b3 \u2192 \u03b2 inst\u271d\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b3 : PolishSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b3 h\u03b3 : OpensMeasurableSpace \u03b3 inst\u271d\u00b9 : Countable \u03b9 f : \u03b9 \u2192 \u03b2 \u2192 \u03b3 hf : \u2200 (i : \u03b9), Measurable (f i) inst\u271d : IsCountablyGenerated \u22a5 \u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd c)} ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.castNum_xor ** \u03b1 : Type u_1 \u22a2 \u2200 (m n : Num), \u2191(m ^^^ n) = \u2191m ^^^ \u2191n ** apply castNum_eq_bitwise PosNum.lxor <;> intros <;> (try cases_type* Bool) <;> rfl ** case pbb \u03b1 : Type u_1 a\u271d b\u271d : Bool m\u271d n\u271d : PosNum \u22a2 PosNum.lxor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d != b\u271d) (PosNum.lxor 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.lxor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d != b\u271d) (PosNum.lxor m\u271d n\u271d) ** cases_type* Bool ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.norm_setToL1_le_norm_setToL1SCLM ** \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 \u22a2 \u2016setToL1 hT\u2016 \u2264 \u21911 * \u2016setToL1SCLM \u03b1 E \u03bc hT\u2016 ** refine'\n  ContinuousLinearMap.op_norm_extend_le (setToL1SCLM \u03b1 E \u03bc hT) (coeToLp \u03b1 E \u211d)\n    (simpleFunc.denseRange one_ne_top) fun x => le_of_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\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 x : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2016x\u2016 = \u21911 * \u2016\u2191(coeToLp \u03b1 E \u211d) x\u2016 ** rw [NNReal.coe_one, one_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\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 x : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2016x\u2016 = \u2016\u2191(coeToLp \u03b1 E \u211d) x\u2016 ** rfl ** \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 \u22a2 \u21911 * \u2016setToL1SCLM \u03b1 E \u03bc hT\u2016 = \u2016setToL1SCLM \u03b1 E \u03bc hT\u2016 ** rw [NNReal.coe_one, one_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos ** 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 \u22a2 0 < \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 ** refine' zero_lt_one.trans_le _ ** 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 \u22a2 1 \u2264 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 ** rw [Content.outerMeasure_eq_iInf] ** 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 \u22a2 1 \u2264 \u2a05 U, \u2a05 (hU : IsOpen U), \u2a05 (_ : \u2191K\u2080 \u2286 U), Content.innerContent (haarContent K\u2080) { carrier := U, is_open' := hU } ** refine' le_iInf\u2082 fun U hU => le_iInf fun hK\u2080 => le_trans _ <| le_iSup\u2082 K\u2080.toCompacts hK\u2080 ** 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 U : Set G hU : IsOpen U hK\u2080 : \u2191K\u2080 \u2286 U \u22a2 1 \u2264 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K\u2080.toCompacts ** exact haarContent_self.ge ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.restrictNonposSeq_subset ** 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 MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n\u271d) \u2286 i ** 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 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) \u2286     i ** exact someExistsOneDivLT_subset' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.summable_measure_toReal ** \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\u03bc : IsFiniteMeasure \u03bc f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) \u22a2 Summable fun x => ENNReal.toReal (\u2191\u2191\u03bc (f x)) ** apply ENNReal.summable_toReal ** case hsum \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\u03bc : IsFiniteMeasure \u03bc f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) \u22a2 \u2211' (x : \u2115), \u2191\u2191\u03bc (f x) \u2260 \u22a4 ** rw [\u2190 MeasureTheory.measure_iUnion hf\u2082 hf\u2081] ** case hsum \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\u03bc : IsFiniteMeasure \u03bc f : \u2115 \u2192 Set \u03b1 hf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i) hf\u2082 : Pairwise (Disjoint on f) \u22a2 \u2191\u2191\u03bc (\u22c3 i, f i) \u2260 \u22a4 ** exact ne_of_lt (measure_lt_top _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.disjSups_inter_subset_right ** 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\u2081 \u2229 t\u2082) \u2286 s \u25cb t\u2081 \u2229 s \u25cb t\u2082 ** simpa only [disjSups, product_inter, filter_inter_distrib] using image_inter_subset _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_pi_Ico_toReal ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211d h : a \u2264 b \u22a2 ENNReal.toReal (\u2191\u2191volume (Set.pi univ fun i => Ico (a i) (b i))) = \u220f i : \u03b9, (b i - a i) ** simp only [volume_pi_Ico, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1to1.exists_enc_dec ** \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Fintype \u0393 \u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a ** letI := Classical.decEq \u0393 ** \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Fintype \u0393 this : DecidableEq \u0393 := Classical.decEq \u0393 \u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a ** let n := Fintype.card \u0393 ** \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Fintype \u0393 this : DecidableEq \u0393 := Classical.decEq \u0393 n : \u2115 := Fintype.card \u0393 \u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a ** obtain \u27e8F\u27e9 := Fintype.truncEquivFin \u0393 ** case mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Fintype \u0393 this : DecidableEq \u0393 := Classical.decEq \u0393 n : \u2115 := Fintype.card \u0393 x\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393)) F : \u0393 \u2243 Fin (Fintype.card \u0393) \u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a ** let G : Fin n \u21aa Fin n \u2192 Bool :=\n  \u27e8fun a b \u21a6 a = b, fun a b h \u21a6\n    Bool.of_decide_true <| (congr_fun h b).trans <| Bool.decide_true rfl\u27e9 ** case mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Fintype \u0393 this : DecidableEq \u0393 := Classical.decEq \u0393 n : \u2115 := Fintype.card \u0393 x\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393)) F : \u0393 \u2243 Fin (Fintype.card \u0393) G : Fin n \u21aa Fin n \u2192 Bool :=   { toFun := fun a b => decide (a = b),     inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) } \u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a ** let H := (F.toEmbedding.trans G).trans (Equiv.vectorEquivFin _ _).symm.toEmbedding ** case mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Fintype \u0393 this : DecidableEq \u0393 := Classical.decEq \u0393 n : \u2115 := Fintype.card \u0393 x\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393)) F : \u0393 \u2243 Fin (Fintype.card \u0393) G : Fin n \u21aa Fin n \u2192 Bool :=   { toFun := fun a b => decide (a = b),     inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) } H : \u0393 \u21aa Vector Bool n :=   Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)     (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm) \u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a ** classical\n  let enc := H.setValue default (Vector.replicate n false)\n  exact \u27e8_, enc, Function.invFun enc, H.setValue_eq _ _, Function.leftInverse_invFun enc.2\u27e9 ** case mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Fintype \u0393 this : DecidableEq \u0393 := Classical.decEq \u0393 n : \u2115 := Fintype.card \u0393 x\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393)) F : \u0393 \u2243 Fin (Fintype.card \u0393) G : Fin n \u21aa Fin n \u2192 Bool :=   { toFun := fun a b => decide (a = b),     inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) } H : \u0393 \u21aa Vector Bool n :=   Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)     (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm) \u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a ** let enc := H.setValue default (Vector.replicate n false) ** case mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Fintype \u0393 this : DecidableEq \u0393 := Classical.decEq \u0393 n : \u2115 := Fintype.card \u0393 x\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393)) F : \u0393 \u2243 Fin (Fintype.card \u0393) G : Fin n \u21aa Fin n \u2192 Bool :=   { toFun := fun a b => decide (a = b),     inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) } H : \u0393 \u21aa Vector Bool n :=   Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)     (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm) enc : \u0393 \u21aa Vector Bool n := Function.Embedding.setValue H default (Vector.replicate n false) \u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a ** exact \u27e8_, enc, Function.invFun enc, H.setValue_eq _ _, Function.leftInverse_invFun enc.2\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.induction_stronglyMeasurable_aux ** \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),     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) 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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc \u22a2 P f ** let f' := (\u27e8f, hf\u27e9 : lpMeas F \u211d m p \u03bc) ** \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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } \u22a2 P f ** let g := lpMeasToLpTrimLie F \u211d p \u03bc hm f' ** \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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' \u22a2 P f ** have hfg : f' = (lpMeasToLpTrimLie F \u211d p \u03bc hm).symm g := by\n  simp only [LinearIsometryEquiv.symm_apply_apply] ** \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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g \u22a2 P f ** change P \u2191f' ** \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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g \u22a2 P \u2191f' ** rw [hfg] ** \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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g \u22a2 P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g) ** refine'\n  @Lp.induction \u03b1 F m _ p (\u03bc.trim hm) _ hp_ne_top\n    (fun g => P ((lpMeasToLpTrimLie F \u211d p \u03bc hm).symm g)) _ _ _ 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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' \u22a2 f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g ** simp only [LinearIsometryEquiv.symm_apply_apply] ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g \u22a2 \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4),     (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g))       \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4) c) ** intro b t ht h\u03bct ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g b : F t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 \u22a2 P     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm))         \u2191(simpleFunc.indicatorConst p ht (_ : \u2191\u2191(Measure.trim \u03bc hm) t \u2260 \u22a4) b)) ** rw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator ht h\u03bct.ne b] ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g b : F t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 \u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b) ** have h\u03bct' : \u03bc t < \u221e := (le_trim hm).trans_lt h\u03bct ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g b : F t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 h\u03bct' : \u2191\u2191\u03bc t < \u22a4 \u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b) ** specialize h_ind b ht h\u03bct' ** case 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\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_add :   \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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g b : F t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 h\u03bct' : \u2191\u2191\u03bc t < \u22a4 h_ind : P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b) \u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b) ** rwa [Lp.simpleFunc.coe_indicatorConst] at h_ind ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g \u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     Disjoint (Function.support f) (Function.support g) \u2192       (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) (Mem\u2112p.toLp f hf) \u2192         (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) (Mem\u2112p.toLp g hg) \u2192           (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g))             (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** intro f g hf hg h_disj hfP hgP ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d } g\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p h_disj : Disjoint (Function.support f) (Function.support g) hfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) hgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)) \u22a2 P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)) ** rw [LinearIsometryEquiv.map_add] ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d } g\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p h_disj : Disjoint (Function.support f) (Function.support g) hfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) hgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)) \u22a2 P     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf) +         \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)) ** push_cast ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d } g\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p h_disj : Disjoint (Function.support f) (Function.support g) hfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) hgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)) \u22a2 P     (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) +       \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))) ** have h_eq :\n  \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p (\u03bc.trim hm)),\n    ((lpMeasToLpTrimLie F \u211d p \u03bc hm).symm (Mem\u2112p.toLp f hf) : Lp F p \u03bc) =\n      (mem\u2112p_of_mem\u2112p_trim hm hf).toLp f :=\n  lpMeasToLpTrimLie_symm_toLp hm ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d } g\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p h_disj : Disjoint (Function.support f) (Function.support g) hfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) hgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)) h_eq :   \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p) \u22a2 P     (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) +       \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))) ** rw [h_eq f hf] at hfP \u22a2 ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d } g\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p h_disj : Disjoint (Function.support f) (Function.support g) hfP : P (Mem\u2112p.toLp f (_ : Mem\u2112p f p)) hgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)) h_eq :   \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p) \u22a2 P (Mem\u2112p.toLp f (_ : Mem\u2112p f p) + \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))) ** rw [h_eq g hg] at hgP \u22a2 ** case 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\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),     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) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d } g\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p h_disj : Disjoint (Function.support f) (Function.support g) hfP : P (Mem\u2112p.toLp f (_ : Mem\u2112p f p)) hgP : P (Mem\u2112p.toLp g (_ : Mem\u2112p g p)) h_eq :   \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),     \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p) \u22a2 P (Mem\u2112p.toLp f (_ : Mem\u2112p f p) + Mem\u2112p.toLp g (_ : Mem\u2112p g p)) ** exact\n  h_add (mem\u2112p_of_mem\u2112p_trim hm hf) (mem\u2112p_of_mem\u2112p_trim hm hg)\n    (aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim hm hf.aestronglyMeasurable)\n    (aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim hm hg.aestronglyMeasurable)\n    h_disj hfP hgP ** case refine'_3 \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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g \u22a2 IsClosed {f | (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) f} ** change IsClosed ((lpMeasToLpTrimLie F \u211d p \u03bc hm).symm \u207b\u00b9' {g : lpMeas F \u211d m p \u03bc | P \u2191g}) ** case refine'_3 \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),     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) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf } g : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f' hfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g \u22a2 IsClosed (\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) \u207b\u00b9' {g | P \u2191g}) ** exact IsClosed.preimage (LinearIsometryEquiv.continuous _) h_closed ** Qed",
        "informal": ""
    },
    {
        "formal": "VitaliFamily.ae_eventually_measure_pos ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a ** set s := {x | \u00ac\u2200\u1da0 a in v.filterAt x, 0 < \u03bc a} with hs ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a ** simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a ** change \u03bc s = 0 ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} \u22a2 \u2191\u2191\u03bc s = 0 ** let f : \u03b1 \u2192 Set (Set \u03b1) := fun _ => {a | \u03bc a = 0} ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} \u22a2 \u2191\u2191\u03bc s = 0 ** have h : v.FineSubfamilyOn f s := by\n  intro x hx \u03b5 \u03b5pos\n  rw [hs] at hx\n  simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx\n  rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9\n  exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9 ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 \u2191\u2191\u03bc s = 0 ** refine' le_antisymm _ bot_le ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 \u2191\u2191\u03bc s \u2264 0 ** calc\n  \u03bc s \u2264 \u2211' x : h.index, \u03bc (h.covering x) := h.measure_le_tsum\n  _ = \u2211' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2\n  _ = 0 := by simp only [tsum_zero, add_zero] ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} \u22a2 FineSubfamilyOn v f s ** intro x hx \u03b5 \u03b5pos ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} x : \u03b1 hx : x \u2208 s \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** rw [hs] at hx ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} x : \u03b1 hx : x \u2208 {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} x : \u03b1 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 hx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0 \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} x : \u03b1 \u03b5 : \u211d \u03b5pos : \u03b5 > 0 hx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0 a : Set \u03b1 a_sets : a \u2208 setsAt v x ax : a \u2286 closedBall x \u03b5 \u03bca : \u2191\u2191\u03bc a = 0 \u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5 ** exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9 ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 \u2211' (x : \u2191(FineSubfamilyOn.index h)), \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = \u2211' (x : \u2191(FineSubfamilyOn.index h)), 0 ** congr ** case e_f \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 (fun x => \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x)) = fun x => 0 ** ext1 x ** case e_f.h \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s x : \u2191(FineSubfamilyOn.index h) \u22a2 \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = 0 ** exact h.covering_mem x.2 ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a} hs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0} f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0} h : FineSubfamilyOn v f s \u22a2 \u2211' (x : \u2191(FineSubfamilyOn.index h)), 0 = 0 ** simp only [tsum_zero, add_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.tendsto_condCdfRat_atBot ** \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 (condCdfRat \u03c1 a) atBot (\ud835\udcdd 0) ** 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 \u22a2 Tendsto (if a \u2208 condCdfSet \u03c1 then fun r => ENNReal.toReal (preCdf \u03c1 r a) else fun r => if r < 0 then 0 else 1) atBot     (\ud835\udcdd 0) ** split_ifs with h ** 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 h : a \u2208 condCdfSet \u03c1 \u22a2 Tendsto (fun r => ENNReal.toReal (preCdf \u03c1 r a)) atBot (\ud835\udcdd 0) ** rw [\u2190 ENNReal.zero_toReal, ENNReal.tendsto_toReal_iff] ** 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 h : a \u2208 condCdfSet \u03c1 \u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0) ** exact (hasCondCdf_of_mem_condCdfSet h).tendsto_atBot_zero ** case pos.hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h : a \u2208 condCdfSet \u03c1 \u22a2 \u2200 (i : \u211a), preCdf \u03c1 i a \u2260 \u22a4 ** have h' := hasCondCdf_of_mem_condCdfSet h ** case pos.hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h : a \u2208 condCdfSet \u03c1 h' : HasCondCdf \u03c1 a \u22a2 \u2200 (i : \u211a), preCdf \u03c1 i a \u2260 \u22a4 ** exact fun r => ((h'.le_one r).trans_lt ENNReal.one_lt_top).ne ** case pos.hx \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h : a \u2208 condCdfSet \u03c1 \u22a2 0 \u2260 \u22a4 ** exact ENNReal.zero_ne_top ** 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 h : \u00aca \u2208 condCdfSet \u03c1 \u22a2 Tendsto (fun r => if r < 0 then 0 else 1) atBot (\ud835\udcdd 0) ** refine' (tendsto_congr' _).mp tendsto_const_nhds ** 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 h : \u00aca \u2208 condCdfSet \u03c1 \u22a2 (fun x => 0) =\u1da0[atBot] fun r => if r < 0 then 0 else 1 ** rw [EventuallyEq, eventually_atBot] ** 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 h : \u00aca \u2208 condCdfSet \u03c1 \u22a2 \u2203 a, \u2200 (b : \u211a), b \u2264 a \u2192 0 = if b < 0 then 0 else 1 ** refine' \u27e8-1, fun q hq => (if_pos (hq.trans_lt _)).symm\u27e9 ** 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 h : \u00aca \u2208 condCdfSet \u03c1 q : \u211a hq : q \u2264 -1 \u22a2 -1 < 0 ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Nat.encard_range ** \u03b1 : Type ?u.102669 s t : Set \u03b1 k : \u2115 \u22a2 encard {i | i < k} = \u2191k ** convert encard_coe_eq_coe_finsetCard (Finset.range k) using 1 ** case h.e'_3 \u03b1 : Type ?u.102669 s t : Set \u03b1 k : \u2115 \u22a2 \u2191k = \u2191(Finset.card (Finset.range k)) ** rw [Finset.card_range] ** case h.e'_2 \u03b1 : Type ?u.102669 s t : Set \u03b1 k : \u2115 \u22a2 encard {i | i < k} = encard \u2191(Finset.range k) ** rw [Finset.coe_range, Iio_def] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.limsup_measure_closed_le_iff_liminf_measure_open_ge ** \u03a9 : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : TopologicalSpace \u03a9 inst\u271d\u00b2 : OpensMeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) \u22a2 (\u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F) \u2194     \u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L ** constructor ** case mp \u03a9 : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : TopologicalSpace \u03a9 inst\u271d\u00b2 : OpensMeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) \u22a2 (\u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F) \u2192     \u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L ** intro h G G_open ** case mp \u03a9 : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : TopologicalSpace \u03a9 inst\u271d\u00b2 : OpensMeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) h : \u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F G : Set \u03a9 G_open : IsOpen G \u22a2 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L ** exact le_measure_liminf_of_limsup_measure_compl_le\n  G_open.measurableSet (h G\u1d9c (isClosed_compl_iff.mpr G_open)) ** case mpr \u03a9 : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : TopologicalSpace \u03a9 inst\u271d\u00b2 : OpensMeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) \u22a2 (\u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L) \u2192     \u2200 (F : Set \u03a9), IsClosed F \u2192 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F ** intro h F F_closed ** case mpr \u03a9 : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : TopologicalSpace \u03a9 inst\u271d\u00b2 : OpensMeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) h : \u2200 (G : Set \u03a9), IsOpen G \u2192 \u2191\u2191\u03bc G \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) G) L F : Set \u03a9 F_closed : IsClosed F \u22a2 limsup (fun i => \u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u03bc F ** exact limsup_measure_le_of_le_liminf_measure_compl\n  F_closed.measurableSet (h F\u1d9c (isOpen_compl_iff.mpr F_closed)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.StronglyMeasurable.mono ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 f g : \u03b1 \u2192 \u03b2 m m' : MeasurableSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 hf : StronglyMeasurable f h_mono : m' \u2264 m \u22a2 StronglyMeasurable f ** let f_approx : \u2115 \u2192 @SimpleFunc \u03b1 m \u03b2 := fun n =>\n  @SimpleFunc.mk \u03b1 m \u03b2\n    (hf.approx n)\n    (fun x => h_mono _ (SimpleFunc.measurableSet_fiber' _ x))\n    (SimpleFunc.finite_range (hf.approx n)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 f g : \u03b1 \u2192 \u03b2 m m' : MeasurableSpace \u03b1 inst\u271d : TopologicalSpace \u03b2 hf : StronglyMeasurable f h_mono : m' \u2264 m f_approx : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 :=   fun n =>     { toFun := \u2191(StronglyMeasurable.approx hf n),       measurableSet_fiber' := (_ : \u2200 (x : \u03b2), MeasurableSet (\u2191(StronglyMeasurable.approx hf n) \u207b\u00b9' {x})),       finite_range' := (_ : Set.Finite (range \u2191(StronglyMeasurable.approx hf n))) } \u22a2 StronglyMeasurable f ** exact \u27e8f_approx, hf.tendsto_approx\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "IsROrC.interval_integral_ofReal ** \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 \ud835\udd5c : Type u_6 inst\u271d : IsROrC \ud835\udd5c a b : \u211d \u03bc : Measure \u211d f : \u211d \u2192 \u211d \u22a2 \u222b (x : \u211d) in a..b, \u2191(f x) \u2202\u03bc = \u2191(\u222b (x : \u211d) in a..b, f x \u2202\u03bc) ** simp only [intervalIntegral, integral_ofReal, IsROrC.ofReal_sub] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.strong_law_aux2 ** \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 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     (fun n =>         \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) =o[atTop]       fun n => \u2191\u230ac ^ n\u230b\u208a ** obtain \u27e8v, -, v_pos, v_lim\u27e9 :\n    \u2203 v : \u2115 \u2192 \u211d, StrictAnti v \u2227 (\u2200 n : \u2115, 0 < v n) \u2227 Tendsto v atTop (\ud835\udcdd 0) :=\n  exists_seq_strictAnti_tendsto (0 : \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 : \u211d c_one : 1 < c v : \u2115 \u2192 \u211d v_pos : \u2200 (n : \u2115), 0 < v n v_lim : Tendsto v atTop (\ud835\udcdd 0) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     (fun n =>         \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) =o[atTop]       fun n => \u2191\u230ac ^ n\u230b\u208a ** have := fun i => strong_law_aux1 X hint hindep hident hnonneg c_one (v_pos i) ** 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 : \u211d c_one : 1 < c v : \u2115 \u2192 \u211d v_pos : \u2200 (n : \u2115), 0 < v n v_lim : Tendsto v atTop (\ud835\udcdd 0) this :   \u2200 (i : \u2115),     \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| <           v i * \u2191\u230ac ^ n\u230b\u208a \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     (fun n =>         \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) =o[atTop]       fun n => \u2191\u230ac ^ n\u230b\u208a ** 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 : \u211d c_one : 1 < c v : \u2115 \u2192 \u211d v_pos : \u2200 (n : \u2115), 0 < v n v_lim : Tendsto v atTop (\ud835\udcdd 0) this :   \u2200 (i : \u2115),     \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| <           v i * \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   \u2200 (i : \u2115),     \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| <         v i * \u2191\u230ac ^ n\u230b\u208a \u22a2 (fun n =>       \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) =o[atTop]     fun n => \u2191\u230ac ^ n\u230b\u208a ** apply Asymptotics.isLittleO_iff.2 fun \u03b5 \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 v : \u2115 \u2192 \u211d v_pos : \u2200 (n : \u2115), 0 < v n v_lim : Tendsto v atTop (\ud835\udcdd 0) this :   \u2200 (i : \u2115),     \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| <           v i * \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   \u2200 (i : \u2115),     \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| <         v i * \u2191\u230ac ^ n\u230b\u208a \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 \u2200\u1da0 (x : \u2115) in atTop,     \u2016\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -           \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a\u2016 \u2264       \u03b5 * \u2016\u2191\u230ac ^ x\u230b\u208a\u2016 ** obtain \u27e8i, hi\u27e9 : \u2203 i, v i < \u03b5 := ((tendsto_order.1 v_lim).2 \u03b5 \u03b5pos).exists ** case 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 : \u211d c_one : 1 < c v : \u2115 \u2192 \u211d v_pos : \u2200 (n : \u2115), 0 < v n v_lim : Tendsto v atTop (\ud835\udcdd 0) this :   \u2200 (i : \u2115),     \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| <           v i * \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   \u2200 (i : \u2115),     \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| <         v i * \u2191\u230ac ^ n\u230b\u208a \u03b5 : \u211d \u03b5pos : 0 < \u03b5 i : \u2115 hi : v i < \u03b5 \u22a2 \u2200\u1da0 (x : \u2115) in atTop,     \u2016\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -           \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a\u2016 \u2264       \u03b5 * \u2016\u2191\u230ac ^ x\u230b\u208a\u2016 ** filter_upwards [h\u03c9 i] with n hn ** 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 v : \u2115 \u2192 \u211d v_pos : \u2200 (n : \u2115), 0 < v n v_lim : Tendsto v atTop (\ud835\udcdd 0) this :   \u2200 (i : \u2115),     \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| <           v i * \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   \u2200 (i : \u2115),     \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| <         v i * \u2191\u230ac ^ n\u230b\u208a \u03b5 : \u211d \u03b5pos : 0 < \u03b5 i : \u2115 hi : v i < \u03b5 n : \u2115 hn :   |\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| <     v i * \u2191\u230ac ^ n\u230b\u208a \u22a2 \u2016\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\u2016 \u2264     \u03b5 * \u2016\u2191\u230ac ^ n\u230b\u208a\u2016 ** simp only [Real.norm_eq_abs, LatticeOrderedGroup.abs_abs, Nat.abs_cast] ** 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 v : \u2115 \u2192 \u211d v_pos : \u2200 (n : \u2115), 0 < v n v_lim : Tendsto v atTop (\ud835\udcdd 0) this :   \u2200 (i : \u2115),     \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| <           v i * \u2191\u230ac ^ n\u230b\u208a \u03c9 : \u03a9 h\u03c9 :   \u2200 (i : \u2115),     \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| <         v i * \u2191\u230ac ^ n\u230b\u208a \u03b5 : \u211d \u03b5pos : 0 < \u03b5 i : \u2115 hi : v i < \u03b5 n : \u2115 hn :   |\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| <     v i * \u2191\u230ac ^ n\u230b\u208a \u22a2 |\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| \u2264     \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** exact hn.le.trans (mul_le_mul_of_nonneg_right hi.le (Nat.cast_nonneg _)) ** 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 : \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 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 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) ** 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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 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 hp_top : \u00acp = \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** have hp1 : 1 \u2264 p.toReal := by\n  rw [\u2190 ENNReal.ofReal_le_iff_le_toReal hp_top, ENNReal.ofReal_one]\n  exact hp ** 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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 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 hp_top : \u00acp = \u22a4 hp1 : 1 \u2264 ENNReal.toReal p \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** have h_cau' : \u2200 N n m : \u2115, N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p.toReal \u03bc < B N := by\n  intro N n m hn hm\n  specialize h_cau N n m hn hm\n  rwa [snorm_eq_snorm' (zero_lt_one.trans_le hp).ne.symm hp_top] at h_cau ** 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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 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 hp_top : \u00acp = \u22a4 hp1 : 1 \u2264 ENNReal.toReal p h_cau' : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** exact ae_tendsto_of_cauchy_snorm' hf hp1 hB h_cau' ** 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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 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 hp_top : p = \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** simp_rw [hp_top] at * ** 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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp : True h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N hp_top : True \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** have h_cau_ae : \u2200\u1d50 x \u2202\u03bc, \u2200 N n m, N \u2264 n \u2192 N \u2264 m \u2192 (\u2016(f n - f m) x\u2016\u208a : \u211d\u22650\u221e) < B N := by\n  simp_rw [ae_all_iff]\n  exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN) ** 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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp : True h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** simp_rw [snorm_exponent_top, snormEssSup] at h_cau ** 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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _ ** 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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N x : \u03b1 hx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N \u22a2 CauchySeq fun n => f n x ** refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _ ** \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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp : True h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N hp_top : True \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N ** simp_rw [ae_all_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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp : True h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm (f n - f m) \u22a4 \u03bc < B N hp_top : True \u22a2 \u2200 (i i_1 i_2 : \u2115), i \u2264 i_1 \u2192 i \u2264 i_2 \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2016(f i_1 - f i_2) a\u2016\u208a < B i ** exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN) ** case pos.refine'_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 inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N x : \u03b1 hx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N \u22a2 \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n x) (f m x) \u2264 (fun n => ENNReal.toReal (B n)) N ** intro n m N hnN hmN ** case pos.refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N x : \u03b1 hx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N n m N : \u2115 hnN : N \u2264 n hmN : N \u2264 m \u22a2 dist (f n x) (f m x) \u2264 (fun n => ENNReal.toReal (B n)) N ** specialize hx N n m hnN hmN ** case pos.refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N x : \u03b1 n m N : \u2115 hnN : N \u2264 n hmN : N \u2264 m hx : \u2191\u2016(f n - f m) x\u2016\u208a < B N \u22a2 dist (f n x) (f m x) \u2264 (fun n => ENNReal.toReal (B n)) N ** rw [dist_eq_norm, \u2190 ENNReal.toReal_ofReal (norm_nonneg _),\n  ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.ne_top_of_tsum_ne_top hB N)] ** case pos.refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N x : \u03b1 n m N : \u2115 hnN : N \u2264 n hmN : N \u2264 m hx : \u2191\u2016(f n - f m) x\u2016\u208a < B N \u22a2 ENNReal.ofReal \u2016f n x - f m x\u2016 \u2264 B N ** rw [\u2190 ofReal_norm_eq_coe_nnnorm] at hx ** case pos.refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N x : \u03b1 n m N : \u2115 hnN : N \u2264 n hmN : N \u2264 m hx : ENNReal.ofReal \u2016(f n - f m) x\u2016 < B N \u22a2 ENNReal.ofReal \u2016f n x - f m x\u2016 \u2264 B N ** exact hx.le ** case pos.refine'_2 \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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N x : \u03b1 hx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N \u22a2 Tendsto (fun n => ENNReal.toReal (B n)) atTop (\ud835\udcdd 0) ** rw [\u2190 ENNReal.zero_toReal] ** case pos.refine'_2 \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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp hp_top : True h_cau_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 essSup (fun x => \u2191\u2016(f n - f m) x\u2016\u208a) \u03bc < B N x : \u03b1 hx : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 \u2191\u2016(f n - f m) x\u2016\u208a < B N \u22a2 Tendsto (fun n => ENNReal.toReal (B n)) atTop (\ud835\udcdd (ENNReal.toReal 0)) ** exact\n  Tendsto.comp (g := ENNReal.toReal) (ENNReal.tendsto_toReal ENNReal.zero_ne_top)\n    (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB) ** \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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 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 hp_top : \u00acp = \u22a4 \u22a2 1 \u2264 ENNReal.toReal p ** rw [\u2190 ENNReal.ofReal_le_iff_le_toReal hp_top, ENNReal.ofReal_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\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 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 hp_top : \u00acp = \u22a4 \u22a2 1 \u2264 p ** exact hp ** \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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 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 hp_top : \u00acp = \u22a4 hp1 : 1 \u2264 ENNReal.toReal p \u22a2 \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N ** intro N n m hn hm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 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 hp_top : \u00acp = \u22a4 hp1 : 1 \u2264 ENNReal.toReal p N n m : \u2115 hn : N \u2264 n hm : N \u2264 m \u22a2 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N ** specialize h_cau N n m hn hm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m\u271d 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 : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 hp_top : \u00acp = \u22a4 hp1 : 1 \u2264 ENNReal.toReal p N n m : \u2115 hn : N \u2264 n hm : N \u2264 m h_cau : snorm (f n - f m) p \u03bc < B N \u22a2 snorm' (f n - f m) (ENNReal.toReal p) \u03bc < B N ** rwa [snorm_eq_snorm' (zero_lt_one.trans_le hp).ne.symm hp_top] at h_cau ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.mkMetric_le_liminf_tsum ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e \u22a2 \u2191\u2191(mkMetric m) s \u2264 liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l ** haveI : \u2200 n, Encodable (\u03b9 n) := fun n => Encodable.ofCountable _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u22a2 \u2191\u2191(mkMetric m) s \u2264 liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l ** simp only [mkMetric_apply] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u22a2 \u2a06 r,       \u2a06 (_ : 0 < r),         \u2a05 t,           \u2a05 (_ : s \u2286 iUnion t),             \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264     liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l ** refine' iSup\u2082_le fun \u03b5 h\u03b5 => _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 \u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264     liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l ** refine' le_of_forall_le_of_dense fun c hc => _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c \u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264     c ** rcases ((frequently_lt_of_liminf_lt (by isBoundedDefault) hc).and_eventually\n      ((hr.eventually (gt_mem_nhds h\u03b5)).and (ht.and hst))).exists with\n  \u27e8n, hn, hrn, htn, hstn\u27e9 ** case intro.intro.intro.intro \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c n : \u03b2 hn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c hrn : r n < \u03b5 htn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hstn : s \u2286 \u22c3 i, t n i \u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264     c ** set u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b ** case intro.intro.intro.intro \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c n : \u03b2 hn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c hrn : r n < \u03b5 htn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hstn : s \u2286 \u22c3 i, t n i u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b \u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264     c ** refine' iInf\u2082_le_of_le u (by rwa [iUnion_decode\u2082]) _ ** case intro.intro.intro.intro \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c n : \u03b2 hn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c hrn : r n < \u03b5 htn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hstn : s \u2286 \u22c3 i, t n i u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b \u22a2 \u2a05 (_ : \u2200 (n : \u2115), diam (u n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (u n)), m (diam (u n)) \u2264 c ** refine' iInf_le_of_le (fun j => _) _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c \u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) l fun n => \u2211' (i : \u03b9 n), m (diam (t n i)) ** isBoundedDefault ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c n : \u03b2 hn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c hrn : r n < \u03b5 htn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hstn : s \u2286 \u22c3 i, t n i u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b \u22a2 s \u2286 iUnion u ** rwa [iUnion_decode\u2082] ** case intro.intro.intro.intro.refine'_1 \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c n : \u03b2 hn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c hrn : r n < \u03b5 htn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hstn : s \u2286 \u22c3 i, t n i u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b j : \u2115 \u22a2 diam (u j) \u2264 \u03b5 ** rw [EMetric.diam_iUnion_mem_option] ** case intro.intro.intro.intro.refine'_1 \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c n : \u03b2 hn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c hrn : r n < \u03b5 htn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hstn : s \u2286 \u22c3 i, t n i u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b j : \u2115 \u22a2 \u2a06 i \u2208 decode\u2082 (\u03b9 n) j, diam (t n i) \u2264 \u03b5 ** exact iSup\u2082_le fun _ _ => (htn _).trans hrn.le ** case intro.intro.intro.intro.refine'_2 \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c n : \u03b2 hn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c hrn : r n < \u03b5 htn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hstn : s \u2286 \u22c3 i, t n i u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b \u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (u n)), m (diam (u n)) \u2264 c ** calc\n  (\u2211' j : \u2115, \u2a06 _ : (u j).Nonempty, m (diam (u j))) = _ :=\n    tsum_iUnion_decode\u2082 (fun t : Set X => \u2a06 _ : t.Nonempty, m (diam t)) (by simp) _\n  _ \u2264 \u2211' i : \u03b9 n, m (diam (t n i)) := (ENNReal.tsum_le_tsum fun b => iSup_le fun _ => le_rfl)\n  _ \u2264 c := hn.le ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2074 : EMetricSpace X inst\u271d\u00b3 : EMetricSpace Y inst\u271d\u00b2 : MeasurableSpace X inst\u271d\u00b9 : BorelSpace X \u03b2 : Type u_4 \u03b9 : \u03b2 \u2192 Type u_5 inst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n) s : Set X l : Filter \u03b2 r : \u03b2 \u2192 \u211d\u22650\u221e hr : Tendsto r l (\ud835\udcdd 0) t : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X ht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 i, t n i m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e this : (n : \u03b2) \u2192 Encodable (\u03b9 n) \u03b5 : \u211d\u22650\u221e h\u03b5 : 0 < \u03b5 c : \u211d\u22650\u221e hc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c n : \u03b2 hn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c hrn : r n < \u03b5 htn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n hstn : s \u2286 \u22c3 i, t n i u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b \u22a2 (fun t => \u2a06 (_ : Set.Nonempty t), m (diam t)) \u2205 = 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "generateFrom_pi_eq ** \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 MeasurableSpace.pi = generateFrom (Set.pi univ '' Set.pi univ C) ** cases nonempty_encodable \u03b9 ** 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)) hC : \u2200 (i : \u03b9), IsCountablySpanning (C i) val\u271d : Encodable \u03b9 \u22a2 MeasurableSpace.pi = generateFrom (Set.pi univ '' Set.pi univ C) ** apply le_antisymm ** case intro.a \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) val\u271d : Encodable \u03b9 \u22a2 MeasurableSpace.pi \u2264 generateFrom (Set.pi univ '' Set.pi univ C) ** refine' iSup_le _ ** case intro.a \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) val\u271d : Encodable \u03b9 \u22a2 \u2200 (i : \u03b9),     MeasurableSpace.comap (fun b => b i) ((fun i => generateFrom (C i)) i) \u2264 generateFrom (Set.pi univ '' Set.pi univ C) ** intro i ** case intro.a \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) val\u271d : Encodable \u03b9 i : \u03b9 \u22a2 MeasurableSpace.comap (fun b => b i) ((fun i => generateFrom (C i)) i) \u2264 generateFrom (Set.pi univ '' Set.pi univ C) ** rw [comap_generateFrom] ** case intro.a \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) val\u271d : Encodable \u03b9 i : \u03b9 \u22a2 generateFrom ((preimage fun b => b i) '' C i) \u2264 generateFrom (Set.pi univ '' Set.pi univ C) ** apply generateFrom_le ** case intro.a.h \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) val\u271d : Encodable \u03b9 i : \u03b9 \u22a2 \u2200 (t : Set ((a : \u03b9) \u2192 (fun i => \u03b1 i) a)), t \u2208 (preimage fun b => b i) '' C i \u2192 MeasurableSet t ** rintro _ \u27e8s, hs, rfl\u27e9 ** case intro.a.h.intro.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)) hC : \u2200 (i : \u03b9), IsCountablySpanning (C i) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i \u22a2 MeasurableSet ((fun b => b i) \u207b\u00b9' s) ** choose t h1t h2t using hC ** case intro.a.h.intro.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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ \u22a2 MeasurableSet ((fun b => b i) \u207b\u00b9' s) ** simp_rw [eval_preimage, \u2190 h2t] ** case intro.a.h.intro.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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ \u22a2 MeasurableSet (Set.pi univ (update (fun i => \u22c3 n, t i n) i s)) ** rw [\u2190 @iUnion_const _ \u2115 _ s] ** case intro.a.h.intro.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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k \u22a2 MeasurableSet (Set.pi univ (update (fun i => \u22c3 n, t i n) i (\u22c3 x, s))) ** rw [this, \u2190 iUnion_univ_pi] ** case intro.a.h.intro.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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k \u22a2 MeasurableSet (\u22c3 x, Set.pi univ fun i_1 => update (fun i' => t i' (x i_1)) i s i_1) ** apply MeasurableSet.iUnion ** case intro.a.h.intro.intro.h \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k \u22a2 \u2200 (b : \u03b9 \u2192 \u2115), MeasurableSet (Set.pi univ fun i_1 => update (fun i' => t i' (b i_1)) i s i_1) ** intro n ** case intro.a.h.intro.intro.h \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 \u22a2 MeasurableSet (Set.pi univ fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) ** apply measurableSet_generateFrom ** case intro.a.h.intro.intro.h.ht \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 \u22a2 (Set.pi univ fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) \u2208 Set.pi univ '' Set.pi univ C ** apply mem_image_of_mem ** case intro.a.h.intro.intro.h.ht.h \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 \u22a2 (fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) \u2208 Set.pi univ C ** intro j _ ** case intro.a.h.intro.intro.h.ht.h \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 j : \u03b9 a\u271d : j \u2208 univ \u22a2 (fun i_1 => update (fun i' => t i' (n i_1)) i s i_1) j \u2208 C j ** dsimp only ** case intro.a.h.intro.intro.h.ht.h \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 j : \u03b9 a\u271d : j \u2208 univ \u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j ** by_cases h : j = i ** case pos \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 j : \u03b9 a\u271d : j \u2208 univ h : j = i \u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j  case neg \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 j : \u03b9 a\u271d : j \u2208 univ h : \u00acj = i \u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j ** subst h ** case pos \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)) val\u271d : Encodable \u03b9 t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ n : \u03b9 \u2192 \u2115 j : \u03b9 a\u271d : j \u2208 univ s : Set ((fun i => \u03b1 i) j) hs : s \u2208 C j this :   Set.pi univ (update (fun i' => iUnion (t i')) j (\u22c3 x, s)) =     Set.pi univ fun k => \u22c3 j_1, update (fun i' => t i' j_1) j s k \u22a2 update (fun i' => t i' (n j)) j s j \u2208 C j  case neg \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 j : \u03b9 a\u271d : j \u2208 univ h : \u00acj = i \u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j ** rwa [update_same] ** case neg \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 j : \u03b9 a\u271d : j \u2208 univ h : \u00acj = i \u22a2 update (fun i' => t i' (n j)) i s j \u2208 C j ** rw [update_noteq h] ** case neg \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ this :   Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k n : \u03b9 \u2192 \u2115 j : \u03b9 a\u271d : j \u2208 univ h : \u00acj = i \u22a2 t j (n j) \u2208 C j ** apply h1t ** \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ \u22a2 Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) = Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k ** ext ** case h \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ x\u271d : (i : \u03b9) \u2192 \u03b1 i \u22a2 x\u271d \u2208 Set.pi univ (update (fun i' => iUnion (t i')) i (\u22c3 x, s)) \u2194     x\u271d \u2208 Set.pi univ fun k => \u22c3 j, update (fun i' => t i' j) i s k ** simp_rw [mem_univ_pi] ** case h \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ x\u271d : (i : \u03b9) \u2192 \u03b1 i \u22a2 (\u2200 (i_1 : \u03b9), x\u271d i_1 \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i_1) \u2194     \u2200 (i_1 : \u03b9), x\u271d i_1 \u2208 \u22c3 j, update (fun i' => t i' j) i s i_1 ** apply forall_congr' ** case h.h \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ x\u271d : (i : \u03b9) \u2192 \u03b1 i \u22a2 \u2200 (a : \u03b9), x\u271d a \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) a \u2194 x\u271d a \u2208 \u22c3 j, update (fun i' => t i' j) i s a ** intro i' ** case h.h \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ x\u271d : (i : \u03b9) \u2192 \u03b1 i i' : \u03b9 \u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i' ** by_cases h : i' = i ** case pos \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ x\u271d : (i : \u03b9) \u2192 \u03b1 i i' : \u03b9 h : i' = i \u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i' ** subst h ** case pos \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)) val\u271d : Encodable \u03b9 t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ x\u271d : (i : \u03b9) \u2192 \u03b1 i i' : \u03b9 s : Set ((fun i => \u03b1 i) i') hs : s \u2208 C i' \u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i' (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i' s i' ** simp ** case neg \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ x\u271d : (i : \u03b9) \u2192 \u03b1 i i' : \u03b9 h : \u00aci' = i \u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i' ** rw [\u2190 Ne.def] at h ** case neg \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)) val\u271d : Encodable \u03b9 i : \u03b9 s : Set ((fun i => \u03b1 i) i) hs : s \u2208 C i t : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1t : \u2200 (i : \u03b9) (n : \u2115), t i n \u2208 C i h2t : \u2200 (i : \u03b9), \u22c3 n, t i n = univ x\u271d : (i : \u03b9) \u2192 \u03b1 i i' : \u03b9 h : i' \u2260 i \u22a2 x\u271d i' \u2208 update (fun i' => iUnion (t i')) i (\u22c3 x, s) i' \u2194 x\u271d i' \u2208 \u22c3 j, update (fun i' => t i' j) i s i' ** simp [h] ** case intro.a \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) val\u271d : Encodable \u03b9 \u22a2 generateFrom (Set.pi univ '' Set.pi univ C) \u2264 MeasurableSpace.pi ** apply generateFrom_le ** case intro.a.h \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) val\u271d : Encodable \u03b9 \u22a2 \u2200 (t : Set ((i : \u03b9) \u2192 \u03b1 i)), t \u2208 Set.pi univ '' Set.pi univ C \u2192 MeasurableSet t ** rintro _ \u27e8s, hs, rfl\u27e9 ** case intro.a.h.intro.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)) hC : \u2200 (i : \u03b9), IsCountablySpanning (C i) val\u271d : Encodable \u03b9 s : (i : \u03b9) \u2192 Set (\u03b1 i) hs : s \u2208 Set.pi univ C \u22a2 MeasurableSet (Set.pi univ s) ** rw [univ_pi_eq_iInter] ** case intro.a.h.intro.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)) hC : \u2200 (i : \u03b9), IsCountablySpanning (C i) val\u271d : Encodable \u03b9 s : (i : \u03b9) \u2192 Set (\u03b1 i) hs : s \u2208 Set.pi univ C \u22a2 MeasurableSet (\u22c2 i, eval i \u207b\u00b9' s i) ** apply MeasurableSet.iInter ** case intro.a.h.intro.intro.h \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) val\u271d : Encodable \u03b9 s : (i : \u03b9) \u2192 Set (\u03b1 i) hs : s \u2208 Set.pi univ C \u22a2 \u2200 (b : \u03b9), MeasurableSet (eval b \u207b\u00b9' s b) ** intro i ** case intro.a.h.intro.intro.h \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) val\u271d : Encodable \u03b9 s : (i : \u03b9) \u2192 Set (\u03b1 i) hs : s \u2208 Set.pi univ C i : \u03b9 \u22a2 MeasurableSet (eval i \u207b\u00b9' s i) ** apply @measurable_pi_apply _ _ (fun i => generateFrom (C i)) ** case intro.a.h.intro.intro.h.a \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) val\u271d : Encodable \u03b9 s : (i : \u03b9) \u2192 Set (\u03b1 i) hs : s \u2208 Set.pi univ C i : \u03b9 \u22a2 MeasurableSet (s i) ** exact measurableSet_generateFrom (hs i (mem_univ i)) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace ** \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_meas_g : Measurable g \u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc ** revert g ** \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f \u22a2 \u2200 {g : \u03a9 \u2192 \u211d\u22650\u221e}, Measurable g \u2192 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc ** have h_measM_f : Measurable f := h_meas_f.mono hMf le_rfl ** \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f \u22a2 \u2200 {g : \u03a9 \u2192 \u211d\u22650\u221e}, Measurable g \u2192 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc ** apply @Measurable.ennreal_induction _ Mg ** case h_ind \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f \u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set \u03a9\u2984,     MeasurableSet s \u2192       \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator s (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c) \u03c9 \u2202\u03bc ** intro c s h_s ** case h_ind \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f c : \u211d\u22650\u221e s : Set \u03a9 h_s : MeasurableSet s \u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * indicator s (fun x => c) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), indicator s (fun x => c) \u03c9 \u2202\u03bc ** apply lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator hMf _ (hMg _ h_s) _ h_meas_f ** \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f c : \u211d\u22650\u221e s : Set \u03a9 h_s : MeasurableSet s \u22a2 IndepSets {s | MeasurableSet s} {s} ** apply indepSets_of_indepSets_of_le_right h_ind ** \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f c : \u211d\u22650\u221e s : Set \u03a9 h_s : MeasurableSet s \u22a2 {s} \u2286 {s | MeasurableSet s} ** rwa [singleton_subset_iff] ** case h_add \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f \u22a2 \u2200 \u2983f_1 g : \u03a9 \u2192 \u211d\u22650\u221e\u2984,     Disjoint (Function.support f_1) (Function.support g) \u2192       Measurable f_1 \u2192         Measurable g \u2192           \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f_1 \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f_1 \u03c9 \u2202\u03bc \u2192             \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc \u2192               \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (f_1 + g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (f_1 + g) \u03c9 \u2202\u03bc ** intro f' g _ h_measMg_f' _ h_ind_f' h_ind_g' ** case h_add \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f f' g : \u03a9 \u2192 \u211d\u22650\u221e a\u271d\u00b9 : Disjoint (Function.support f') (Function.support g) h_measMg_f' : Measurable f' a\u271d : Measurable g h_ind_f' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc h_ind_g' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc \u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (f' + g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc ** have h_measM_f' : Measurable f' := h_measMg_f'.mono hMg le_rfl ** case h_add \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f f' g : \u03a9 \u2192 \u211d\u22650\u221e a\u271d\u00b9 : Disjoint (Function.support f') (Function.support g) h_measMg_f' : Measurable f' a\u271d : Measurable g h_ind_f' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc h_ind_g' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc h_measM_f' : Measurable f' \u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (f' + g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (f' + g) \u03c9 \u2202\u03bc ** simp_rw [Pi.add_apply, left_distrib] ** case h_add \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f f' g : \u03a9 \u2192 \u211d\u22650\u221e a\u271d\u00b9 : Disjoint (Function.support f') (Function.support g) h_measMg_f' : Measurable f' a\u271d : Measurable g h_ind_f' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 \u2202\u03bc h_ind_g' : \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc h_measM_f' : Measurable f' \u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' \u03c9 + f \u03c9 * g \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' \u03c9 + g \u03c9 \u2202\u03bc ** rw [lintegral_add_left h_measM_f', lintegral_add_left (h_measM_f.mul h_measM_f'), left_distrib,\n  h_ind_f', h_ind_g'] ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f \u22a2 \u2200 \u2983f_1 : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e\u2984,     (\u2200 (n : \u2115), Measurable (f_1 n)) \u2192       Monotone f_1 \u2192         (\u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f_1 n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f_1 n \u03c9 \u2202\u03bc) \u2192           \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (fun x => \u2a06 n, f_1 n x) \u03c9 \u2202\u03bc =             (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f_1 n x) \u03c9 \u2202\u03bc ** intro f' h_meas_f' h_mono_f' h_ind_f' ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f f' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f' : \u2200 (n : \u2115), Measurable (f' n) h_mono_f' : Monotone f' h_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc \u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc ** have h_measM_f' : \u2200 n, Measurable (f' n) := fun n => (h_meas_f' n).mono hMg le_rfl ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f f' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f' : \u2200 (n : \u2115), Measurable (f' n) h_mono_f' : Monotone f' h_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc h_measM_f' : \u2200 (n : \u2115), Measurable (f' n) \u22a2 \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), (fun x => \u2a06 n, f' n x) \u03c9 \u2202\u03bc ** simp_rw [ENNReal.mul_iSup] ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f f' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f' : \u2200 (n : \u2115), Measurable (f' n) h_mono_f' : Monotone f' h_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc h_measM_f' : \u2200 (n : \u2115), Measurable (f' n) \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2a06 i, f \u03c9 * f' i \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), \u2a06 n, f' n \u03c9 \u2202\u03bc ** rw [lintegral_iSup, lintegral_iSup h_measM_f' h_mono_f', ENNReal.mul_iSup] ** case h_iSup \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f f' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f' : \u2200 (n : \u2115), Measurable (f' n) h_mono_f' : Monotone f' h_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc h_measM_f' : \u2200 (n : \u2115), Measurable (f' n) \u22a2 \u2a06 n, \u222b\u207b (a : \u03a9), f a * f' n a \u2202\u03bc = \u2a06 i, (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (a : \u03a9), f' i a \u2202\u03bc ** simp_rw [\u2190 h_ind_f'] ** case h_iSup.hf \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f f' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f' : \u2200 (n : \u2115), Measurable (f' n) h_mono_f' : Monotone f' h_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc h_measM_f' : \u2200 (n : \u2115), Measurable (f' n) \u22a2 \u2200 (n : \u2115), Measurable fun \u03c9 => f \u03c9 * f' n \u03c9 ** exact fun n => h_measM_f.mul (h_measM_f' n) ** case h_iSup.h_mono \u03a9 : Type u_1 m\u03a9\u271d : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 f : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d Mf Mg m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 hMf : Mf \u2264 m\u03a9 hMg : Mg \u2264 m\u03a9 h_ind : Indep Mf Mg h_meas_f : Measurable f h_measM_f : Measurable f f' : \u2115 \u2192 \u03a9 \u2192 \u211d\u22650\u221e h_meas_f' : \u2200 (n : \u2115), Measurable (f' n) h_mono_f' : Monotone f' h_ind_f' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), f \u03c9 * f' n \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), f' n \u03c9 \u2202\u03bc h_measM_f' : \u2200 (n : \u2115), Measurable (f' n) \u22a2 Monotone fun i \u03c9 => f \u03c9 * f' i \u03c9 ** exact fun n m (h_le : n \u2264 m) a => mul_le_mul_left' (h_mono_f' h_le a) _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.erase_union_singleton ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b : \u03b2 a s : Finmap \u03b2 h : lookup a s = some b x : \u03b1 \u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s ** by_cases h' : x = a ** case pos \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b : \u03b2 a s : Finmap \u03b2 h : lookup a s = some b x : \u03b1 h' : x = a \u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s ** subst a ** case pos \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 s : Finmap \u03b2 x : \u03b1 b : \u03b2 x h : lookup x s = some b \u22a2 lookup x (erase x s \u222a singleton x b) = lookup x s ** rw [lookup_union_right not_mem_erase_self, lookup_singleton_eq, h] ** case neg \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b : \u03b2 a s : Finmap \u03b2 h : lookup a s = some b x : \u03b1 h' : \u00acx = a \u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s ** have : x \u2209 singleton a b := by rwa [mem_singleton] ** case neg \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b : \u03b2 a s : Finmap \u03b2 h : lookup a s = some b x : \u03b1 h' : \u00acx = a this : \u00acx \u2208 singleton a b \u22a2 lookup x (erase a s \u222a singleton a b) = lookup x s ** rw [lookup_union_left_of_not_in this, lookup_erase_ne h'] ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b : \u03b2 a s : Finmap \u03b2 h : lookup a s = some b x : \u03b1 h' : \u00acx = a \u22a2 \u00acx \u2208 singleton a b ** rwa [mem_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "QPF.Cofix.bisim_aux ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x 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 ** intro x ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x : Cofix F \u22a2 \u2200 (y : Cofix F), r x y \u2192 x = y ** apply Quot.inductionOn (motive := _) x ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x : Cofix F \u22a2 \u2200 (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y \u2192 Quot.mk Mcongr a = y ** clear x ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y \u22a2 \u2200 (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y \u2192 Quot.mk Mcongr a = y ** intro 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x : PFunctor.M (P F) y : Cofix F \u22a2 r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y ** apply Quot.inductionOn (motive := _) 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x : PFunctor.M (P F) y : Cofix F \u22a2 \u2200 (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a ** clear 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x : PFunctor.M (P F) \u22a2 \u2200 (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a ** intro 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u22a2 Quot.mk Mcongr x = Quot.mk Mcongr y ** apply Quot.sound ** 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u22a2 Mcongr x y ** let r' x y := r (Quot.mk _ x) (Quot.mk _ y) ** 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := 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    Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =\n      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=\n    h _ _ r'ab\n  have h\u2081 : \u2200 u v : q.P.M, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v := by\n    intro u v cuv\n    apply Quot.sound\n    simp only\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; apply Quot.inductionOn (motive := _) c; clear c\n        intro c d; apply Quot.inductionOn (motive := _) d; 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, PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map,\n    abs_map, h\u2080]\n  rw [PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map] ** 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) this : IsPrecongr r' \u22a2 Mcongr x y ** refine' \u27e8r', this, rxy\u27e9 ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u22a2 IsPrecongr r' ** intro a b r'ab ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b \u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b) ** have h\u2080 :\n  Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =\n    Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=\n  h _ _ r'ab ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) \u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b) ** have h\u2081 : \u2200 u v : q.P.M, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v := by\n  intro u v cuv\n  apply Quot.sound\n  simp only\n  rw [Quot.sound cuv]\n  apply 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v \u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b) ** let f : Quot r \u2192 Quot r' :=\n  Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n    (by\n      intro c; apply Quot.inductionOn (motive := _) c; clear c\n      intro c d; apply Quot.inductionOn (motive := _) d; clear d\n      intro d rcd; apply Quot.sound; apply rcd) ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), 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), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) \u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b) ** have : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' := rfl ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), 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), 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 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b) ** rw [\u2190 this, PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map,\n  abs_map, h\u2080] ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), 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), 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 f <$> Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) =     abs ((f \u2218 Quot.mk r \u2218 Quot.mk Mcongr) <$> PFunctor.M.dest b) ** rw [PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map] ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) \u22a2 \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v ** intro u v cuv ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) u v : PFunctor.M (P F) cuv : Mcongr u v \u22a2 Quot.mk r' u = Quot.mk r' v ** apply Quot.sound ** 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) u v : PFunctor.M (P F) cuv : Mcongr u v \u22a2 r' u v ** simp only ** 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) u v : PFunctor.M (P F) cuv : Mcongr u v \u22a2 r (Quot.mk Mcongr u) (Quot.mk Mcongr v) ** rw [Quot.sound cuv] ** 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) u v : PFunctor.M (P F) cuv : Mcongr u v \u22a2 r (Quot.mk Mcongr v) (Quot.mk Mcongr v) ** apply 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 : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v \u22a2 \u2200 (a b : Cofix F), r a b \u2192 Quot.lift (Quot.mk r') h\u2081 a = Quot.lift (Quot.mk r') h\u2081 b ** intro c ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : Cofix F \u22a2 \u2200 (b : Cofix F), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b ** apply Quot.inductionOn (motive := _) c ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : Cofix F \u22a2 \u2200 (a : PFunctor.M (P F)) (b : Cofix F),     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 ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v \u22a2 \u2200 (a : PFunctor.M (P F)) (b : Cofix F),     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 ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : PFunctor.M (P F) d : Cofix F \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 (motive := _) d ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : PFunctor.M (P F) d : Cofix F \u22a2 \u2200 (a : PFunctor.M (P F)),     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 ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : PFunctor.M (P F) \u22a2 \u2200 (a : PFunctor.M (P F)),     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 ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c d : PFunctor.M (P F) 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 F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h' : \u2200 (x : Cofix F), r x x h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y x y : PFunctor.M (P F) rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : PFunctor.M (P F) r'ab : r' a b h\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) h\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c d : PFunctor.M (P F) rcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d) \u22a2 r' c d ** apply rcd ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.dvd_add_iff_right ** k m n : Nat h : k \u2223 m d : Nat x\u271d : k \u2223 k * d + n e : Nat he : k * d + n = k * e \u22a2 n = k * (e - d) ** rw [Nat.mul_sub_left_distrib, \u2190 he, Nat.add_sub_cancel_left] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrable_of_integrable_trim ** \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 H : Type u_5 inst\u271d : NormedAddCommGroup H m0 : MeasurableSpace \u03b1 \u03bc' : Measure \u03b1 f : \u03b1 \u2192 H hm : m \u2264 m0 hf_int : Integrable f \u22a2 Integrable f ** obtain \u27e8hf_meas_ae, hf\u27e9 := hf_int ** case 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 H : Type u_5 inst\u271d : NormedAddCommGroup H m0 : MeasurableSpace \u03b1 \u03bc' : Measure \u03b1 f : \u03b1 \u2192 H hm : m \u2264 m0 hf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm) hf : HasFiniteIntegral f \u22a2 Integrable f ** refine' \u27e8aestronglyMeasurable_of_aestronglyMeasurable_trim hm hf_meas_ae, _\u27e9 ** case 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 H : Type u_5 inst\u271d : NormedAddCommGroup H m0 : MeasurableSpace \u03b1 \u03bc' : Measure \u03b1 f : \u03b1 \u2192 H hm : m \u2264 m0 hf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm) hf : HasFiniteIntegral f \u22a2 HasFiniteIntegral f ** rw [HasFiniteIntegral] at hf \u22a2 ** case 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 H : Type u_5 inst\u271d : NormedAddCommGroup H m0 : MeasurableSpace \u03b1 \u03bc' : Measure \u03b1 f : \u03b1 \u2192 H hm : m \u2264 m0 hf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm) hf : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202Measure.trim \u03bc' hm < \u22a4 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc' < \u22a4 ** rwa [lintegral_trim_ae hm _] at 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\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 H : Type u_5 inst\u271d : NormedAddCommGroup H m0 : MeasurableSpace \u03b1 \u03bc' : Measure \u03b1 f : \u03b1 \u2192 H hm : m \u2264 m0 hf_meas_ae : AEStronglyMeasurable f (Measure.trim \u03bc' hm) hf : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202Measure.trim \u03bc' hm < \u22a4 \u22a2 AEMeasurable fun a => \u2191\u2016f a\u2016\u208a ** exact AEStronglyMeasurable.ennnorm hf_meas_ae ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.lintegral_join ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 \u222b\u207b (x : \u03b1), f x \u2202join m = \u222b\u207b (\u03bc : Measure \u03b1), \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2202m ** simp_rw [lintegral_eq_iSup_eapprox_lintegral hf, SimpleFunc.lintegral,\n  join_apply (SimpleFunc.measurableSet_preimage _ _)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),     (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192       (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192         \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m ** intro s f hf hm ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f\u271d : \u03b1 \u2192 \u211d\u22650\u221e hf\u271d : Measurable f\u271d s : \u2115 \u2192 Finset \u211d\u22650\u221e f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r) hm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc \u22a2 \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m ** rw [lintegral_iSup _ hm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f\u271d : \u03b1 \u2192 \u211d\u22650\u221e hf\u271d : Measurable f\u271d s : \u2115 \u2192 Finset \u211d\u22650\u221e f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r) hm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc \u22a2 \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u2a06 n, \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m  \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f\u271d : \u03b1 \u2192 \u211d\u22650\u221e hf\u271d : Measurable f\u271d s : \u2115 \u2192 Finset \u211d\u22650\u221e f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r) hm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc \u22a2 \u2200 (n : \u2115), Measurable fun \u03bc => \u2211 r in s n, r * f n r \u03bc ** swap ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f\u271d : \u03b1 \u2192 \u211d\u22650\u221e hf\u271d : Measurable f\u271d s : \u2115 \u2192 Finset \u211d\u22650\u221e f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r) hm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc \u22a2 \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u2a06 n, \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m ** congr ** case e_s \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f\u271d : \u03b1 \u2192 \u211d\u22650\u221e hf\u271d : Measurable f\u271d s : \u2115 \u2192 Finset \u211d\u22650\u221e f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r) hm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc \u22a2 (fun n => \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m) = fun n => \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m ** funext n ** case e_s.h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f\u271d : \u03b1 \u2192 \u211d\u22650\u221e hf\u271d : Measurable f\u271d s : \u2115 \u2192 Finset \u211d\u22650\u221e f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r) hm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc n : \u2115 \u22a2 \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (a : Measure \u03b1), \u2211 r in s n, r * f n r a \u2202m ** rw [lintegral_finset_sum (s n)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f this :   \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),     (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192       (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192         \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m \u22a2 \u2a06 n,       \u2211 x in SimpleFunc.range (SimpleFunc.eapprox f n),         x * \u222b\u207b (\u03bc : Measure \u03b1), \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {x}) \u2202m =     \u222b\u207b (\u03bc : Measure \u03b1),       \u2a06 n, \u2211 x in SimpleFunc.range (SimpleFunc.eapprox f n), x * \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {x}) \u2202m ** refine'\n  this (fun n => SimpleFunc.range (SimpleFunc.eapprox f n))\n    (fun n r \u03bc => \u03bc (SimpleFunc.eapprox f n \u207b\u00b9' {r})) _ _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f this :   \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),     (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192       (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192         \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m \u22a2 \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable ((fun n r \u03bc => \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {r})) n r) ** exact fun n r => measurable_coe (SimpleFunc.measurableSet_preimage _ _) ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f this :   \u2200 (s : \u2115 \u2192 Finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e),     (\u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r)) \u2192       (Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc) \u2192         \u2a06 n, \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u222b\u207b (\u03bc : Measure \u03b1), \u2a06 n, \u2211 r in s n, r * f n r \u03bc \u2202m \u22a2 Monotone fun n \u03bc =>     \u2211 r in (fun n => SimpleFunc.range (SimpleFunc.eapprox f n)) n,       r * (fun n r \u03bc => \u2191\u2191\u03bc (\u2191(SimpleFunc.eapprox f n) \u207b\u00b9' {r})) n r \u03bc ** exact fun n m h \u03bc => SimpleFunc.lintegral_mono (SimpleFunc.monotone_eapprox _ h) le_rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f\u271d : \u03b1 \u2192 \u211d\u22650\u221e hf\u271d : Measurable f\u271d s : \u2115 \u2192 Finset \u211d\u22650\u221e f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r) hm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc \u22a2 \u2200 (n : \u2115), Measurable fun \u03bc => \u2211 r in s n, r * f n r \u03bc ** exact fun n => Finset.measurable_sum _ fun r _ => (hf _ _).const_mul _ ** case e_s.h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f\u271d : \u03b1 \u2192 \u211d\u22650\u221e hf\u271d : Measurable f\u271d s : \u2115 \u2192 Finset \u211d\u22650\u221e f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r) hm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc n : \u2115 \u22a2 \u2211 r in s n, r * \u222b\u207b (\u03bc : Measure \u03b1), f n r \u03bc \u2202m = \u2211 b in s n, \u222b\u207b (a : Measure \u03b1), b * f n b a \u2202m ** simp_rw [lintegral_const_mul _ (hf _ _)] ** case e_s.h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure (Measure \u03b1) f\u271d : \u03b1 \u2192 \u211d\u22650\u221e hf\u271d : Measurable f\u271d s : \u2115 \u2192 Finset \u211d\u22650\u221e f : \u2115 \u2192 \u211d\u22650\u221e \u2192 Measure \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115) (r : \u211d\u22650\u221e), Measurable (f n r) hm : Monotone fun n \u03bc => \u2211 r in s n, r * f n r \u03bc n : \u2115 \u22a2 \u2200 (b : \u211d\u22650\u221e), b \u2208 s n \u2192 Measurable fun a => b * f n b a ** exact fun r _ => (hf _ _).const_mul _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.bind_bind ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 m : Measure \u03b1 f : \u03b1 \u2192 Measure \u03b2 g : \u03b2 \u2192 Measure \u03b3 hf : Measurable f hg : Measurable g \u22a2 bind (bind m f) g = bind m fun a => bind (f a) g ** ext1 s hs ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 m : Measure \u03b1 f : \u03b1 \u2192 Measure \u03b2 g : \u03b2 \u2192 Measure \u03b3 hf : Measurable f hg : Measurable g s : Set \u03b3 hs : MeasurableSet s \u22a2 \u2191\u2191(bind (bind m f) g) s = \u2191\u2191(bind m fun a => bind (f a) g) s ** erw [bind_apply hs hg, bind_apply hs ((measurable_bind' hg).comp hf),\n  lintegral_bind hf ((measurable_coe hs).comp hg)] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 m : Measure \u03b1 f : \u03b1 \u2192 Measure \u03b2 g : \u03b2 \u2192 Measure \u03b3 hf : Measurable f hg : Measurable g s : Set \u03b3 hs : MeasurableSet s \u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (x : \u03b2), ((fun \u03bc => \u2191\u2191\u03bc s) \u2218 g) x \u2202f a \u2202m = \u222b\u207b (a : \u03b1), \u2191\u2191(((fun m => bind m g) \u2218 f) a) s \u2202m ** conv_rhs => enter [2, a]; erw [bind_apply hs hg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.count_injective_image ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 s\u271d : Set \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b2 \u2192 \u03b1 hf : Function.Injective f s : Set \u03b2 \u22a2 \u2191\u2191count (f '' s) = \u2191\u2191count s ** by_cases hs : s.Finite ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 s\u271d : Set \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b2 \u2192 \u03b1 hf : Function.Injective f s : Set \u03b2 hs : \u00acSet.Finite s \u22a2 \u2191\u2191count (f '' s) = \u2191\u2191count s ** rw [count_apply_infinite hs] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 s\u271d : Set \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b2 \u2192 \u03b1 hf : Function.Injective f s : Set \u03b2 hs : \u00acSet.Finite s \u22a2 \u2191\u2191count (f '' s) = \u22a4 ** rw [\u2190 finite_image_iff <| hf.injOn _] at hs ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 s\u271d : Set \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b2 \u2192 \u03b1 hf : Function.Injective f s : Set \u03b2 hs : \u00acSet.Finite (f '' s) \u22a2 \u2191\u2191count (f '' s) = \u22a4 ** rw [count_apply_infinite hs] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 s\u271d : Set \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b2 \u2192 \u03b1 hf : Function.Injective f s : Set \u03b2 hs : Set.Finite s \u22a2 \u2191\u2191count (f '' s) = \u2191\u2191count s ** exact count_injective_image' hf hs.measurableSet (Finite.image f hs).measurableSet ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AnalyticSet.iUnion ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (n : \u03b9), AnalyticSet (s n) \u22a2 AnalyticSet (\u22c3 n, s n) ** choose \u03b2 h\u03b2 h'\u03b2 f f_cont f_range using fun n =>\n  analyticSet_iff_exists_polishSpace_range.1 (hs n) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (n : \u03b9), AnalyticSet (s n) \u03b2 : \u03b9 \u2192 Type h\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n) h'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n) f : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1 f_cont : \u2200 (n : \u03b9), Continuous (f n) f_range : \u2200 (n : \u03b9), range (f n) = s n \u22a2 AnalyticSet (\u22c3 n, s n) ** let \u03b3 := \u03a3n, \u03b2 n ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (n : \u03b9), AnalyticSet (s n) \u03b2 : \u03b9 \u2192 Type h\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n) h'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n) f : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1 f_cont : \u2200 (n : \u03b9), Continuous (f n) f_range : \u2200 (n : \u03b9), range (f n) = s n \u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n \u22a2 AnalyticSet (\u22c3 n, s n) ** let F : \u03b3 \u2192 \u03b1 := fun \u27e8n, x\u27e9 \u21a6 f n x ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (n : \u03b9), AnalyticSet (s n) \u03b2 : \u03b9 \u2192 Type h\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n) h'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n) f : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1 f_cont : \u2200 (n : \u03b9), Continuous (f n) f_range : \u2200 (n : \u03b9), range (f n) = s n \u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n F : \u03b3 \u2192 \u03b1 :=   fun x =>     match x with     | { fst := n, snd := x } => f n x \u22a2 AnalyticSet (\u22c3 n, s n) ** have F_cont : Continuous F := continuous_sigma f_cont ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (n : \u03b9), AnalyticSet (s n) \u03b2 : \u03b9 \u2192 Type h\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n) h'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n) f : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1 f_cont : \u2200 (n : \u03b9), Continuous (f n) f_range : \u2200 (n : \u03b9), range (f n) = s n \u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n F : \u03b3 \u2192 \u03b1 :=   fun x =>     match x with     | { fst := n, snd := x } => f n x F_cont : Continuous F \u22a2 AnalyticSet (\u22c3 n, s n) ** have F_range : range F = \u22c3 n, s n := by\n  simp only [range_sigma_eq_iUnion_range, f_range] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (n : \u03b9), AnalyticSet (s n) \u03b2 : \u03b9 \u2192 Type h\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n) h'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n) f : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1 f_cont : \u2200 (n : \u03b9), Continuous (f n) f_range : \u2200 (n : \u03b9), range (f n) = s n \u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n F : \u03b3 \u2192 \u03b1 :=   fun x =>     match x with     | { fst := n, snd := x } => f n x F_cont : Continuous F F_range : range F = \u22c3 n, s n \u22a2 AnalyticSet (\u22c3 n, s n) ** rw [\u2190 F_range] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (n : \u03b9), AnalyticSet (s n) \u03b2 : \u03b9 \u2192 Type h\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n) h'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n) f : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1 f_cont : \u2200 (n : \u03b9), Continuous (f n) f_range : \u2200 (n : \u03b9), range (f n) = s n \u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n F : \u03b3 \u2192 \u03b1 :=   fun x =>     match x with     | { fst := n, snd := x } => f n x F_cont : Continuous F F_range : range F = \u22c3 n, s n \u22a2 AnalyticSet (range F) ** exact analyticSet_range_of_polishSpace F_cont ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : TopologicalSpace \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (n : \u03b9), AnalyticSet (s n) \u03b2 : \u03b9 \u2192 Type h\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n) h'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n) f : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1 f_cont : \u2200 (n : \u03b9), Continuous (f n) f_range : \u2200 (n : \u03b9), range (f n) = s n \u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n F : \u03b3 \u2192 \u03b1 :=   fun x =>     match x with     | { fst := n, snd := x } => f n x F_cont : Continuous F \u22a2 range F = \u22c3 n, s n ** simp only [range_sigma_eq_iUnion_range, f_range] ** Qed",
        "informal": ""
    },
    {
        "formal": "WellFounded.fix_eq_fixC ** \u22a2 @fix = @WellFounded.fixC ** funext \u03b1 C r hwf F x ** case h.h.h.h.h.h \u03b1 : Sort u_1 C : \u03b1 \u2192 Sort u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop hwf : WellFounded r F : (x : \u03b1) \u2192 ((y : \u03b1) \u2192 r y x \u2192 C y) \u2192 C x x : \u03b1 \u22a2 fix hwf F x = WellFounded.fixC hwf F x ** rw [fix, fixF_eq_fixFC, fixC] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_countable ** \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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f \u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a ** have hi : Countable { x // x \u2208 s } := Iff.mpr countable_coe_iff hs ** \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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } hf' : Integrable fun x => f \u2191x \u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a ** rw [set_integral_eq_subtype' hs.measurableSet, integral_countable' 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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } hf' : Integrable fun x => f \u2191x \u22a2 \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191(Measure.comap Subtype.val \u03bc) {a}) \u2022 f \u2191a =     \u2211' (a : \u2191s), ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a ** congr 1 with a : 1 ** case e_f.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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } hf' : Integrable fun x => f \u2191x a : \u2191s \u22a2 ENNReal.toReal (\u2191\u2191(Measure.comap Subtype.val \u03bc) {a}) \u2022 f \u2191a = ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a ** rw [Measure.comap_apply Subtype.val Subtype.coe_injective\n  (fun s' hs' => MeasurableSet.subtype_image (Countable.measurableSet hs) hs') _\n  (MeasurableSet.singleton a)] ** case e_f.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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } hf' : Integrable fun x => f \u2191x a : \u2191s \u22a2 ENNReal.toReal (\u2191\u2191\u03bc (Subtype.val '' {a})) \u2022 f \u2191a = ENNReal.toReal (\u2191\u2191\u03bc {\u2191a}) \u2022 f \u2191a ** simp ** \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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } \u22a2 Integrable fun x => f \u2191x ** rw [\u2190 map_comap_subtype_coe, integrable_map_measure] 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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable (f \u2218 Subtype.val) hi : Countable { x // x \u2208 s } \u22a2 Integrable fun x => f \u2191x  case 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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } \u22a2 AEStronglyMeasurable f (Measure.map Subtype.val (Measure.comap Subtype.val \u03bc))  case 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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } \u22a2 AEMeasurable Subtype.val  case hs \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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } \u22a2 MeasurableSet s ** apply hf ** case 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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } \u22a2 AEStronglyMeasurable f (Measure.map Subtype.val (Measure.comap Subtype.val \u03bc)) ** exact Integrable.aestronglyMeasurable hf ** case 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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } \u22a2 AEMeasurable Subtype.val ** exact Measurable.aemeasurable measurable_subtype_coe ** case hs \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 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : Set.Countable s hf : Integrable f hi : Countable { x // x \u2208 s } \u22a2 MeasurableSet s ** exact Countable.measurableSet hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.simpleFunc.neg_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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc E p \u03bc } \u22a2 \u2191(toSimpleFunc (-f)) =\u1d50[\u03bc] -\u2191(toSimpleFunc f) ** filter_upwards [toSimpleFunc_eq_toFun (-f), toSimpleFunc_eq_toFun f,\n  Lp.coeFn_neg (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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc E p \u03bc } a\u271d : \u03b1 \u22a2 \u2191(toSimpleFunc (-f)) a\u271d = \u2191\u2191\u2191(-f) a\u271d \u2192     \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(-\u2191f) a\u271d = (-\u2191\u2191\u2191f) a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = (-\u2191(toSimpleFunc f)) a\u271d ** simp only [Pi.neg_apply, AddSubgroup.coe_neg] ** 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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc E p \u03bc } a\u271d : \u03b1 \u22a2 \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d \u2192     \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = -\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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc E p \u03bc } a\u271d : \u03b1 h\u271d : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d h : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u22a2 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\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\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc E p \u03bc } a\u271d : \u03b1 h\u271d\u00b9 : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d h\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d h : \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u22a2 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d ** rw [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.map_some_eraseNone ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq (Option \u03b1) s : Finset (Option \u03b1) \u22a2 map Embedding.some (\u2191eraseNone s) = erase s none ** rw [map_eq_image, Embedding.some_apply, image_some_eraseNone] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.perm_of_nodup_nodup_toFinset_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 l l' : List \u03b1 a : \u03b1 hl : Nodup l hl' : Nodup l' h : toFinset l = toFinset l' \u22a2 l ~ l' ** rw [\u2190 Multiset.coe_eq_coe] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 l l' : List \u03b1 a : \u03b1 hl : Nodup l hl' : Nodup l' h : toFinset l = toFinset l' \u22a2 \u2191l = \u2191l' ** exact Multiset.Nodup.toFinset_inj hl hl' h ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.vars_bind\u2081 ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 vars (\u2191(bind\u2081 f) \u03c6) \u2286 Finset.biUnion (vars \u03c6) fun i => vars (f i) ** calc\n  (bind\u2081 f \u03c6).vars = (\u03c6.support.sum fun x : \u03c3 \u2192\u2080 \u2115 => (bind\u2081 f) (monomial x (coeff x \u03c6))).vars :=\n    by rw [\u2190 AlgHom.map_sum, \u2190 \u03c6.as_sum]\n  _ \u2264 \u03c6.support.biUnion fun i : \u03c3 \u2192\u2080 \u2115 => ((bind\u2081 f) (monomial i (coeff i \u03c6))).vars :=\n    (vars_sum_subset _ _)\n  _ = \u03c6.support.biUnion fun d : \u03c3 \u2192\u2080 \u2115 => vars (C (coeff d \u03c6) * \u220f i in d.support, f i ^ d i) := by\n    simp only [bind\u2081_monomial]\n  _ \u2264 \u03c6.support.biUnion fun d : \u03c3 \u2192\u2080 \u2115 => d.support.biUnion fun i => vars (f i) := ?_\n  _ \u2264 \u03c6.vars.biUnion fun i : \u03c3 => vars (f i) := ?_ ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 vars (\u2191(bind\u2081 f) \u03c6) = vars (\u2211 x in support \u03c6, \u2191(bind\u2081 f) (\u2191(monomial x) (coeff x \u03c6))) ** rw [\u2190 AlgHom.map_sum, \u2190 \u03c6.as_sum] ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 (Finset.biUnion (support \u03c6) fun i => vars (\u2191(bind\u2081 f) (\u2191(monomial i) (coeff i \u03c6)))) =     Finset.biUnion (support \u03c6) fun d => vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i) ** simp only [bind\u2081_monomial] ** case calc_1 \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 (Finset.biUnion (support \u03c6) fun d => vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i)) \u2264     Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i) ** apply Finset.biUnion_mono ** case calc_1.h \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 \u2200 (a : \u03c3 \u2192\u2080 \u2115),     a \u2208 support \u03c6 \u2192 vars (\u2191C (coeff a \u03c6) * \u220f i in a.support, f i ^ \u2191a i) \u2286 Finset.biUnion a.support fun i => vars (f i) ** intro d _hd ** case calc_1.h \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 _hd : d \u2208 support \u03c6 \u22a2 vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i) \u2286 Finset.biUnion d.support fun i => vars (f i) ** calc\n  vars (C (coeff d \u03c6) * \u220f i : \u03c3 in d.support, f i ^ d i) \u2264\n      (C (coeff d \u03c6)).vars \u222a (\u220f i : \u03c3 in d.support, f i ^ d i).vars :=\n    vars_mul _ _\n  _ \u2264 (\u220f i : \u03c3 in d.support, f i ^ d i).vars := by\n    simp only [Finset.empty_union, vars_C, Finset.le_iff_subset, Finset.Subset.refl]\n  _ \u2264 d.support.biUnion fun i : \u03c3 => vars (f i ^ d i) := (vars_prod _)\n  _ \u2264 d.support.biUnion fun i : \u03c3 => (f i).vars := ?_ ** case calc_1.h \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 _hd : d \u2208 support \u03c6 \u22a2 (Finset.biUnion d.support fun i => vars (f i ^ \u2191d i)) \u2264 Finset.biUnion d.support fun i => vars (f i) ** apply Finset.biUnion_mono ** case calc_1.h.h \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 _hd : d \u2208 support \u03c6 \u22a2 \u2200 (a : \u03c3), a \u2208 d.support \u2192 vars (f a ^ \u2191d a) \u2286 vars (f a) ** intro i _hi ** case calc_1.h.h \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 _hd : d \u2208 support \u03c6 i : \u03c3 _hi : i \u2208 d.support \u22a2 vars (f i ^ \u2191d i) \u2286 vars (f i) ** apply vars_pow ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 _hd : d \u2208 support \u03c6 \u22a2 vars (\u2191C (coeff d \u03c6)) \u222a vars (\u220f i in d.support, f i ^ \u2191d i) \u2264 vars (\u220f i in d.support, f i ^ \u2191d i) ** simp only [Finset.empty_union, vars_C, Finset.le_iff_subset, Finset.Subset.refl] ** case calc_2 \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 (Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)) \u2264     Finset.biUnion (vars \u03c6) fun i => vars (f i) ** intro j ** case calc_2 \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R j : \u03c4 \u22a2 (j \u2208 Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)) \u2192     j \u2208 Finset.biUnion (vars \u03c6) fun i => vars (f i) ** simp_rw [Finset.mem_biUnion] ** case calc_2 \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R j : \u03c4 \u22a2 (\u2203 a, a \u2208 support \u03c6 \u2227 \u2203 a_1, a_1 \u2208 a.support \u2227 j \u2208 vars (f a_1)) \u2192 \u2203 a, a \u2208 vars \u03c6 \u2227 j \u2208 vars (f a) ** rintro \u27e8d, hd, \u27e8i, hi, hj\u27e9\u27e9 ** case calc_2.intro.intro.intro.intro \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R inst\u271d : DecidableEq \u03c4 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R j : \u03c4 d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support \u03c6 i : \u03c3 hi : i \u2208 d.support hj : j \u2208 vars (f i) \u22a2 \u2203 a, a \u2208 vars \u03c6 \u2227 j \u2208 vars (f a) ** exact \u27e8i, (mem_vars _).mpr \u27e8d, hd, hi\u27e9, hj\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.foldlM_eq_foldlM_data ** m : Type u_1 \u2192 Type u_2 \u03b2 : Type u_1 \u03b1 : Type u_3 inst\u271d : Monad m f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2 init : \u03b2 arr : Array \u03b1 \u22a2 foldlM f init arr 0 (size arr) = List.foldlM f init arr.data ** simp [foldlM, foldlM_eq_foldlM_data.aux] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.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' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9) \u03bc ** 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 : AEStronglyMeasurable f \u03bc \u22a2 AEStronglyMeasurable' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9) \u03bc ** have h := 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 : m \u2293 m\u03a9 \u2264 m\u03a9)) ** \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 h :   AEStronglyMeasurable' (MeasurableSpace.comap id (m \u2293 m\u03a9))     (fun a => \u222b (y : \u03a9), f (id a, y).2 \u2202\u2191(condDistrib id id \u03bc) (id a)) \u03bc \u22a2 AEStronglyMeasurable' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9) \u03bc ** rw [MeasurableSpace.comap_id] at h ** \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 h : AEStronglyMeasurable' (m \u2293 m\u03a9) (fun a => \u222b (y : \u03a9), f (id a, y).2 \u2202\u2191(condDistrib id id \u03bc) (id a)) \u03bc \u22a2 AEStronglyMeasurable' m (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9) \u03bc ** exact AEStronglyMeasurable'.mono h inf_le_left ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_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)) \u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) ** rw [tendsto_iff_forall_testAgainstNN_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 (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) \u22a2 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f)) ** exact fun f =>\n  tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass \u03bcs_lim mass_lim f ** Qed",
        "informal": ""
    },
    {
        "formal": "measurableSet_setOf ** \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 inst\u271d : MeasurableSpace \u03b1 p : \u03b1 \u2192 Prop h : MeasurableSet {a | p a} \u22a2 MeasurableSet (p \u207b\u00b9' {True}) ** simpa only [preimage_singleton_true] ** \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 inst\u271d : MeasurableSpace \u03b1 p : \u03b1 \u2192 Prop h : Measurable p \u22a2 MeasurableSet {a | p a} ** simpa using h (measurableSet_singleton True) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEStronglyMeasurable.integral_condKernel ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2079 : TopologicalSpace \u03a9 inst\u271d\u2078 : PolishSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : BorelSpace \u03a9 inst\u271d\u2075 : Nonempty \u03a9 \u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d E : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d : IsFiniteMeasure \u03c1 f : \u03b1 \u00d7 \u03a9 \u2192 E hf : AEStronglyMeasurable f \u03c1 \u22a2 AEStronglyMeasurable (fun x => \u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x) (Measure.fst \u03c1) ** rw [measure_eq_compProd \u03c1] at hf ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2079 : TopologicalSpace \u03a9 inst\u271d\u2078 : PolishSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : BorelSpace \u03a9 inst\u271d\u2075 : Nonempty \u03a9 \u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d E : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d : IsFiniteMeasure \u03c1 f : \u03b1 \u00d7 \u03a9 \u2192 E hf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) \u22a2 AEStronglyMeasurable (fun x => \u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x) (Measure.fst \u03c1) ** exact AEStronglyMeasurable.integral_kernel_compProd hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.prehaar_sup_eq ** 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\u2081 K\u2082 : Compacts G hU : Set.Nonempty (interior U) h : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9) \u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) = prehaar (\u2191K\u2080) U K\u2081 + 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 K\u2081 K\u2082 : Compacts G hU : Set.Nonempty (interior U) h : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9) \u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) / \u2191(index (\u2191K\u2080) U) = \u2191(index (\u2191K\u2081) U) / \u2191(index (\u2191K\u2080) U) + \u2191(index (\u2191K\u2082) U) / \u2191(index (\u2191K\u2080) U) ** rw [div_add_div_same] ** 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\u2081 K\u2082 : Compacts G hU : Set.Nonempty (interior U) h : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9) \u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) / \u2191(index (\u2191K\u2080) U) = (\u2191(index (\u2191K\u2081) U) + \u2191(index (\u2191K\u2082) U)) / \u2191(index (\u2191K\u2080) U) ** refine congr_arg (fun x : \u211d => x / index K\u2080 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 K\u2081 K\u2082 : Compacts G hU : Set.Nonempty (interior U) h : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9) \u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) = \u2191(index (\u2191K\u2081) U) + \u2191(index (\u2191K\u2082) U) ** exact_mod_cast index_union_eq K\u2081 K\u2082 hU h ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.some_mul_some ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Mul \u03b1 a b : \u03b1 \u22a2 some a * some b = some (a * b) ** simp [mul_def] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.withDensity_smul_measure ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 withDensity (r \u2022 \u03bc) f = r \u2022 withDensity \u03bc f ** ext s hs ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(withDensity (r \u2022 \u03bc) f) s = \u2191\u2191(r \u2022 withDensity \u03bc f) s ** rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,\n  smul_eq_mul, set_lintegral_smul_measure] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Nonempty.cons_induction ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1\u271d s\u271d t u v : Finset \u03b1\u271d a b : \u03b1\u271d \u03b1 : Type u_4 p : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop h\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a}) h\u2081 :   \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),     p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h)) s : Finset \u03b1 hs : Finset.Nonempty s \u22a2 p s hs ** induction' s using Finset.cons_induction with a t ha h ** case cons \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1\u271d s\u271d t\u271d u v : Finset \u03b1\u271d a\u271d b : \u03b1\u271d \u03b1 : Type u_4 p : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop h\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a}) h\u2081 :   \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),     p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h)) s : Finset \u03b1 hs\u271d : Finset.Nonempty s a : \u03b1 t : Finset \u03b1 ha : \u00aca \u2208 t h : \u2200 (hs : Finset.Nonempty t), p t hs hs : Finset.Nonempty (cons a t ha) \u22a2 p (cons a t ha) hs ** obtain rfl | ht := t.eq_empty_or_nonempty ** case empty \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1\u271d s\u271d t u v : Finset \u03b1\u271d a b : \u03b1\u271d \u03b1 : Type u_4 p : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop h\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a}) h\u2081 :   \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),     p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h)) s : Finset \u03b1 hs\u271d : Finset.Nonempty s hs : Finset.Nonempty \u2205 \u22a2 p \u2205 hs ** exact (not_nonempty_empty hs).elim ** case cons.inl \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1\u271d s\u271d t u v : Finset \u03b1\u271d a\u271d b : \u03b1\u271d \u03b1 : Type u_4 p : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop h\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a}) h\u2081 :   \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),     p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h)) s : Finset \u03b1 hs\u271d : Finset.Nonempty s a : \u03b1 ha : \u00aca \u2208 \u2205 h : \u2200 (hs : Finset.Nonempty \u2205), p \u2205 hs hs : Finset.Nonempty (cons a \u2205 ha) \u22a2 p (cons a \u2205 ha) hs ** exact h\u2080 a ** case cons.inr \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1\u271d s\u271d t\u271d u v : Finset \u03b1\u271d a\u271d b : \u03b1\u271d \u03b1 : Type u_4 p : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop h\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a}) h\u2081 :   \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),     p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h)) s : Finset \u03b1 hs\u271d : Finset.Nonempty s a : \u03b1 t : Finset \u03b1 ha : \u00aca \u2208 t h : \u2200 (hs : Finset.Nonempty t), p t hs hs : Finset.Nonempty (cons a t ha) ht : Finset.Nonempty t \u22a2 p (cons a t ha) hs ** exact h\u2081 t ha ht (h ht) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.disjSups_assoc ** 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 : DistribLattice \u03b1 inst\u271d\u00b9 : OrderBot \u03b1 inst\u271d : DecidableRel Disjoint s t u v : Finset \u03b1 \u22a2 \u2200 (s t u : Finset \u03b1), s \u25cb t \u25cb u = s \u25cb (t \u25cb u) ** refine' associative_of_commutative_of_le disjSups_comm _ ** 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 : DistribLattice \u03b1 inst\u271d\u00b9 : OrderBot \u03b1 inst\u271d : DecidableRel Disjoint s t u v : Finset \u03b1 \u22a2 \u2200 (a b c : Finset \u03b1), a \u25cb b \u25cb c \u2264 a \u25cb (b \u25cb c) ** simp only [le_eq_subset, disjSups_subset_iff, 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 : DistribLattice \u03b1 inst\u271d\u00b9 : OrderBot \u03b1 inst\u271d : DecidableRel Disjoint s t u v : Finset \u03b1 \u22a2 \u2200 (a b c : Finset \u03b1) (a_1 : \u03b1),     (\u2203 a_2, a_2 \u2208 a \u2227 \u2203 b_1, b_1 \u2208 b \u2227 Disjoint a_2 b_1 \u2227 a_2 \u2294 b_1 = a_1) \u2192       \u2200 (b_1 : \u03b1),         b_1 \u2208 c \u2192           Disjoint a_1 b_1 \u2192             \u2203 a_5,               a_5 \u2208 a \u2227                 \u2203 b_2, (\u2203 a, a \u2208 b \u2227 \u2203 b, b \u2208 c \u2227 Disjoint a b \u2227 a \u2294 b = b_2) \u2227 Disjoint a_5 b_2 \u2227 a_5 \u2294 b_2 = a_1 \u2294 b_1 ** rintro s t u _ \u27e8a, ha, b, hb, hab, rfl\u27e9 c hc habc ** case intro.intro.intro.intro.intro 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 : DistribLattice \u03b1 inst\u271d\u00b9 : OrderBot \u03b1 inst\u271d : DecidableRel Disjoint s\u271d t\u271d u\u271d v s t u : Finset \u03b1 a : \u03b1 ha : a \u2208 s b : \u03b1 hb : b \u2208 t hab : Disjoint a b c : \u03b1 hc : c \u2208 u habc : Disjoint (a \u2294 b) c \u22a2 \u2203 a_1,     a_1 \u2208 s \u2227 \u2203 b_1, (\u2203 a, a \u2208 t \u2227 \u2203 b, b \u2208 u \u2227 Disjoint a b \u2227 a \u2294 b = b_1) \u2227 Disjoint a_1 b_1 \u2227 a_1 \u2294 b_1 = a \u2294 b \u2294 c ** rw [disjoint_sup_left] at habc ** case intro.intro.intro.intro.intro 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 : DistribLattice \u03b1 inst\u271d\u00b9 : OrderBot \u03b1 inst\u271d : DecidableRel Disjoint s\u271d t\u271d u\u271d v s t u : Finset \u03b1 a : \u03b1 ha : a \u2208 s b : \u03b1 hb : b \u2208 t hab : Disjoint a b c : \u03b1 hc : c \u2208 u habc : Disjoint a c \u2227 Disjoint b c \u22a2 \u2203 a_1,     a_1 \u2208 s \u2227 \u2203 b_1, (\u2203 a, a \u2208 t \u2227 \u2203 b, b \u2208 u \u2227 Disjoint a b \u2227 a \u2294 b = b_1) \u2227 Disjoint a_1 b_1 \u2227 a_1 \u2294 b_1 = a \u2294 b \u2294 c ** exact \u27e8a, ha, _, \u27e8b, hb, c, hc, habc.2, rfl\u27e9, hab.sup_right habc.1, sup_assoc.symm\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_smul_measure_of_ne_zero_of_ne_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 : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192 F c : \u211d\u22650\u221e \u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc ** simp_rw [snorm_eq_snorm' hp_ne_zero hp_ne_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 : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192 F c : \u211d\u22650\u221e \u22a2 snorm' f (ENNReal.toReal p) (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm' f (ENNReal.toReal p) \u03bc ** rw [snorm'_smul_measure 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 hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192 F c : \u211d\u22650\u221e \u22a2 c ^ (1 / ENNReal.toReal p) * snorm' f (ENNReal.toReal p) \u03bc =     c ^ ENNReal.toReal (1 / p) \u2022 snorm' f (ENNReal.toReal p) \u03bc ** congr ** case e_a.e_a \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 hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192 F c : \u211d\u22650\u221e \u22a2 1 / ENNReal.toReal p = ENNReal.toReal (1 / p) ** simp_rw [one_div] ** case e_a.e_a \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 hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192 F c : \u211d\u22650\u221e \u22a2 (ENNReal.toReal p)\u207b\u00b9 = ENNReal.toReal p\u207b\u00b9 ** rw [ENNReal.toReal_inv] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_singleton_inter ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 \u22a2 ncard ({a} \u2229 s) \u2264 1 ** rw [\u2190Nat.cast_le (\u03b1 := \u2115\u221e), (toFinite _).cast_ncard_eq, Nat.cast_one] ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 \u22a2 encard ({a} \u2229 s) \u2264 1 ** apply encard_singleton_inter ** Qed",
        "informal": ""
    },
    {
        "formal": "iUnion_Ici_eq_Ici_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 has_least_elem : \u2a05 i, f i \u2208 range f \u22a2 \u22c3 i, Ici (f i) = Ici (\u2a05 i, f i) ** simp only [\u2190 IsGLB.biUnion_Ici_eq_Ici (@isGLB_iInf _ _ _ f) has_least_elem, mem_range,\n  iUnion_exists, iUnion_iUnion_eq'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.sup ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b3 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 f g : \u03b9 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \u2131 \u03bc hg : Submartingale g \u2131 \u03bc \u22a2 Submartingale (f \u2294 g) \u2131 \u03bc ** refine' \u27e8fun i => @StronglyMeasurable.sup _ _ _ _ (\u2131 i) _ _ _ (hf.adapted i) (hg.adapted i),\n  fun i j hij => _, fun i => Integrable.sup (hf.integrable _) (hg.integrable _)\u27e9 ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b3 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 f g : \u03b9 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \u2131 \u03bc hg : Submartingale g \u2131 \u03bc i j : \u03b9 hij : i \u2264 j \u22a2 (f \u2294 g) i \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i] ** refine' EventuallyLE.sup_le _ _ ** case refine'_1 \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b3 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 f g : \u03b9 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \u2131 \u03bc hg : Submartingale g \u2131 \u03bc i j : \u03b9 hij : i \u2264 j \u22a2 (fun i_1 => f i i_1) \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i] ** exact EventuallyLE.trans (hf.2.1 i j hij)\n  (condexp_mono (hf.integrable _) (Integrable.sup (hf.integrable j) (hg.integrable j))\n    (eventually_of_forall fun x => le_max_left _ _)) ** case refine'_2 \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b3 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 f g : \u03b9 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \u2131 \u03bc hg : Submartingale g \u2131 \u03bc i j : \u03b9 hij : i \u2264 j \u22a2 (fun i_1 => g i i_1) \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i] ** exact EventuallyLE.trans (hg.2.1 i j hij)\n  (condexp_mono (hg.integrable _) (Integrable.sup (hf.integrable j) (hg.integrable j))\n    (eventually_of_forall fun x => le_max_right _ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Ioc_eq_cons_Ioo ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 h : a < b \u22a2 Ioc a b = cons b (Ioo a b) (_ : \u00acb \u2208 Ioo a b) ** classical rw [cons_eq_insert, Ioo_insert_right h] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 h : a < b \u22a2 Ioc a b = cons b (Ioo a b) (_ : \u00acb \u2208 Ioo a b) ** rw [cons_eq_insert, Ioo_insert_right h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.cast_int_covby_iff ** a b : \u2115 \u22a2 \u2191a \u22d6 \u2191b \u2194 a \u22d6 b ** rw [Nat.covby_iff_succ_eq, Int.covby_iff_succ_eq] ** a b : \u2115 \u22a2 \u2191a + 1 = \u2191b \u2194 a + 1 = b ** exact Int.coe_nat_inj' ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.mkRat_mul_right ** n : Int d a : Nat a0 : a \u2260 0 \u22a2 mkRat (n * \u2191a) (d * a) = mkRat n d ** rw [\u2190 mkRat_mul_left (d := d) a0] ** n : Int d a : Nat a0 : a \u2260 0 \u22a2 mkRat (n * \u2191a) (d * a) = mkRat (\u2191a * n) (a * d) ** congr 1 <;> [apply Int.mul_comm; apply Nat.mul_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.DList.toList_push ** \u03b1 : Type u x : \u03b1 l : DList \u03b1 \u22a2 toList (push l x) = toList l ++ [x] ** cases' l with _ l_invariant ** case mk \u03b1 : Type u x : \u03b1 apply\u271d : List \u03b1 \u2192 List \u03b1 l_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l \u22a2 toList (push { apply := apply\u271d, invariant := l_invariant } x) =     toList { apply := apply\u271d, invariant := l_invariant } ++ [x] ** simp ** case mk \u03b1 : Type u x : \u03b1 apply\u271d : List \u03b1 \u2192 List \u03b1 l_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l \u22a2 apply\u271d [x] = apply\u271d [] ++ [x] ** rw [l_invariant] ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.add_succ ** \u03b1 : Type u_1 n : Num \u22a2 0 + succ n = succ (0 + n) ** simp [zero_add] ** \u03b1 : Type u_1 p : PosNum \u22a2 pos (p + 1) = succ (pos p + 0) ** rw [PosNum.add_one, add_zero] ** \u03b1 : Type u_1 p : PosNum \u22a2 pos (PosNum.succ p) = succ (pos p) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.integral_const ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedAddCommGroup F' inst\u271d : NormedSpace \u211d F' m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 y : F \u22a2 integral \u03bc (const \u03b1 y) = ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 y ** classical\ncalc\n  (const \u03b1 y).integral \u03bc = \u2211 z in {y}, (\u03bc (const \u03b1 y \u207b\u00b9' {z})).toReal \u2022 z :=\n    integral_eq_sum_of_subset <| (filter_subset _ _).trans (range_const_subset _ _)\n  _ = (\u03bc univ).toReal \u2022 y := by simp [Set.preimage] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedAddCommGroup F' inst\u271d : NormedSpace \u211d F' m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 y : F \u22a2 integral \u03bc (const \u03b1 y) = ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 y ** calc\n  (const \u03b1 y).integral \u03bc = \u2211 z in {y}, (\u03bc (const \u03b1 y \u207b\u00b9' {z})).toReal \u2022 z :=\n    integral_eq_sum_of_subset <| (filter_subset _ _).trans (range_const_subset _ _)\n  _ = (\u03bc univ).toReal \u2022 y := by simp [Set.preimage] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : NormedAddCommGroup F' inst\u271d : NormedSpace \u211d F' m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 y : F \u22a2 \u2211 z in {y}, ENNReal.toReal (\u2191\u2191\u03bc (\u2191(const \u03b1 y) \u207b\u00b9' {z})) \u2022 z = ENNReal.toReal (\u2191\u2191\u03bc Set.univ) \u2022 y ** simp [Set.preimage] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpL2_indicator_nonneg ** \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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) \u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) ** have h : AEStronglyMeasurable' m (condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) \u03bc :=\n  aeStronglyMeasurable'_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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc \u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) ** refine' EventuallyLE.trans_eq _ h.ae_eq_mk.symm ** \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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc \u22a2 0 \u2264\u1d50[\u03bc] AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h ** refine' @ae_le_of_ae_le_trim _ _ _ _ _ _ hm (0 : \u03b1 \u2192 \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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc \u22a2 0 \u2264\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h ** refine' ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite _ _ ** 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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc \u22a2 \u2200 (s_1 : Set \u03b1),     MeasurableSet s_1 \u2192       \u2191\u2191(Measure.trim \u03bc hm) s_1 < \u22a4 \u2192         IntegrableOn (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) s_1 ** rintro t - - ** 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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 \u22a2 IntegrableOn (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) t ** refine @Integrable.integrableOn _ _ m _ _ _ _ ?_ ** 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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 \u22a2 Integrable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) ** refine' Integrable.trim hm _ _ ** case refine'_1.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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 \u22a2 Integrable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) ** rw [integrable_congr h.ae_eq_mk.symm] ** case refine'_1.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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 \u22a2 Integrable \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) ** exact integrable_condexpL2_indicator hm hs h\u03bcs _ ** case refine'_1.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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 \u22a2 StronglyMeasurable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) ** exact h.stronglyMeasurable_mk ** 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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc \u22a2 \u2200 (s_1 : Set \u03b1),     MeasurableSet s_1 \u2192       \u2191\u2191(Measure.trim \u03bc hm) s_1 < \u22a4 \u2192         0 \u2264           \u222b (x : \u03b1) in s_1,             AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202Measure.trim \u03bc hm ** intro t ht h\u03bct ** 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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 \u22a2 0 \u2264     \u222b (x : \u03b1) in t,       AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202Measure.trim \u03bc hm ** rw [\u2190 set_integral_trim hm h.stronglyMeasurable_mk ht] ** 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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 \u22a2 0 \u2264 \u222b (x : \u03b1) in t, AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202\u03bc ** have h_ae :\n  \u2200\u1d50 x \u2202\u03bc, x \u2208 t \u2192 h.mk _ x = (condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) x := by\n  filter_upwards [h.ae_eq_mk] with x hx\n  exact fun _ => hx.symm ** 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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 h_ae :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     x \u2208 t \u2192       AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =         \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x \u22a2 0 \u2264 \u222b (x : \u03b1) in t, AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202\u03bc ** rw [set_integral_congr_ae (hm t ht) h_ae,\n  set_integral_condexpL2_indicator ht hs ((le_trim hm).trans_lt h\u03bct).ne 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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 h_ae :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     x \u2208 t \u2192       AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =         \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x \u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2229 t)) ** exact ENNReal.toReal_nonneg ** \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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     x \u2208 t \u2192       AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =         \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x ** filter_upwards [h.ae_eq_mk] with x hx ** case h \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\u00b2\u2070 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2079 : NormedAddCommGroup E inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2077 : CompleteSpace E inst\u271d\u00b9\u2076 : NormedAddCommGroup E' inst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2074 : CompleteSpace E' inst\u271d\u00b9\u00b3 : NormedSpace \u211d E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup F inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u2070 : NormedAddCommGroup G inst\u271d\u2079 : NormedAddCommGroup G' inst\u271d\u2078 : NormedSpace \u211d G' inst\u271d\u2077 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2076 : IsROrC \ud835\udd5c' inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b3 : CompleteSpace E'' inst\u271d\u00b2 : NormedSpace \u211d E'' inst\u271d\u00b9 : NormedSpace \u211d G hm\u271d hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) h : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4 x : \u03b1 hx :   \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x =     AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u22a2 x \u2208 t \u2192     AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =       \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x ** exact fun _ => hx.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.shiftRight_eq_div_pow ** m n : \u2115 \u22a2 \u2191m >>> \u2191n = \u2191m / \u2191(2 ^ n) ** rw [shiftRight_coe_nat, Nat.shiftRight_eq_div_pow _ _] ** m n : \u2115 \u22a2 \u2191(m / 2 ^ n) = \u2191m / \u2191(2 ^ n) ** simp ** m n : \u2115 \u22a2 -[m+1] >>> \u2191n = -[m+1] / \u2191(2 ^ n) ** rw [shiftRight_negSucc, negSucc_ediv, Nat.shiftRight_eq_div_pow] ** m n : \u2115 \u22a2 -[m / 2 ^ n+1] = -(div \u2191m \u2191(2 ^ n) + 1)  case H m n : \u2115 \u22a2 0 < \u2191(2 ^ n) ** rfl ** case H m n : \u2115 \u22a2 0 < \u2191(2 ^ n) ** exact ofNat_lt_ofNat_of_lt (pow_pos (by decide) _) ** m n : \u2115 \u22a2 0 < 2 ** decide ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_mul_Iic_of_neg ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a c : \u03b1 h : c < 0 \u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Iic a = Ici (a / c) ** simpa only [mul_comm] using preimage_mul_const_Iic_of_neg a h ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Nat.antidiagonal_succ_succ' ** n : \u2115 \u22a2 \u00ac(n + 2, 0) \u2208       map         (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }           { toFun := Nat.succ, inj' := Nat.succ_injective })         (antidiagonal n) ** simp ** n : \u2115 \u22a2 \u00ac(0, n + 2) \u2208       cons (n + 2, 0)         (map           (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }             { toFun := Nat.succ, inj' := Nat.succ_injective })           (antidiagonal n))         (_ :           \u00ac(n + 2, 0) \u2208               map                 (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }                   { toFun := Nat.succ, inj' := Nat.succ_injective })                 (antidiagonal n)) ** simp ** n : \u2115 \u22a2 antidiagonal (n + 2) =     cons (0, n + 2)       (cons (n + 2, 0)         (map           (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }             { toFun := Nat.succ, inj' := Nat.succ_injective })           (antidiagonal n))         (_ :           \u00ac(n + 2, 0) \u2208               map                 (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }                   { toFun := Nat.succ, inj' := Nat.succ_injective })                 (antidiagonal n)))       (_ :         \u00ac(0, n + 2) \u2208             cons (n + 2, 0)               (map                 (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }                   { toFun := Nat.succ, inj' := Nat.succ_injective })                 (antidiagonal n))               (_ :                 \u00ac(n + 2, 0) \u2208                     map                       (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }                         { toFun := Nat.succ, inj' := Nat.succ_injective })                       (antidiagonal n))) ** simp_rw [antidiagonal_succ (n + 1), antidiagonal_succ', Finset.map_cons, map_map] ** n : \u2115 \u22a2 cons (0, n + 1 + 1)       (cons (\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0))         (map           (Embedding.trans (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })             (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))           (antidiagonal n))         (_ :           \u00ac\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0) \u2208               map                 (Embedding.trans                   (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })                   (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))                 (antidiagonal n)))       (_ :         \u00ac(0, n + 1 + 1) \u2208             cons (\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0))               (map                 (Embedding.trans                   (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })                   (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))                 (antidiagonal n))               (_ :                 \u00ac\u2191(Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)) (n + 1, 0) \u2208                     map                       (Embedding.trans                         (Embedding.prodMap (Embedding.refl \u2115) { toFun := Nat.succ, inj' := Nat.succ_injective })                         (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Embedding.refl \u2115)))                       (antidiagonal n))) =     cons (0, n + 2)       (cons (n + 2, 0)         (map           (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }             { toFun := Nat.succ, inj' := Nat.succ_injective })           (antidiagonal n))         (_ :           \u00ac(n + 2, 0) \u2208               map                 (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }                   { toFun := Nat.succ, inj' := Nat.succ_injective })                 (antidiagonal n)))       (_ :         \u00ac(0, n + 2) \u2208             cons (n + 2, 0)               (map                 (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }                   { toFun := Nat.succ, inj' := Nat.succ_injective })                 (antidiagonal n))               (_ :                 \u00ac(n + 2, 0) \u2208                     map                       (Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }                         { toFun := Nat.succ, inj' := Nat.succ_injective })                       (antidiagonal n))) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "integral_smul_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2079 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c\u271d : Type u_6 inst\u271d\u2078 : IsROrC \ud835\udd5c\u271d inst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F p : \u211d\u22650\u221e inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : NormedSpace \u211d F \ud835\udd5c : Type u_7 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : CompleteSpace E f : \u03b1 \u2192 \ud835\udd5c c : E \u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2079 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c\u271d : Type u_6 inst\u271d\u2078 : IsROrC \ud835\udd5c\u271d inst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F p : \u211d\u22650\u221e inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : NormedSpace \u211d F \ud835\udd5c : Type u_7 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : CompleteSpace E f : \u03b1 \u2192 \ud835\udd5c c : E hf : Integrable f \u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c ** exact ((1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c).smulRight c).integral_comp_comm hf ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2079 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c\u271d : Type u_6 inst\u271d\u2078 : IsROrC \ud835\udd5c\u271d inst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F p : \u211d\u22650\u221e inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : NormedSpace \u211d F \ud835\udd5c : Type u_7 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : CompleteSpace E f : \u03b1 \u2192 \ud835\udd5c c : E hf : \u00acIntegrable f \u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c ** by_cases hc : c = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2079 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c\u271d : Type u_6 inst\u271d\u2078 : IsROrC \ud835\udd5c\u271d inst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F p : \u211d\u22650\u221e inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : NormedSpace \u211d F \ud835\udd5c : Type u_7 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : CompleteSpace E f : \u03b1 \u2192 \ud835\udd5c c : E hf : \u00acIntegrable f hc : \u00acc = 0 \u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c ** rw [integral_undef hf, integral_undef, zero_smul] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2079 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c\u271d : Type u_6 inst\u271d\u2078 : IsROrC \ud835\udd5c\u271d inst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F p : \u211d\u22650\u221e inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : NormedSpace \u211d F \ud835\udd5c : Type u_7 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : CompleteSpace E f : \u03b1 \u2192 \ud835\udd5c c : E hf : \u00acIntegrable f hc : \u00acc = 0 \u22a2 \u00acIntegrable fun x => f x \u2022 c ** rw [integrable_smul_const hc] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2079 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c\u271d : Type u_6 inst\u271d\u2078 : IsROrC \ud835\udd5c\u271d inst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F p : \u211d\u22650\u221e inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : NormedSpace \u211d F \ud835\udd5c : Type u_7 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : CompleteSpace E f : \u03b1 \u2192 \ud835\udd5c c : E hf : \u00acIntegrable f hc : \u00acc = 0 \u22a2 \u00acIntegrable fun x => f x ** simp_rw [hf] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2079 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c\u271d : Type u_6 inst\u271d\u2078 : IsROrC \ud835\udd5c\u271d inst\u271d\u2077 : NormedSpace \ud835\udd5c\u271d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c\u271d F p : \u211d\u22650\u221e inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : NormedSpace \u211d F \ud835\udd5c : Type u_7 inst\u271d\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : CompleteSpace E f : \u03b1 \u2192 \ud835\udd5c c : E hf : \u00acIntegrable f hc : c = 0 \u22a2 \u222b (x : \u03b1), f x \u2022 c \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) \u2022 c ** simp only [hc, integral_zero, smul_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.rev_le_rev ** n : Nat i j : Fin n \u22a2 rev i \u2264 rev j \u2194 j \u2264 i ** simp only [le_def, val_rev, Nat.sub_le_sub_iff_left (Nat.succ_le.2 j.is_lt)] ** n : Nat i j : Fin n \u22a2 Nat.succ \u2191j \u2264 \u2191i + 1 \u2194 \u2191j \u2264 \u2191i ** exact Nat.succ_le_succ_iff ** Qed",
        "informal": ""
    },
    {
        "formal": "Equiv.piCongrLeft_symm_preimage_univ_pi ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 f : \u03b9' \u2243 \u03b9 t : (i : \u03b9) \u2192 Set (\u03b1 i) \u22a2 (\u2191(piCongrLeft \u03b1 f).symm \u207b\u00b9' pi univ fun i' => t (\u2191f i')) = pi univ t ** simpa [f.surjective.range_eq] using piCongrLeft_symm_preimage_pi f univ t ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae ** \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 \u22a2 \u2203 ns, StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** have h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d) (_ : 0 < \u03b5), \u2203 k : \u2115, 2 * (2 : \u211d)\u207b\u00b9 ^ k < \u03b5 := by\n  intro \u03b5 h\u03b5\n  obtain \u27e8k, h_k\u27e9 : \u2203 k : \u2115, (2 : \u211d)\u207b\u00b9 ^ k < \u03b5 := exists_pow_lt_of_lt_one h\u03b5 (by norm_num)\n  refine' \u27e8k + 1, (le_of_eq _).trans_lt h_k\u27e9\n  rw [pow_add]; ring ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 \u22a2 \u2203 ns, StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** set ns := ExistsSeqTendstoAe.seqTendstoAeSeq hfg ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg \u22a2 \u2203 ns, StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** use ns ** case h \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg \u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** let S := fun k => { x | (2 : \u211d)\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x) } ** case h \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} \u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** have h\u03bcS_le : \u2200 k, \u03bc (S k) \u2264 (2 : \u211d\u22650\u221e)\u207b\u00b9 ^ k :=\n  fun k => ExistsSeqTendstoAe.seqTendstoAeSeq_spec hfg k (ns k) le_rfl ** case h \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k \u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** set s := Filter.atTop.limsup S with hs ** case h \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop \u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** have h\u03bcs : \u03bc s = 0 := by\n  refine' measure_limsup_eq_zero (ne_of_lt <| lt_of_le_of_lt (ENNReal.tsum_le_tsum h\u03bcS_le) _)\n  simp only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, inv_inv] ** case h \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** have h_tendsto : \u2200 x \u2208 s\u1d9c, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) := by\n  refine' fun x hx => Metric.tendsto_atTop.mpr fun \u03b5 h\u03b5 => _\n  rw [hs, limsup_eq_iInf_iSup_of_nat] at hx\n  simp only [Set.iSup_eq_iUnion, Set.iInf_eq_iInter, Set.compl_iInter, Set.compl_iUnion,\n    Set.mem_iUnion, Set.mem_iInter, Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hx\n  obtain \u27e8N, hNx\u27e9 := hx\n  obtain \u27e8k, hk_lt_\u03b5\u27e9 := h_lt_\u03b5_real \u03b5 h\u03b5\n  refine' \u27e8max N (k - 1), fun n hn_ge => lt_of_le_of_lt _ hk_lt_\u03b5\u27e9\n  specialize hNx n ((le_max_left _ _).trans hn_ge)\n  have h_inv_n_le_k : (2 : \u211d)\u207b\u00b9 ^ n \u2264 2 * (2 : \u211d)\u207b\u00b9 ^ k := by\n    rw [mul_comm, \u2190 inv_mul_le_iff' (zero_lt_two' \u211d)]\n    conv_lhs =>\n      congr\n      rw [\u2190 pow_one (2 : \u211d)\u207b\u00b9]\n    rw [\u2190 pow_add, add_comm]\n    exact pow_le_pow_of_le_one (one_div (2 : \u211d) \u25b8 one_half_pos.le) (inv_le_one one_le_two)\n      ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans\n        (add_le_add_right hn_ge 1))\n  exact le_trans hNx.le h_inv_n_le_k ** case h \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 h_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) \u22a2 StrictMono ns \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** rw [ae_iff] ** case h \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 h_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) \u22a2 StrictMono ns \u2227 \u2191\u2191\u03bc {a | \u00acTendsto (fun i => f (ns i) a) atTop (\ud835\udcdd (g a))} = 0 ** refine' \u27e8ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono hfg, measure_mono_null (fun x => _) h\u03bcs\u27e9 ** case h \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 h_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) x : \u03b1 \u22a2 x \u2208 {a | \u00acTendsto (fun i => f (ns i) a) atTop (\ud835\udcdd (g a))} \u2192 x \u2208 s ** rw [Set.mem_setOf_eq, \u2190 @Classical.not_not (x \u2208 s), not_imp_not] ** case h \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 h_tendsto : \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) x : \u03b1 \u22a2 \u00acx \u2208 s \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** exact h_tendsto x ** \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 \u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ** intro \u03b5 h\u03b5 ** \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 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ** obtain \u27e8k, h_k\u27e9 : \u2203 k : \u2115, (2 : \u211d)\u207b\u00b9 ^ k < \u03b5 := exists_pow_lt_of_lt_one h\u03b5 (by norm_num) ** case intro \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 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 k : \u2115 h_k : 2\u207b\u00b9 ^ k < \u03b5 \u22a2 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ** refine' \u27e8k + 1, (le_of_eq _).trans_lt h_k\u27e9 ** case intro \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 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 k : \u2115 h_k : 2\u207b\u00b9 ^ k < \u03b5 \u22a2 2 * 2\u207b\u00b9 ^ (k + 1) = 2\u207b\u00b9 ^ k ** rw [pow_add] ** case intro \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 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 k : \u2115 h_k : 2\u207b\u00b9 ^ k < \u03b5 \u22a2 2 * (2\u207b\u00b9 ^ k * 2\u207b\u00b9 ^ 1) = 2\u207b\u00b9 ^ k ** ring ** \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 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 2\u207b\u00b9 < 1 ** norm_num ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop \u22a2 \u2191\u2191\u03bc s = 0 ** refine' measure_limsup_eq_zero (ne_of_lt <| lt_of_le_of_lt (ENNReal.tsum_le_tsum h\u03bcS_le) _) ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop \u22a2 \u2211' (a : \u2115), 2\u207b\u00b9 ^ a < \u22a4 ** simp only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, inv_inv] ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 \u22a2 \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 Tendsto (fun i => f (ns i) x) atTop (\ud835\udcdd (g x)) ** refine' fun x hx => Metric.tendsto_atTop.mpr fun \u03b5 h\u03b5 => _ ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 hx : x \u2208 s\u1d9c \u03b5 : \u211d h\u03b5 : \u03b5 > 0 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5 ** rw [hs, limsup_eq_iInf_iSup_of_nat] at hx ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 hx : x \u2208 (\u2a05 n, \u2a06 i, \u2a06 (_ : i \u2265 n), S i)\u1d9c \u03b5 : \u211d h\u03b5 : \u03b5 > 0 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5 ** simp only [Set.iSup_eq_iUnion, Set.iInf_eq_iInter, Set.compl_iInter, Set.compl_iUnion,\n  Set.mem_iUnion, Set.mem_iInter, Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hx ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 hx : \u2203 i, \u2200 (i_1 : \u2115), i_1 \u2265 i \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i_1) x) (g x) < 2\u207b\u00b9 ^ i_1 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5 ** obtain \u27e8N, hNx\u27e9 := hx ** case intro \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 N : \u2115 hNx : \u2200 (i : \u2115), i \u2265 N \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2\u207b\u00b9 ^ i \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5 ** obtain \u27e8k, hk_lt_\u03b5\u27e9 := h_lt_\u03b5_real \u03b5 h\u03b5 ** case intro.intro \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 N : \u2115 hNx : \u2200 (i : \u2115), i \u2265 N \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2\u207b\u00b9 ^ i k : \u2115 hk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f (ns n) x) (g x) < \u03b5 ** refine' \u27e8max N (k - 1), fun n hn_ge => lt_of_le_of_lt _ hk_lt_\u03b5\u27e9 ** case intro.intro \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 N : \u2115 hNx : \u2200 (i : \u2115), i \u2265 N \u2192 dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2\u207b\u00b9 ^ i k : \u2115 hk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5 n : \u2115 hn_ge : n \u2265 max N (k - 1) \u22a2 dist (f (ns n) x) (g x) \u2264 2 * 2\u207b\u00b9 ^ k ** specialize hNx n ((le_max_left _ _).trans hn_ge) ** case intro.intro \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 N k : \u2115 hk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5 n : \u2115 hn_ge : n \u2265 max N (k - 1) hNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n \u22a2 dist (f (ns n) x) (g x) \u2264 2 * 2\u207b\u00b9 ^ k ** have h_inv_n_le_k : (2 : \u211d)\u207b\u00b9 ^ n \u2264 2 * (2 : \u211d)\u207b\u00b9 ^ k := by\n  rw [mul_comm, \u2190 inv_mul_le_iff' (zero_lt_two' \u211d)]\n  conv_lhs =>\n    congr\n    rw [\u2190 pow_one (2 : \u211d)\u207b\u00b9]\n  rw [\u2190 pow_add, add_comm]\n  exact pow_le_pow_of_le_one (one_div (2 : \u211d) \u25b8 one_half_pos.le) (inv_le_one one_le_two)\n    ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans\n      (add_le_add_right hn_ge 1)) ** case intro.intro \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 N k : \u2115 hk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5 n : \u2115 hn_ge : n \u2265 max N (k - 1) hNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n h_inv_n_le_k : 2\u207b\u00b9 ^ n \u2264 2 * 2\u207b\u00b9 ^ k \u22a2 dist (f (ns n) x) (g x) \u2264 2 * 2\u207b\u00b9 ^ k ** exact le_trans hNx.le h_inv_n_le_k ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 N k : \u2115 hk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5 n : \u2115 hn_ge : n \u2265 max N (k - 1) hNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n \u22a2 2\u207b\u00b9 ^ n \u2264 2 * 2\u207b\u00b9 ^ k ** rw [mul_comm, \u2190 inv_mul_le_iff' (zero_lt_two' \u211d)] ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 N k : \u2115 hk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5 n : \u2115 hn_ge : n \u2265 max N (k - 1) hNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n \u22a2 2\u207b\u00b9 * 2\u207b\u00b9 ^ n \u2264 2\u207b\u00b9 ^ k ** conv_lhs =>\n  congr\n  rw [\u2190 pow_one (2 : \u211d)\u207b\u00b9] ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 N k : \u2115 hk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5 n : \u2115 hn_ge : n \u2265 max N (k - 1) hNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n \u22a2 2\u207b\u00b9 ^ 1 * 2\u207b\u00b9 ^ n \u2264 2\u207b\u00b9 ^ k ** rw [\u2190 pow_add, add_comm] ** \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 h_lt_\u03b5_real : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 k, 2 * 2\u207b\u00b9 ^ k < \u03b5 ns : \u2115 \u2192 \u2115 := ExistsSeqTendstoAe.seqTendstoAeSeq hfg S : \u2115 \u2192 Set \u03b1 := fun k => {x | 2\u207b\u00b9 ^ k \u2264 dist (f (ns k) x) (g x)} h\u03bcS_le : \u2200 (k : \u2115), \u2191\u2191\u03bc (S k) \u2264 2\u207b\u00b9 ^ k s : Set \u03b1 := limsup S atTop hs : s = limsup S atTop h\u03bcs : \u2191\u2191\u03bc s = 0 x : \u03b1 \u03b5 : \u211d h\u03b5 : \u03b5 > 0 N k : \u2115 hk_lt_\u03b5 : 2 * 2\u207b\u00b9 ^ k < \u03b5 n : \u2115 hn_ge : n \u2265 max N (k - 1) hNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2\u207b\u00b9 ^ n \u22a2 2\u207b\u00b9 ^ (n + 1) \u2264 2\u207b\u00b9 ^ k ** exact pow_le_pow_of_le_one (one_div (2 : \u211d) \u25b8 one_half_pos.le) (inv_le_one one_le_two)\n  ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans\n    (add_le_add_right hn_ge 1)) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.lintegral_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 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 f : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (b : \u03b2), \u222b\u207b (c : \u03b3), f (b, c) \u2202\u2191\u03b7 (a, b) \u2202\u2191\u03ba a ** let g := Function.curry f ** \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 f : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e hf : Measurable f g : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f \u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (b : \u03b2), \u222b\u207b (c : \u03b3), f (b, c) \u2202\u2191\u03b7 (a, b) \u2202\u2191\u03ba a ** change \u222b\u207b bc, f bc \u2202(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b b, \u222b\u207b c, g b c \u2202\u03b7 (a, b) \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 f : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e hf : Measurable f g : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f \u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (b : \u03b2), \u222b\u207b (c : \u03b3), g b c \u2202\u2191\u03b7 (a, b) \u2202\u2191\u03ba a ** rw [\u2190 lintegral_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 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 f : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e hf : Measurable f g : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f \u22a2 \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), f bc \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (bc : \u03b2 \u00d7 \u03b3), g bc.1 bc.2 \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a ** simp_rw [Function.curry_apply] ** case hf \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 f : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e hf : Measurable f g : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f \u22a2 Measurable (Function.uncurry fun b c => g b c) ** simp_rw [Function.uncurry_curry] ** case hf \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 f : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e hf : Measurable f g : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e := Function.curry f \u22a2 Measurable f ** exact hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsStoppingTime.add_const_nat ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i : \u2115 \u22a2 IsStoppingTime f fun \u03c9 => \u03c4 \u03c9 + i ** refine' isStoppingTime_of_measurableSet_eq fun j => _ ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i j : \u2115 \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j} ** by_cases hij : i \u2264 j ** case pos \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i j : \u2115 hij : i \u2264 j \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j} ** simp_rw [eq_comm, \u2190 Nat.sub_eq_iff_eq_add hij, eq_comm] ** case pos \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i j : \u2115 hij : i \u2264 j \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 = j - i} ** exact f.mono (j.sub_le i) _ (h\u03c4.measurableSet_eq (j - i)) ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i j : \u2115 hij : \u00aci \u2264 j \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j} ** rw [not_le] at hij ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i j : \u2115 hij : j < i \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 + i = j} ** convert @MeasurableSet.empty _ (f.1 j) ** case h.e'_3 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i j : \u2115 hij : j < i \u22a2 {\u03c9 | \u03c4 \u03c9 + i = j} = \u2205 ** ext \u03c9 ** case h.e'_3.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i j : \u2115 hij : j < i \u03c9 : \u03a9 \u22a2 \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 + i = j} \u2194 \u03c9 \u2208 \u2205 ** simp only [Set.mem_empty_iff_false, iff_false_iff] ** case h.e'_3.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i j : \u2115 hij : j < i \u03c9 : \u03a9 \u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 + i = j} ** rintro (hx : \u03c4 \u03c9 + i = j) ** case h.e'_3.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m \u03c4 : \u03a9 \u2192 \u2115 h\u03c4 : IsStoppingTime f \u03c4 i j : \u2115 hij : j < i \u03c9 : \u03a9 hx : \u03c4 \u03c9 + i = j \u22a2 False ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "USize.mod_lt ** a b : USize h : 0 < b \u22a2 \u2191b.1 > 0 ** simp at h ** a b : USize h : 0 < toNat b \u22a2 \u2191b.1 > 0 ** exact h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5 ** set s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) \u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5 ** have : \u2200 n, \u03bc (s n) < \u221e := fun n =>\n  (measure_mono <| disjointed_subset _ _).trans_lt (measure_spanningSets_lt_top \u03bc n) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) this : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4 \u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5 ** obtain \u27e8\u03b4, \u03b4pos, \u03b4sum\u27e9 : \u2203 \u03b4 : \u2115 \u2192 \u211d\u22650, (\u2200 i, 0 < \u03b4 i) \u2227 (\u2211' i, \u03bc (s i) * \u03b4 i) < \u03b5 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) this : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4 \u22a2 \u2203 \u03b4, (\u2200 (i : \u2115), 0 < \u03b4 i) \u2227 \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5  case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) this : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4 \u03b4 : \u2115 \u2192 \u211d\u22650 \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i \u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5 \u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5 ** exact ENNReal.exists_pos_tsum_mul_lt_of_countable \u03b50 _ fun n => (this n).ne ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) this : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4 \u03b4 : \u2115 \u2192 \u211d\u22650 \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i \u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5 \u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5 ** set N : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) this : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4 \u03b4 : \u2115 \u2192 \u211d\u22650 \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i \u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5 N : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc \u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5 ** have hN_meas : Measurable N := measurable_spanningSetsIndex \u03bc ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) this : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4 \u03b4 : \u2115 \u2192 \u211d\u22650 \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i \u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5 N : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc hN_meas : Measurable N \u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5 ** have hNs : \u2200 n, N \u207b\u00b9' {n} = s n := preimage_spanningSetsIndex_singleton \u03bc ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) this : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4 \u03b4 : \u2115 \u2192 \u211d\u22650 \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i \u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5 N : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc hN_meas : Measurable N hNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n \u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5 ** refine' \u27e8\u03b4 \u2218 N, fun x => \u03b4pos _, measurable_from_nat.comp hN_meas, _\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) this : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4 \u03b4 : \u2115 \u2192 \u211d\u22650 \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i \u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5 N : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc hN_meas : Measurable N hNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n \u22a2 \u222b\u207b (x : \u03b1), \u2191((\u03b4 \u2218 N) x) \u2202\u03bc < \u03b5 ** erw [lintegral_comp measurable_from_nat.coe_nnreal_ennreal hN_meas] ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4\u271d : Type u_4 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc) this : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4 \u03b4 : \u2115 \u2192 \u211d\u22650 \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i \u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5 N : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc hN_meas : Measurable N hNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n \u22a2 \u222b\u207b (a : \u2115), \u2191(\u03b4 a) \u2202Measure.map N \u03bc < \u03b5 ** simpa [hNs, lintegral_countable', measurable_spanningSetsIndex, mul_comm] using \u03b4sum ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Primrec.casesOn1 ** f : \u2115 \u2192 \u2115 m : \u2115 hf : Nat.Primrec f \u22a2 \u2200 (n : \u2115), Nat.rec m (fun y IH => f (unpair (Nat.pair y IH)).1) n = Nat.casesOn n m f ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.prod_subset_prod_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 \u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 ** cases' (s \u00d7\u02e2 t).eq_empty_or_nonempty with h h ** case inr \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 h : Set.Nonempty (s \u00d7\u02e2 t) \u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 ** have st : s.Nonempty \u2227 t.Nonempty := by rwa [prod_nonempty_iff] at h ** case inr \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 h : Set.Nonempty (s \u00d7\u02e2 t) st : Set.Nonempty s \u2227 Set.Nonempty t \u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 ** refine' \u27e8fun H => Or.inl \u27e8_, _\u27e9, _\u27e9 ** case inl \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 h : s \u00d7\u02e2 t = \u2205 \u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 ** simp [h, prod_eq_empty_iff.1 h] ** \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 h : Set.Nonempty (s \u00d7\u02e2 t) \u22a2 Set.Nonempty s \u2227 Set.Nonempty t ** rwa [prod_nonempty_iff] at h ** case inr.refine'_1 \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 h : Set.Nonempty (s \u00d7\u02e2 t) st : Set.Nonempty s \u2227 Set.Nonempty t H : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u22a2 s \u2286 s\u2081 ** have := image_subset (Prod.fst : \u03b1 \u00d7 \u03b2 \u2192 \u03b1) H ** case inr.refine'_1 \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 h : Set.Nonempty (s \u00d7\u02e2 t) st : Set.Nonempty s \u2227 Set.Nonempty t H : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 this : Prod.fst '' s \u00d7\u02e2 t \u2286 Prod.fst '' s\u2081 \u00d7\u02e2 t\u2081 \u22a2 s \u2286 s\u2081 ** rwa [fst_image_prod _ st.2, fst_image_prod _ (h.mono H).snd] at this ** case inr.refine'_2 \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 h : Set.Nonempty (s \u00d7\u02e2 t) st : Set.Nonempty s \u2227 Set.Nonempty t H : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u22a2 t \u2286 t\u2081 ** have := image_subset (Prod.snd : \u03b1 \u00d7 \u03b2 \u2192 \u03b2) H ** case inr.refine'_2 \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 h : Set.Nonempty (s \u00d7\u02e2 t) st : Set.Nonempty s \u2227 Set.Nonempty t H : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 this : Prod.snd '' s \u00d7\u02e2 t \u2286 Prod.snd '' s\u2081 \u00d7\u02e2 t\u2081 \u22a2 t \u2286 t\u2081 ** rwa [snd_image_prod st.1, snd_image_prod (h.mono H).fst] at this ** case inr.refine'_3 \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 h : Set.Nonempty (s \u00d7\u02e2 t) st : Set.Nonempty s \u2227 Set.Nonempty t \u22a2 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 \u2192 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 ** intro H ** case inr.refine'_3 \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 h : Set.Nonempty (s \u00d7\u02e2 t) st : Set.Nonempty s \u2227 Set.Nonempty t H : s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 \u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 ** simp only [st.1.ne_empty, st.2.ne_empty, or_false_iff] at H ** case inr.refine'_3 \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 h : Set.Nonempty (s \u00d7\u02e2 t) st : Set.Nonempty s \u2227 Set.Nonempty t H : s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 ** exact prod_mono H.1 H.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snormEssSup_indicator_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\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 \u22a2 snormEssSup (Set.indicator s fun x => c) \u03bc \u2264 \u2191\u2016c\u2016\u208a ** by_cases h\u03bc0 : \u03bc = 0 ** 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 h\u03bc0 : \u03bc = 0 \u22a2 snormEssSup (Set.indicator s fun x => c) \u03bc \u2264 \u2191\u2016c\u2016\u208a ** rw [h\u03bc0, snormEssSup_measure_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\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G h\u03bc0 : \u03bc = 0 \u22a2 0 \u2264 \u2191\u2016c\u2016\u208a ** exact zero_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 c\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G h\u03bc0 : \u00ac\u03bc = 0 \u22a2 snormEssSup (Set.indicator s fun x => c) \u03bc \u2264 \u2191\u2016c\u2016\u208a ** exact (snormEssSup_indicator_le s fun _ => c).trans (snormEssSup_const c h\u03bc0).le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendstoInMeasure_of_tendsto_ae ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : MetricSpace E f : \u2115 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u22a2 TendstoInMeasure \u03bc f atTop g ** have hg : AEStronglyMeasurable g \u03bc := aestronglyMeasurable_of_tendsto_ae _ hf hfg ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : MetricSpace E f : \u2115 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) hg : AEStronglyMeasurable g \u03bc \u22a2 TendstoInMeasure \u03bc f atTop 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\u00b9 : MetricSpace E f : \u2115 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) hg : AEStronglyMeasurable g \u03bc \u22a2 TendstoInMeasure \u03bc (fun i => AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) \u03bc)) atTop     (AEStronglyMeasurable.mk g hg) ** refine' tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable\n  (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\u00b9 : MetricSpace E f : \u2115 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hfg : \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 => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop       (\ud835\udcdd (AEStronglyMeasurable.mk g hg x)) ** have hf_eq_ae : \u2200\u1d50 x \u2202\u03bc, \u2200 n, (hf n).mk (f n) x = f n x :=\n  ae_all_iff.mpr fun n => (hf n).ae_eq_mk.symm ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : MetricSpace E f : \u2115 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) hg : AEStronglyMeasurable g \u03bc hf_eq_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop       (\ud835\udcdd (AEStronglyMeasurable.mk g hg x)) ** filter_upwards [hf_eq_ae, hg.ae_eq_mk, hfg] with x hxf hxg hxfg ** case h \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : MetricSpace E f : \u2115 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) hg : AEStronglyMeasurable g \u03bc hf_eq_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x x : \u03b1 hxf : \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x hxg : g x = AEStronglyMeasurable.mk g hg x hxfg : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x) atTop     (\ud835\udcdd (AEStronglyMeasurable.mk g hg x)) ** rw [\u2190 hxg, funext fun n => hxf n] ** case h \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : MetricSpace E f : \u2115 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) hg : AEStronglyMeasurable g \u03bc hf_eq_ae : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x x : \u03b1 hxf : \u2200 (n : \u2115), AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) x = f n x hxg : g x = AEStronglyMeasurable.mk g hg x hxfg : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u22a2 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) ** exact hxfg ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.smul_neg ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b3 : Monoid \u03b1 inst\u271d\u00b2 : AddGroup \u03b2 inst\u271d\u00b9 : DistribMulAction \u03b1 \u03b2 inst\u271d : DecidableEq \u03b2 a : \u03b1 s : Finset \u03b1 t : Finset \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\u00b3 : Monoid \u03b1 inst\u271d\u00b2 : AddGroup \u03b2 inst\u271d\u00b9 : DistribMulAction \u03b1 \u03b2 inst\u271d : DecidableEq \u03b2 a : \u03b1 s : Finset \u03b1 t : Finset \u03b2 \u22a2 s \u2022 image (fun x => -x) t = image (fun x => -x) (s \u2022 t) ** exact image_image\u2082_right_comm smul_neg ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.le_measure_compl_liminf_of_limsup_measure_le ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E \u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L ** rcases L.eq_or_neBot with rfl | hne ** case inr \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L \u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L ** have meas_Ec : \u03bc E\u1d9c = 1 - \u03bc E := by\n  simpa only [measure_univ] using measure_compl E_mble (measure_lt_top \u03bc E).ne ** case inr \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L meas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E \u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L ** have meas_i_Ec : \u2200 i, \u03bcs i E\u1d9c = 1 - \u03bcs i E := by\n  intro i\n  simpa only [measure_univ] using measure_compl E_mble (measure_lt_top (\u03bcs i) E).ne ** case inr \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L meas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E meas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E \u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) L ** simp_rw [meas_Ec, meas_i_Ec] ** case inr \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L meas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E meas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E \u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L ** have obs :\n  (L.liminf fun i : \u03b9 => 1 - \u03bcs i E) = L.liminf ((fun x => 1 - x) \u2218 fun i : \u03b9 => \u03bcs i E) := rfl ** case inr \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L meas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E meas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E obs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L \u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L ** rw [obs] ** case inr \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L meas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E meas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E obs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L \u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L ** have := antitone_const_tsub.map_limsup_of_continuousAt (F := L)\n  (fun i => \u03bcs i E) (ENNReal.continuous_sub_left ENNReal.one_ne_top).continuousAt ** case inr \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L meas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E meas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E obs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L this : 1 - limsup (fun i => \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L \u22a2 1 - \u2191\u2191\u03bc E \u2264 liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L ** simp_rw [\u2190 this] ** case inr \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L meas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E meas_i_Ec : \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E obs : liminf (fun i => 1 - \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L this : 1 - limsup (fun i => \u2191\u2191(\u03bcs i) E) L = liminf ((fun x => 1 - x) \u2218 fun i => \u2191\u2191(\u03bcs i) E) L \u22a2 1 - \u2191\u2191\u03bc E \u2264 1 - limsup (fun i => \u2191\u2191(\u03bcs i) E) L ** exact antitone_const_tsub h ** case inl \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) \u22a5 \u2264 \u2191\u2191\u03bc E \u22a2 \u2191\u2191\u03bc E\u1d9c \u2264 liminf (fun i => \u2191\u2191(\u03bcs i) E\u1d9c) \u22a5 ** simp only [liminf_bot, le_top] ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L \u22a2 \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E ** simpa only [measure_univ] using measure_compl E_mble (measure_lt_top \u03bc E).ne ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L meas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E \u22a2 \u2200 (i : \u03b9), \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E ** intro i ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 \u03b9 : Type u_2 L : Filter \u03b9 \u03bc : Measure \u03a9 \u03bcs : \u03b9 \u2192 Measure \u03a9 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc inst\u271d : \u2200 (i : \u03b9), IsProbabilityMeasure (\u03bcs i) E : Set \u03a9 E_mble : MeasurableSet E h : limsup (fun i => \u2191\u2191(\u03bcs i) E) L \u2264 \u2191\u2191\u03bc E hne : NeBot L meas_Ec : \u2191\u2191\u03bc E\u1d9c = 1 - \u2191\u2191\u03bc E i : \u03b9 \u22a2 \u2191\u2191(\u03bcs i) E\u1d9c = 1 - \u2191\u2191(\u03bcs i) E ** simpa only [measure_univ] using measure_compl E_mble (measure_lt_top (\u03bcs i) E).ne ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_mul_Ici_of_neg ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a c : \u03b1 h : c < 0 \u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Ici a = Iic (a / c) ** simpa only [mul_comm] using preimage_mul_const_Ici_of_neg a h ** Qed",
        "informal": ""
    },
    {
        "formal": "FinEnum.pi.mem_enum ** \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u v) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) f : (a : \u03b1) \u2192 \u03b2 a \u22a2 f \u2208 enum \u03b2 ** simp [pi.enum] ** \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u v) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) f : (a : \u03b1) \u2192 \u03b2 a \u22a2 \u2203 a, (a \u2208 pi (toList \u03b1) fun x => toList (\u03b2 x)) \u2227 (fun x => a x (_ : x \u2208 toList \u03b1)) = f ** refine' \u27e8fun a _ => f a, mem_pi _ _, rfl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.liftR_map ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 \u03b2 : TypeVec.{u_1} n F' : TypeVec.{u_1} n \u2192 Type u inst\u271d\u00b9 : MvFunctor F' inst\u271d : LawfulMvFunctor F' R : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop x : F' \u03b1 f g : \u03b1 \u27f9 \u03b2 h : \u03b1 \u27f9 Subtype_ R hh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag \u22a2 LiftR' R (f <$$> x) (g <$$> x) ** rw [LiftR_def] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 \u03b2 : TypeVec.{u_1} n F' : TypeVec.{u_1} n \u2192 Type u inst\u271d\u00b9 : MvFunctor F' inst\u271d : LawfulMvFunctor F' R : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop x : F' \u03b1 f g : \u03b1 \u27f9 \u03b2 h : \u03b1 \u27f9 Subtype_ R hh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag \u22a2 \u2203 u, (prod.fst \u229a subtypeVal R) <$$> u = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> u = g <$$> x ** exists h <$$> x ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 \u03b2 : TypeVec.{u_1} n F' : TypeVec.{u_1} n \u2192 Type u inst\u271d\u00b9 : MvFunctor F' inst\u271d : LawfulMvFunctor F' R : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop x : F' \u03b1 f g : \u03b1 \u27f9 \u03b2 h : \u03b1 \u27f9 Subtype_ R hh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag \u22a2 (prod.fst \u229a subtypeVal R) <$$> h <$$> x = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> h <$$> x = g <$$> x ** rw [MvFunctor.map_map, comp_assoc, hh, \u2190 comp_assoc, fst_prod_mk, comp_assoc, fst_diag] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 \u03b2 : TypeVec.{u_1} n F' : TypeVec.{u_1} n \u2192 Type u inst\u271d\u00b9 : MvFunctor F' inst\u271d : LawfulMvFunctor F' R : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop x : F' \u03b1 f g : \u03b1 \u27f9 \u03b2 h : \u03b1 \u27f9 Subtype_ R hh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag \u22a2 (f \u229a TypeVec.id) <$$> x = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> h <$$> x = g <$$> x ** rw [MvFunctor.map_map, comp_assoc, hh, \u2190 comp_assoc, snd_prod_mk, comp_assoc, snd_diag] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 \u03b2 : TypeVec.{u_1} n F' : TypeVec.{u_1} n \u2192 Type u inst\u271d\u00b9 : MvFunctor F' inst\u271d : LawfulMvFunctor F' R : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop x : F' \u03b1 f g : \u03b1 \u27f9 \u03b2 h : \u03b1 \u27f9 Subtype_ R hh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag \u22a2 (f \u229a TypeVec.id) <$$> x = f <$$> x \u2227 (g \u229a TypeVec.id) <$$> x = g <$$> x ** dsimp [LiftR'] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 \u03b2 : TypeVec.{u_1} n F' : TypeVec.{u_1} n \u2192 Type u inst\u271d\u00b9 : MvFunctor F' inst\u271d : LawfulMvFunctor F' R : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop x : F' \u03b1 f g : \u03b1 \u27f9 \u03b2 h : \u03b1 \u27f9 Subtype_ R hh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag \u22a2 f <$$> x = f <$$> x \u2227 g <$$> x = g <$$> x ** constructor <;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Ordered.balRight ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 lv : All (fun x => cmpLT cmp x v) l vr : All (fun x => cmpLT cmp v x) r hl : Ordered cmp l hr : Ordered cmp r \u22a2 Ordered cmp (balRight l v r) ** unfold balRight ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 lv : All (fun x => cmpLT cmp x v) l vr : All (fun x => cmpLT cmp v x) r hl : Ordered cmp l hr : Ordered cmp r \u22a2 Ordered cmp     (match r with     | node red b y c => node red l v (node black b y c)     | r =>       match l with       | node black a x b => balance1 (node red a x b) v r       | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)       | l => node red l v r) ** split ** case h_2 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 lv : All (fun x => cmpLT cmp x v) l vr : All (fun x => cmpLT cmp v x) r hl : Ordered cmp l hr : Ordered cmp r l\u271d : RBNode \u03b1 x\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False \u22a2 Ordered cmp     (match l with     | node black a x b => balance1 (node red a x b) v r     | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)     | l => node red l v r) ** split ** case h_1 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering l : RBNode \u03b1 v : \u03b1 lv : All (fun x => cmpLT cmp x v) l hl : Ordered cmp l l\u271d a\u271d : RBNode \u03b1 x\u271d : \u03b1 b\u271d : RBNode \u03b1 vr : All (fun x => cmpLT cmp v x) (node red a\u271d x\u271d b\u271d) hr : Ordered cmp (node red a\u271d x\u271d b\u271d) \u22a2 Ordered cmp (node red l v (node black a\u271d x\u271d b\u271d)) ** exact \u27e8lv, vr, hl, hr\u27e9 ** case h_2.h_1 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 r : RBNode \u03b1 vr : All (fun x => cmpLT cmp v x) r hr : Ordered cmp r l\u271d\u00b9 : RBNode \u03b1 x\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False l\u271d a\u271d : RBNode \u03b1 x\u271d : \u03b1 b\u271d : RBNode \u03b1 lv : All (fun x => cmpLT cmp x v) (node black a\u271d x\u271d b\u271d) hl : Ordered cmp (node black a\u271d x\u271d b\u271d) \u22a2 Ordered cmp (balance1 (node red a\u271d x\u271d b\u271d) v r) ** exact hl.balance1 lv vr hr ** case h_2.h_2 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 r : RBNode \u03b1 vr : All (fun x => cmpLT cmp v x) r hr : Ordered cmp r l\u271d\u00b9 : RBNode \u03b1 x\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False l\u271d a\u271d : RBNode \u03b1 x\u271d : \u03b1 b\u271d : RBNode \u03b1 y\u271d : \u03b1 c\u271d : RBNode \u03b1 lv : All (fun x => cmpLT cmp x v) (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d)) hl : Ordered cmp (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d)) \u22a2 Ordered cmp (node red (balance1 (setRed a\u271d) x\u271d b\u271d) y\u271d (node black c\u271d v r)) ** have \u27e8yv, _, cv\u27e9 := lv.2.2 ** case h_2.h_2 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 r : RBNode \u03b1 vr : All (fun x => cmpLT cmp v x) r hr : Ordered cmp r l\u271d\u00b9 : RBNode \u03b1 x\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False l\u271d a\u271d : RBNode \u03b1 x\u271d : \u03b1 b\u271d : RBNode \u03b1 y\u271d : \u03b1 c\u271d : RBNode \u03b1 lv : All (fun x => cmpLT cmp x v) (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d)) hl : Ordered cmp (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d)) yv : (fun x => cmpLT cmp x v) y\u271d left\u271d : All (fun x => cmpLT cmp x v) b\u271d cv : All (fun x => cmpLT cmp x v) c\u271d \u22a2 Ordered cmp (node red (balance1 (setRed a\u271d) x\u271d b\u271d) y\u271d (node black c\u271d v r)) ** have \u27e8ax, \u27e8xy, xb, _\u27e9, ha, by_, yc, hb, hc\u27e9 := hl ** case h_2.h_2 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 r : RBNode \u03b1 vr : All (fun x => cmpLT cmp v x) r hr : Ordered cmp r l\u271d\u00b9 : RBNode \u03b1 x\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False l\u271d a\u271d : RBNode \u03b1 x\u271d : \u03b1 b\u271d : RBNode \u03b1 y\u271d : \u03b1 c\u271d : RBNode \u03b1 lv : All (fun x => cmpLT cmp x v) (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d)) hl : Ordered cmp (node red a\u271d x\u271d (node black b\u271d y\u271d c\u271d)) yv : (fun x => cmpLT cmp x v) y\u271d left\u271d : All (fun x => cmpLT cmp x v) b\u271d cv : All (fun x => cmpLT cmp x v) c\u271d ax : All (fun x => cmpLT cmp x x\u271d) a\u271d xy : (fun x => cmpLT cmp x\u271d x) y\u271d xb : All (fun x => cmpLT cmp x\u271d x) b\u271d right\u271d : All (fun x => cmpLT cmp x\u271d x) c\u271d ha : Ordered cmp a\u271d by_ : All (fun x => cmpLT cmp x y\u271d) b\u271d yc : All (fun x => cmpLT cmp y\u271d x) c\u271d hb : Ordered cmp b\u271d hc : Ordered cmp c\u271d \u22a2 Ordered cmp (node red (balance1 (setRed a\u271d) x\u271d b\u271d) y\u271d (node black c\u271d v r)) ** exact \u27e8balance1_All.2 \u27e8xy, (xy.trans_r ax).setRed, by_\u27e9, \u27e8yv, yc, yv.trans_l vr\u27e9,\n  (Ordered.setRed.2 ha).balance1 ax.setRed xb hb, cv, vr, hc, hr\u27e9 ** case h_2.h_3 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 lv : All (fun x => cmpLT cmp x v) l vr : All (fun x => cmpLT cmp v x) r hl : Ordered cmp l hr : Ordered cmp r l\u271d\u00b9 : RBNode \u03b1 x\u271d\u00b2 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), r = node red a x b \u2192 False l\u271d : RBNode \u03b1 x\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), l = node black a x b \u2192 False x\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), l = node red a x (node black b y c) \u2192 False \u22a2 Ordered cmp (node red l v r) ** exact \u27e8lv, vr, hl, hr\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrable_smul_const ** \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\u2076 : MeasurableSpace \u03b4 inst\u271d\u2075 : NormedAddCommGroup \u03b2 inst\u271d\u2074 : NormedAddCommGroup \u03b3 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : CompleteSpace \ud835\udd5c E : Type u_6 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c c : E hc : c \u2260 0 \u22a2 (Integrable fun x => f x \u2022 c) \u2194 Integrable f ** simp_rw [Integrable, aestronglyMeasurable_smul_const_iff (f := f) hc, and_congr_right_iff,\n  HasFiniteIntegral, nnnorm_smul, ENNReal.coe_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\u2076 : MeasurableSpace \u03b4 inst\u271d\u2075 : NormedAddCommGroup \u03b2 inst\u271d\u2074 : NormedAddCommGroup \u03b3 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : CompleteSpace \ud835\udd5c E : Type u_6 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c c : E hc : c \u2260 0 \u22a2 AEStronglyMeasurable f \u03bc \u2192 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a * \u2191\u2016c\u2016\u208a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4) ** 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\u2076 : MeasurableSpace \u03b4 inst\u271d\u2075 : NormedAddCommGroup \u03b2 inst\u271d\u2074 : NormedAddCommGroup \u03b3 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : CompleteSpace \ud835\udd5c E : Type u_6 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c c : E hc : c \u2260 0 a\u271d : AEStronglyMeasurable f \u03bc \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a * \u2191\u2016c\u2016\u208a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 ** rw [lintegral_mul_const' _ _ ENNReal.coe_ne_top, ENNReal.mul_lt_top_iff] ** \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\u2076 : MeasurableSpace \u03b4 inst\u271d\u2075 : NormedAddCommGroup \u03b2 inst\u271d\u2074 : NormedAddCommGroup \u03b3 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : CompleteSpace \ud835\udd5c E : Type u_6 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c c : E hc : c \u2260 0 a\u271d : AEStronglyMeasurable f \u03bc \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 \u2227 \u2191\u2016c\u2016\u208a < \u22a4 \u2228 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = 0 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 ** have : \u2200 x : \u211d\u22650\u221e, x = 0 \u2192 x < \u221e := by simp ** \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\u2076 : MeasurableSpace \u03b4 inst\u271d\u2075 : NormedAddCommGroup \u03b2 inst\u271d\u2074 : NormedAddCommGroup \u03b3 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : CompleteSpace \ud835\udd5c E : Type u_6 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c c : E hc : c \u2260 0 a\u271d : AEStronglyMeasurable f \u03bc this : \u2200 (x : \u211d\u22650\u221e), x = 0 \u2192 x < \u22a4 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 \u2227 \u2191\u2016c\u2016\u208a < \u22a4 \u2228 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = 0 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2194 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 ** simp [hc, or_iff_left_of_imp (this _)] ** \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\u2076 : MeasurableSpace \u03b4 inst\u271d\u2075 : NormedAddCommGroup \u03b2 inst\u271d\u2074 : NormedAddCommGroup \u03b3 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : CompleteSpace \ud835\udd5c E : Type u_6 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f : \u03b1 \u2192 \ud835\udd5c c : E hc : c \u2260 0 a\u271d : AEStronglyMeasurable f \u03bc \u22a2 \u2200 (x : \u211d\u22650\u221e), x = 0 \u2192 x < \u22a4 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "WithBot.image_coe_Ioc ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ioc a b = Ioc \u2191a \u2191b ** rw [\u2190 preimage_coe_Ioc, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Subset.trans Ioc_subset_Ioi_self <| Ioi_subset_Ioi bot_le)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.induction_on_inter ** \u03b1 : Type u_1 C : Set \u03b1 \u2192 Prop s : Set (Set \u03b1) m : MeasurableSpace \u03b1 h_eq : m = generateFrom s h_inter : IsPiSystem s h_empty : C \u2205 h_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t h_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c h_union :   \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i) \u22a2 MeasurableSet = DynkinSystem.GenerateHas s ** rfl ** \u03b1 : Type u_1 C : Set \u03b1 \u2192 Prop s : Set (Set \u03b1) m : MeasurableSpace \u03b1 h_eq : m = generateFrom s h_inter : IsPiSystem s h_empty : C \u2205 h_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t h_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c h_union :   \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i) eq : MeasurableSet = DynkinSystem.GenerateHas s t : Set \u03b1 ht : MeasurableSet t \u22a2 DynkinSystem.GenerateHas s t ** rwa [eq] at ht ** \u03b1 : Type u_1 C : Set \u03b1 \u2192 Prop s : Set (Set \u03b1) m : MeasurableSpace \u03b1 h_eq : m = generateFrom s h_inter : IsPiSystem s h_empty : C \u2205 h_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t h_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c h_union :   \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i) eq : MeasurableSet = DynkinSystem.GenerateHas s t\u271d : Set \u03b1 ht\u271d : MeasurableSet t\u271d this : DynkinSystem.GenerateHas s t\u271d t : Set \u03b1 ht : DynkinSystem.GenerateHas s t \u22a2 MeasurableSet t ** rw [eq] ** \u03b1 : Type u_1 C : Set \u03b1 \u2192 Prop s : Set (Set \u03b1) m : MeasurableSpace \u03b1 h_eq : m = generateFrom s h_inter : IsPiSystem s h_empty : C \u2205 h_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t h_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c h_union :   \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i) eq : MeasurableSet = DynkinSystem.GenerateHas s t\u271d : Set \u03b1 ht\u271d : MeasurableSet t\u271d this : DynkinSystem.GenerateHas s t\u271d t : Set \u03b1 ht : DynkinSystem.GenerateHas s t \u22a2 DynkinSystem.GenerateHas s t ** exact ht ** \u03b1 : Type u_1 C : Set \u03b1 \u2192 Prop s : Set (Set \u03b1) m : MeasurableSpace \u03b1 h_eq : m = generateFrom s h_inter : IsPiSystem s h_empty : C \u2205 h_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t h_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c h_union :   \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i) eq : MeasurableSet = DynkinSystem.GenerateHas s t : Set \u03b1 ht\u271d : MeasurableSet t this : DynkinSystem.GenerateHas s t f : \u2115 \u2192 Set \u03b1 hf : Pairwise (Disjoint on f) ht : \u2200 (i : \u2115), DynkinSystem.GenerateHas s (f i) i : \u2115 \u22a2 MeasurableSet (f i) ** rw [eq] ** \u03b1 : Type u_1 C : Set \u03b1 \u2192 Prop s : Set (Set \u03b1) m : MeasurableSpace \u03b1 h_eq : m = generateFrom s h_inter : IsPiSystem s h_empty : C \u2205 h_basic : \u2200 (t : Set \u03b1), t \u2208 s \u2192 C t h_compl : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 C t \u2192 C t\u1d9c h_union :   \u2200 (f : \u2115 \u2192 Set \u03b1), Pairwise (Disjoint on f) \u2192 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), C (f i)) \u2192 C (\u22c3 i, f i) eq : MeasurableSet = DynkinSystem.GenerateHas s t : Set \u03b1 ht\u271d : MeasurableSet t this : DynkinSystem.GenerateHas s t f : \u2115 \u2192 Set \u03b1 hf : Pairwise (Disjoint on f) ht : \u2200 (i : \u2115), DynkinSystem.GenerateHas s (f i) i : \u2115 \u22a2 DynkinSystem.GenerateHas s (f i) ** exact ht _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.tr_eval ** \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 b\u2081 : \u03c3\u2081 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab : b\u2081 \u2208 eval f\u2081 a\u2081 \u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082 ** cases' mem_eval.1 ab with ab b0 ** case intro \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 b\u2081 : \u03c3\u2081 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d : b\u2081 \u2208 eval f\u2081 a\u2081 ab : Reaches f\u2081 a\u2081 b\u2081 b0 : f\u2081 b\u2081 = none \u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082 ** rcases tr_reaches H aa ab with \u27e8b\u2082, bb, ab\u27e9 ** case intro.intro.intro \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 b\u2081 : \u03c3\u2081 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081 ab\u271d : Reaches f\u2081 a\u2081 b\u2081 b0 : f\u2081 b\u2081 = none b\u2082 : \u03c3\u2082 bb : tr b\u2081 b\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 \u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082 ** refine' \u27e8_, bb, mem_eval.2 \u27e8ab, _\u27e9\u27e9 ** case intro.intro.intro \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 b\u2081 : \u03c3\u2081 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081 ab\u271d : Reaches f\u2081 a\u2081 b\u2081 b0 : f\u2081 b\u2081 = none b\u2082 : \u03c3\u2082 bb : tr b\u2081 b\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 \u22a2 f\u2082 b\u2082 = none ** have := H bb ** case intro.intro.intro \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 b\u2081 : \u03c3\u2081 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 ab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081 ab\u271d : Reaches f\u2081 a\u2081 b\u2081 b0 : f\u2081 b\u2081 = none b\u2082 : \u03c3\u2082 bb : tr b\u2081 b\u2082 ab : Reaches f\u2082 a\u2082 b\u2082 this :   match f\u2081 b\u2081 with   | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1   | none => f\u2082 b\u2082 = none \u22a2 f\u2082 b\u2082 = none ** rwa [b0] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "image_circleMap_Ioc ** E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d \u22a2 circleMap c R '' Ioc 0 (2 * \u03c0) = sphere c |R| ** rw [\u2190 range_circleMap, \u2190 (periodic_circleMap c R).image_Ioc Real.two_pi_pos 0, zero_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haarMeasure_self ** G : Type u_1 inst\u271d\u2075 : Group G inst\u271d\u2074 : TopologicalSpace G inst\u271d\u00b3 : T2Space G inst\u271d\u00b2 : TopologicalGroup G inst\u271d\u00b9 : MeasurableSpace G inst\u271d : BorelSpace G K\u2080 : PositiveCompacts G \u22a2 \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080 = 1 ** haveI : LocallyCompactSpace G := K\u2080.locallyCompactSpace_of_group ** G : Type u_1 inst\u271d\u2075 : Group G inst\u271d\u2074 : TopologicalSpace G inst\u271d\u00b3 : T2Space G inst\u271d\u00b2 : TopologicalGroup G inst\u271d\u00b9 : MeasurableSpace G inst\u271d : BorelSpace G K\u2080 : PositiveCompacts G this : LocallyCompactSpace G \u22a2 \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080 = 1 ** rw [haarMeasure_apply K\u2080.isCompact.measurableSet, ENNReal.div_self] ** case h0 G : Type u_1 inst\u271d\u2075 : Group G inst\u271d\u2074 : TopologicalSpace G inst\u271d\u00b3 : T2Space G inst\u271d\u00b2 : TopologicalGroup G inst\u271d\u00b9 : MeasurableSpace G inst\u271d : BorelSpace G K\u2080 : PositiveCompacts G this : LocallyCompactSpace G \u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 \u2260 0 ** rw [\u2190 pos_iff_ne_zero] ** case h0 G : Type u_1 inst\u271d\u2075 : Group G inst\u271d\u2074 : TopologicalSpace G inst\u271d\u00b3 : T2Space G inst\u271d\u00b2 : TopologicalGroup G inst\u271d\u00b9 : MeasurableSpace G inst\u271d : BorelSpace G K\u2080 : PositiveCompacts G this : LocallyCompactSpace G \u22a2 0 < \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 ** exact haarContent_outerMeasure_self_pos ** case hI G : Type u_1 inst\u271d\u2075 : Group G inst\u271d\u2074 : TopologicalSpace G inst\u271d\u00b3 : T2Space G inst\u271d\u00b2 : TopologicalGroup G inst\u271d\u00b9 : MeasurableSpace G inst\u271d : BorelSpace G K\u2080 : PositiveCompacts G this : LocallyCompactSpace G \u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 \u2260 \u22a4 ** exact (Content.outerMeasure_lt_top_of_isCompact _ K\u2080.isCompact).ne ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.valMinAbs_spec ** n : \u2115 inst\u271d : NeZero n x : ZMod n y : \u2124 \u22a2 valMinAbs x = y \u2192 x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n ** rintro rfl ** n : \u2115 inst\u271d : NeZero n x : ZMod n \u22a2 x = \u2191(valMinAbs x) \u2227 valMinAbs x * 2 \u2208 Set.Ioc (-\u2191n) \u2191n ** exact \u27e8x.coe_valMinAbs.symm, x.valMinAbs_mem_Ioc\u27e9 ** n : \u2115 inst\u271d : NeZero n x : ZMod n y : \u2124 h : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n \u22a2 valMinAbs x = y ** rw [\u2190 sub_eq_zero] ** n : \u2115 inst\u271d : NeZero n x : ZMod n y : \u2124 h : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n \u22a2 valMinAbs x - y = 0 ** apply @Int.eq_zero_of_abs_lt_dvd n ** case h2 n : \u2115 inst\u271d : NeZero n x : ZMod n y : \u2124 h : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n \u22a2 |valMinAbs x - y| < \u2191n ** rw [\u2190 mul_lt_mul_right (@zero_lt_two \u2124 _ _ _ _ _)] ** case h2 n : \u2115 inst\u271d : NeZero n x : ZMod n y : \u2124 h : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n \u22a2 |valMinAbs x - y| * 2 < \u2191n * 2 ** nth_rw 1 [\u2190 abs_eq_self.2 (@zero_le_two \u2124 _ _ _ _)] ** case h2 n : \u2115 inst\u271d : NeZero n x : ZMod n y : \u2124 h : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n \u22a2 |valMinAbs x - y| * |2| < \u2191n * 2 ** rw [\u2190 abs_mul, sub_mul, abs_lt] ** case h2 n : \u2115 inst\u271d : NeZero n x : ZMod n y : \u2124 h : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n \u22a2 -(\u2191n * 2) < valMinAbs x * 2 - y * 2 \u2227 valMinAbs x * 2 - y * 2 < \u2191n * 2 ** constructor <;> linarith only [x.valMinAbs_mem_Ioc.1, x.valMinAbs_mem_Ioc.2, h.2.1, h.2.2] ** case h1 n : \u2115 inst\u271d : NeZero n x : ZMod n y : \u2124 h : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n \u22a2 \u2191n \u2223 valMinAbs x - y ** rw [\u2190 int_cast_zmod_eq_zero_iff_dvd, Int.cast_sub, coe_valMinAbs, h.1, sub_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.setToSimpleFunc_const' ** \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 : Measure \u03b1 inst\u271d : Nonempty \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' x : F m : MeasurableSpace \u03b1 \u22a2 setToSimpleFunc T (const \u03b1 x) = \u2191(T Set.univ) x ** simp only [setToSimpleFunc, range_const, Set.mem_singleton, preimage_const_of_mem,\n  sum_singleton, \u2190 Function.const_def, coe_const] ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.rev_lt_rev ** n : Nat i j : Fin n \u22a2 rev i < rev j \u2194 j < i ** rw [\u2190 Fin.not_le, \u2190 Fin.not_le, rev_le_rev] ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.mem_toSubtype_iff ** \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 p : \u03b2 \u2192 Prop f : \u03b1 \u2192 \u03b2 a : \u03b1 b : Subtype p \u22a2 b \u2208 toSubtype p f a \u2194 \u2191b = f a ** rw [toSubtype_apply, Part.mem_mk_iff, exists_subtype_mk_eq_iff, eq_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Function.Semiconj.bijOn_range ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 fa : \u03b1 \u2192 \u03b1 fb : \u03b2 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 \u03b3 s t : Set \u03b1 h : Semiconj f fa fb ha : Bijective fa hf : Injective f \u22a2 BijOn fb (range f) (range f) ** rw [\u2190 image_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 fa : \u03b1 \u2192 \u03b1 fb : \u03b2 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 \u03b3 s t : Set \u03b1 h : Semiconj f fa fb ha : Bijective fa hf : Injective f \u22a2 BijOn fb (f '' univ) (f '' univ) ** exact h.bijOn_image (bijective_iff_bijOn_univ.1 ha) (hf.injOn univ) ** Qed",
        "informal": ""
    },
    {
        "formal": "BoundedContinuousFunction.lintegral_lt_top_of_nnreal ** X : Type u_1 inst\u271d\u00b3 : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : OpensMeasurableSpace X \u03bc : Measure X inst\u271d : IsFiniteMeasure \u03bc f : X \u2192\u1d47 \u211d\u22650 \u22a2 \u222b\u207b (x : X), \u2191(\u2191f x) \u2202\u03bc < \u22a4 ** apply IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal ** case f_bdd X : Type u_1 inst\u271d\u00b3 : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : OpensMeasurableSpace X \u03bc : Measure X inst\u271d : IsFiniteMeasure \u03bc f : X \u2192\u1d47 \u211d\u22650 \u22a2 \u2203 c, \u2200 (x : X), \u2191(\u2191f x) \u2264 \u2191c ** refine \u27e8nndist f 0, fun x \u21a6 ?_\u27e9 ** case f_bdd X : Type u_1 inst\u271d\u00b3 : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : OpensMeasurableSpace X \u03bc : Measure X inst\u271d : IsFiniteMeasure \u03bc f : X \u2192\u1d47 \u211d\u22650 x : X \u22a2 \u2191(\u2191f x) \u2264 \u2191(nndist f 0) ** have key := BoundedContinuousFunction.NNReal.upper_bound f x ** case f_bdd X : Type u_1 inst\u271d\u00b3 : MeasurableSpace X inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : OpensMeasurableSpace X \u03bc : Measure X inst\u271d : IsFiniteMeasure \u03bc f : X \u2192\u1d47 \u211d\u22650 x : X key : \u2191f x \u2264 nndist f 0 \u22a2 \u2191(\u2191f x) \u2264 \u2191(nndist f 0) ** rwa [ENNReal.coe_le_coe] ** Qed",
        "informal": ""
    }
]